HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS …€¦ · HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS Volume 42 Issue 3 Jun 2013 A Peer Reviewed Journal Published Bimonthly

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HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

A Bimonthly PublicationVolume 42 Issue 3

2013

ISSN 1303 5010

HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

Volume 42 Issue 3

Jun 2013

A Peer Reviewed Journal

Published Bimonthly by the

Faculty of Science of Hacettepe University

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ISSN 1303 5010

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Hacettepe Journal of Mathematics and Statistics

Cilt 42 Sayı 3 (2013)

ISSN 1303 – 5010

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Hacettepe Journal of Mathematics and Statistics

A Bimonthly Publication – Volume 42 Issue 3 (2013)

ISSN 1303 – 5010

EDITORIAL BOARD

HONORARY EDITOR :

Lawrence Micheal Brown

Editors in Chief:

Mathematics:

Yucel Tıras (Hacettepe University - Algebra - [email protected])

Statistics:

Cem Kadılar (Hacettepe University-Statistics - [email protected])

MEMBERS

Ali Allahverdi (Operational research statistics, [email protected])

Olcay Arslan (Robust statistics, [email protected])

N. Balakrishnan (Statistics, [email protected])

Gary F. Birkenmeier (Algebra, [email protected])

G. C. L. Brummer (Topology, [email protected])

Sat Gupta (Sampling, Time Series, [email protected])

Varga Kalantarov (Appl. Math., [email protected])

Vladimir Levchuk (Algebra, [email protected])

Cihan Orhan (Analysis, [email protected])

Ivan Reilly (Topology, [email protected])

Patrick Smith (Algebra, [email protected] )

Bulent Sarac (Algebra, [email protected])

Alexander P. Sostak (Analysis, [email protected])

Agacık Zafer (Appl. Math., [email protected])

Published by Hacettepe UniversityFaculty of Science

CONTENTS

Mathematics

R. A. Bandaliev

On One Weighted Inequalities for Convolution Type Operator . . . . . . . . . . . . . . . 199

Zafer Siar and Refik Keskin

Some New Identities Concerning Generalized Fibonacci

and Lucas Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211

Akbar Azam, Muhammad Arshad and Pasquale Vetro

On Edelstein Type Multivalued Random Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Jae Won Lee

Anti-Invariant ξ⊥-Riemannian Submersions from

Almost Contact Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Bo-Yan Xi and Feng Qi

Some Hermite-Hadamard Type Inequalities for Differentiable

Convex Functions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Sadulla Z. Jafarov

Approximation by Fejer Sums of Fourier Trigonometric Series in

Weighted Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Nihat Gokhan Gogus

Operator Valued Dirichlet Problem in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Y. S. Kim, A. K. Rathie, U. Pandey and R. B. Paris

Some Summation Formulas for the Hypergeometric Series r+2Fr+1(12 ) . . . . . . 281

Wenjun Liu

New Integral Inequalities Via (α,m)-Convexity and Quasi-Convexity . . . . . . . 289

Statistics

Tulay Kesemen

On the Semi-Markovian Random Walk with Delay and

Weibull Distributed Interference of Chance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299

MATHEMATICS

Hacettepe Journal of Mathematics and StatisticsVolume 42 (3) (2013), 199 – 210

ON ONE WEIGHTED INEQUALITIES

FOR CONVOLUTION TYPE OPERATOR

R. A. Bandaliev ∗

Received 26 : 09 : 2011 : Accepted 14 : 04 : 2012

Abstract

In this paper we prove the boundedness of certain convolution operatorin a weighted Lebesgue space with kernel satisfying the generalizedHormander’s condition. The sufficient conditions for the pair of generalweights ensuring the validity of two-weight inequalities of a strong typeand of a weak type for convolution operator with kernel satisfying thegeneralized Hormander’s condition are found.

Keywords: Weighted Lebesgue space, Singular integral, Kernel, Generalized Horman-der’s condition, Boundedness.

2000 AMS Classification: 42B20, 42B25, 46B50

1. Introduction.

Let Rn be n-dimensional Euclidean spaces of points x = (x1, . . . , xn) , where n ∈ Nand Rn0 = Rn \ 0. Suppose that ω is a non-negative, Lebesgue measurable and realfunction defined on Rn, i.e., ω is a weight function defined on Rn. By Lp, ω (Rn) wedenote the weighted Lebesgue space of measurable functions f on Rn such that

‖f‖Lp, ω(Rn) = ‖f‖p, ω =

∫Rn

|f(x)|p ω(x) dx

1/p

<∞, 1 ≤ p <∞.

In the case p =∞, the norm on the space L∞,ω (Rn) is defined as

‖f‖L∞,ω(Rn) = ‖f‖∞ = ess supx∈Rn

|f(x)|.

For ω = 1 we obtain the nonweighted Lp spaces, i.e., ‖f‖Lp, 1(Rn) = ‖f‖Lp(Rn) = ‖f‖p.Our aim in this paper is to show the boundedness of certain convolution operator in

a weighted Lebesgue space with kernel satisfying the generalized Hormander’s condition.

∗Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan Az

1141, Baku, Azerbaijan F.Agayev str.,9E-mail: [email protected]

200 R. A. Bandaliev

The sufficient conditions for the pair of general weights ensuring the validity of two-weight inequalities of a strong type and of a weak type for convolution operator withkernel satisfying the generalized Hormander’s condition are found. In particular, is givena class B(u, v) of weight pair which is generalized earlier obtained results (see below).Also, in this paper we give a weight pairs which satisfy the condition of obtain results.

Now we give a chronological development of earlier results. Let K : Rn0 → R, K ∈Lloc1 (Rn0 ) is a function satisfy following conditions:

1) K(tx) ≡ K (tx1, . . . , txn) = t−nK(x) for all t > 0 and x ∈ Rn0 ;

2)

∫|ξ|=1

K(ξ) dσ(ξ) = 0;

3)

1∫0

w(t)

tdt <∞, where w(t) = sup

|ξ−η|≤t|K(ξ)−K(η)| for |ξ| = |η| = 1.

We consider the following singular integral

(1.1) Af(x) = limε→+0

∫|x−y|>ε

K(x− y) f(y) dy = p.v.

∫Rn

K(x− y) f(y) dy,

where f ∈ C∞0 (Rn) and last integral is understood in the sense of principal value.The following Calderon-Zygmund Theorem is valid.

1.1. Theorem. [3, 4] Let 1 < p < ∞ and A be a singular integral operator with kernelK satisfying conditions 1)-3). Then singular integral Af is exist for almost every (a.e.)x ∈ Rn and the inequality

‖Tf‖p ≤ C‖f‖p

holds, where a constant C > 0 is independent of f ∈ Lp (Rn) .

Further development of this theory is closely related the boundedness of Calderon-Zygmund singular integral operator in the weighted Lebesgue space with power weights.Namely, in the paper [13] Stein proved the following Theorem.

1.2. Theorem. [13] Let 1 < p < ∞, −n < α < n(p − 1) and A be singular integraloperator (1.1) with kernel K satisfying conditions 1)-3). Then singular integral Af isexist for a.e. x ∈ Rn and the inequality

‖Tf‖p, |x|α ≤ C‖f‖p, |x|α

holds, where a constant C > 0 is independent on f ∈ Lp, |x|α (Rn) .

Further Hormander in the paper [9] replacing the condition 3) weaker condition provedthe following Theorem.

1.3. Theorem. [9] Let 1 2|y|

|K(x− y)−K(x)| dx ≤ C1,

where C1 > 0 doesn’t depend on y ∈ Rn0 . Then singular integral Af is exist for a.e.x ∈ Rn and the inequality

‖Af‖p ≤ C‖f‖p

holds, where a constant C > 0 is independent of f ∈ Lp (Rn) .

On one weighted inequalities for convolution type operator 201

On the other hand, the convolution operators whose kernels do not satisfy Horman-der’s condition (1.2) have been widely considered (for example, oscillatory and othersingular integral) (see [5]).

Now we formulated the known results connected with generalized Hormander’s con-dition.

1.4. Definition. [7] A positive measurable and locally integrable function g is said tosatisfy the reverse Holder RH∞ condition or g ∈ RH∞ (Rn) if

0 < supx∈B

g(x) ≤ C 1

|B|

∫B

g(x) dx,

where B is an arbitrary ball centered at the origin and C > 0 is a constant independentof B.

Let K ∈ L2 (Rn) is a function satisfy the following conditions:

(a)∥∥∥K∥∥∥

∞≤ C;

(b) there exist functions A1, . . . , Am and Φ = ϕ1, . . . , ϕm such that ϕi ∈ L∞ (Rn)

and |det [ϕj (yi)]|2 ∈ RH∞ (Rnm) , yi ∈ Rn, i, j = 1, . . . ,m;(c) for a fixed γ > 0 and for any |x| > 2|y| > 0 the inequality

∫|x|>2|y|

∣∣∣∣∣K(x− y)−m∑i=1

Ai(x)ϕi(y)

∣∣∣∣∣ dx ≤ Cis valid;

(d) |K(x)| ≤ C

|x|n .

It is obvious that condition (c) is a generalization the condition (1.2) for m = 1,A1(x) = K(x) and ϕ1(x) ≡ 1.

For f ∈ C∞0 (Rn) we define the convolution operator associated to the kernel K by

(1.3) Tf(x) =

∫Rn

K(x− y) f(y) dy.

1.5. Theorem. [7] Let 1 < p < ∞ and T be a convolution operator with kernel Ksatisfying (a)-(c). Then the inequality

‖Tf‖p ≤ C‖f‖p

holds, where a constant C depend only on p, n and the constant in the RH∞-conditionfor the functions ϕj .

For p = 1 there exists a constant C such that

|x : |Tf(x)| > λ| ≤ C

λ

∫Rn

|f(x)| dx,

for every smooth function f with compact support and λ > 0.

Note that Theorem 1.3 is particular case of Theorem 1.5 for m = 1, A1(x) = K(x)and ϕ1(x) ≡ 1.

202 R. A. Bandaliev

2. Preliminaries

2.1. Remark. It is clear that from condition RH∞ implies the well known reverse Holderinequality 1

|B|

∫B

[g(x)]1+ε dx

11+ε

≤ C

1

|B|

∫B

g(x) dx

,

where ε > 0. It is well known that the reverse Holder condition be characterized thecondition Ap (Rn) (see [5]).

2.2. Example. Let m = 2, K(x) =sin x

x, x ∈ R \ 0, A1(x) =

eix

2i xA2(x) = − e

ix

2i x,

ϕ1(y) = e−iy and ϕ2(y) = eiy. Then the conditions (a)-(d) hold (see [2]).

We will also need the following theorem.

2.3. Theorem. [12] Let 1 < q < p < ∞ and u(t) and v(t) be positive functions on(0, ∞). Suppose that F : (0, ∞) 7→ R be a Lebesgue measurable function.

1. For the validity of the inequality ∞∫0

u(t)

∣∣∣∣∣∣t∫

0

F (τ) dτ

∣∣∣∣∣∣q

dt

1/q

≤ C1

∞∫0

|F (t)|p v(t) dt

1/p

it is necessary and sufficient that

∞∫0

∞∫t

u(τ) dτ

t∫0

v1−p′(τ) dτ

q−1pp−q

v1−p′(t) dt <∞,

where C1 > 0 is independent of F.2. For the validity of the inequality ∞∫

0

u(t)

∣∣∣∣∣∣∞∫t

F (τ) dτ

∣∣∣∣∣∣q

dt

1/q

≤ C2

∞∫0

|F (t)|p v(t) dt

1/p

it is necessary and sufficient that

∞∫0

t∫0

u(τ) dτ

∞∫t

v1−p′(τ) dτ

q−1pp−q

v1−p′(t) dt <∞,

where C2 > 0 is independent of F.

For q = 1 the following Lemma is valid.

2.4. Lemma. [11] Let p > 1 and u(t) and v(t) be positive functions on (0, ∞).1. If a pair (u, v) satisfies the condition

∞∫0

∞∫t

u(τ) dτ

p′

v1−p′(t) dt <∞,

then there exists a positive constant C1 such that for an arbitrary function F : (0, ∞) 7→ Rthe inequality

∞∫0

u(t)

∣∣∣∣∣∣t∫

0

F (τ) dτ

∣∣∣∣∣∣ dt ≤ C1

∞∫0

|F (t)|p v(t) dt

1/p


holds.2. If a pair (u, v) satisfies the condition

∞∫0

t∫0

u(τ) dτ

p′

v1−p′(t) dt <∞,

then there exists a positive constant C2 such that for an arbitrary function F : (0, ∞) 7→ Rthe inequality

∞∫0

u(t)

∣∣∣∣∣∣∞∫t

F (τ) dτ

∣∣∣∣∣∣ dt ≤ C2

∞∫0

|F (t)|p v(t) dt

1/p

holds.

2.5. Theorem. [10] Let 1 ≤ q < p < ∞ and u(x) and v(x) be weight functions on Rn.Then the condition

(2.1) A =

∫Rn

[u(x)]pp−q [v(x)]

− qp−q dx <∞

is necessary and sufficient for the validity of the inequality

(2.2)

∫Rn

|f(x)|q u(x) dx

1/q

≤ A1q− 1p

∫Rn

|f(x)|p v(x) dx

1/p

.

3. Main results

Let Z = 0,±1,±2, . . . , . By Bu, v we denote the pair (u, v) satisfy the condition

(3.1)

∑k∈Z

sup2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q dx

1/q

≤ C

∫Rn

|f(x)|p v(x) dx

1/p

,

where the constant C independent of k ∈ Z.

3.1. Remark. Let (u, v) ∈ Bu,v. It is clear that∫Rn

|f(x)|q u(x) dx

1/q

=

∑k∈Z

∫2k<|x|≤2k+1

|f(x)|q u(x) dx

1/q

≤

≤

∑k∈Z

sup2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q dx

1/q

≤ C

∫Rn

|f(x)|p v(x) dx

1/p

.

Therefore, the weight pair (u, v) satisfies the condition (2.1).

3.2. Lemma. Let 1 ≤ q < p < ∞ and u(x) and v(x) be weight functions on Rn. Letthere exists a constant M such that for any k ∈ Z the inequality

sup2k<|x|≤2k+1

u(x) ≤M inf2k<|x|≤2k+1

u(x)

holds. Then the conditions (2.2) and (3.1) is equivalent.

204 R. A. Bandaliev

Proof. (2.2)⇒ (3.1). We have∑k∈Z

sup2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q dx

1/q

=

∑k∈Z

sup2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q u(x)[u(x)]−1 dx

1/q

≤

∑k∈Z

sup2k<|x|≤2k+1

u(x)

inf2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q u(x) dx

1/q

≤M1/q

∑k∈Z

∫2k<|x|≤2k+1

|f(x)|q u(x) dx

1/q

= M1/q ‖f‖q, u ≤M1/q A1q− 1p ‖f‖p, v.

The fact (3.1)⇒ (2.2) automatically implies from Remark 2.

3.3. Lemma. Let 1 ≤ q < p < ∞, u(x) and v(x) be weight functions on Rn andv ∈ L1 (Rn) . Let there exists a constant M1 such that for any k ∈ Z the inequality

sup2k<|x|≤2k+1

u(x) ≤M1 inf2k<|x|≤2k+1

v(x)

holds. Then the inequality∑k∈Z

sup2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q dx

1/q

≤

≤ M1/q1

∫Rn

v(x) dx

1q− 1p

‖f‖p, v

is valid.

Proof. Indeed, we have∑k∈Z

sup2k<|x|≤2k+1

u(x)

∫2k<|x|≤2k+1

|f(x)|q dx

1/q

≤

≤M1/q1

∑k∈Z

inf2k<|x|≤2k+1

v(x)

∫2k<|x|≤2k+1

|f(x)|q dx

1/q

=

= M1/q1

∑k∈Z

∫2k<|x|≤2k+1

|f(x)|q inf2k<|x|≤2k+1

v(x) dx

1/q

≤


≤M1/q1

∑k∈Z

∫2k<|x|≤2k+1

|f(x)|q v(x) dx

1/q

= M1/q1

∫Rn

|f(x)|q v(x) dx

1/q

≤

≤M1/q1

∫Rn

v(x) dx

1q− 1p

‖f‖p, v.

The sufficient condition for pair of general weights guaranteeing the two-weight in-equalities of a strong type (p, q) for convolution operator (1.3) are proved in the followingTheorem.

3.4. Theorem. Let 1 < q < p <∞ and the kernel of convolution operator (1.3) satisfiesthe conditions (a)-(d). Let ω and ω1 be weight functions on Rn. Suppose that the weightpair (ω1, ω) satisfy the following conditions:

1)

∫Rn

∫|y|>|x|

ω1(y)

|y|nq dy

∫|y|<|x|

ω1−p′(y) dy

q−1

pp−q

ω1−p′(x) dx <∞;

2)

∫Rn

∫|y|<|x|

ω1(y) dy

∫|y|>|x|

ω1−p′(y)

|y|np′dx

q−1

pp−q

ω1−p′(x)

|x|np′dx <∞;

3) there exists a constantd > 0 such that for any f ∈ Lp, ω (Rn) the inequality

∑k∈Z

sup2k−1<|x|≤2k+2

ω1(x)

∫2k−1<|x|≤2k+2

|f(x)|q dx

1/q

≤

d

∫Rn

|f(x)|p ω(x) dx

1/p

holds. Then

(3.2) ‖Tf‖Lq, ω1(Rn) ≤ C‖f‖Lp, ω(Rn),

where the constant C > 0 is independent of f.

Proof. Estimate the left-hand side of inequality (3.2). We have∫Rn

|Tf(x)|q ω1(x) dx

1/q

=

∑k∈Z

∫2k<|x|≤2k+1

|Tf(x)|q ω1(x) dx

1/q

=

=

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣T (f · χ|y|≤2k−1)

(x) + T(f · χ2k−1<|y|≤2k+2

)(x)+

+T(f · χ|y|>2k+2

)(x)∣∣∣q ω1(x) dx

)1/q≤

206 R. A. Bandaliev

≤ 41/q′

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣T (f · χ|y|≤2k−1)

(x)∣∣∣q ω1(x) dx

1/q

+

+41/q′

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣T (f · χ2k−1<|y|≤2k+2)

(x)∣∣∣q ω1(x) dx

1/q

+

+41/q′

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣T (f · χ|y|>2k+2)

(x)∣∣∣q ω1(x) dx

1/q

=

= 41/q′ (A1 +A2 +A3) .

Now we estimate A1. If 2k < |x| ≤ 2k+1 and |y| ≤ 2k−1, then |y| ≤ 2k−1 ≤ |x|2≤ |x| and

|x− y| ≥ |x| − |y| ≥ |x| − |x|2

=|x|2. We have

A1 =

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣∣∣∣∫Rn

K(x− y) f(y)χ|z|≤2k−1(y) dy

∣∣∣∣∣∣q

ω1(x) dx

1/q

≤

≤ C

∑k∈Z

∫2k<|x|≤2k+1

∫|y|≤2k−1

|f(y)||x− y|n dy

q

ω1(x) dx

1/q

≤

≤ C1

∑k∈Z

∫2k<|x|≤2k+1

ω1(x)

|x|nq

∫|y|≤|x|

|f(y)| dy

q

dx

1/q

=

= C2

∫Rn

ω1(x)

|x|nq

∫|y|≤|x|

|f(y)| dy

q

dx

1/q

=

= C2

∫Rn

ω1(x)

|x|nq

|x|∫0

sn−1

∫|ξ|=1

|f(sξ)| dξ

ds

q

dx

1/q

=

= C2

∞∫0

tn(1−q)−1

∫|η|=1

ω1(tη) dη

t∫

0

sn−1

∫|ξ|=1

|f(sξ)| dξ

ds

q

dt

1/q

.

Taking

u(t) = tn(1−q)−1

∫|η|=1

ω1(tη) dη

, F (t) = tn−1

∫|ξ|=1

|f(tξ)| dξ

,


v(t) = t−(n−1)(p−1)

∫|ξ|=1

ω1−p′(tξ) dξ

1−p

and using the Theorem 2.3 (part one), we

get

A1 ≤ C3

∞∫0

t(n−1)p

∫|ξ|=1

|f(tξ)| dξ

p ∫|ξ|=1

ω1−p′

(tξ) dξ

1−p

t−(n−1)(p−1)

dt

1/p

=

= C3

∞∫0

tn−1

∫|ξ|=1

|f(tξ)| dξ

p ∫|ξ|=1

ω1−p′

(tξ) dξ

1−p

dt

1/p

.

Applying the Holder’s inequality, we have∞∫0

tn−1

∫|ξ|=1

|f(tξ)| dξ

p ∫|ξ|=1

ω1−p′

(tξ) dξ

1−p

dt

1/p

=

=

∞∫0

tn−1

∫|ξ|=1

[|f(tξ)|ω

1p (tξ)

]ω− 1p (tξ)dξ

p ∫|ξ|=1

ω1−p′

(tξ) dξ

1−p

dt

1/p

≤

≤

∞∫0

tn−1

∫|ξ|=1

|f(tξ)|p ω(tξ) dξ

∫|ξ|=1

ω− p′p (tξ) dξ

p/p′ ∫

|ξ|=1

ω1−p′

(tξ) dξ

1−p

dt

1/p

=

∞∫0

tn−1

∫|ξ|=1


∫|ξ|=1

ω1−p′

(tξ) dξ

p−1 ∫

|ξ|=1

ω1−p′

(tξ) dξ

1−p

dt

1/p

=

∞∫0

tn−1

∫|ξ|=1


dt

1/p

=

∫Rn|f(x)|p ω(x) dx

1/p

.

Therefore A1 ≤ C3

∫Rn

|f(x)|p ω(x) dx

1/p

and by condition 1) of Theorem 3.4

∞∫0

∞∫t

u(τ) dτ

t∫0

v1−p′(τ) dτ

q−1pp−q

v1−p′(t) dt =

=

∫Rn

∫|y|>|x|

ω1(y)

|y|nq dy

∫|y|<|x|

ω1−p′(y) dy

q−1

pp−q

ω1−p′(x) dx <∞.

Now we estimate A3. Note that if 2k < |x| ≤ 2k+1 and |y| > 2k+2, then |x| ≤ |y|2

and

|x− y| ≥ |y| − |x| ≥ |y| − |y|2

=|y|2. We get

A3 =

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣∣∣∣∫Rn

K(x− y) f(y)χ|z|>2k+2(y) dy

∣∣∣∣∣∣q

ω1(x) dx

1/q

≤

≤ C

∑k∈Z

∫2k<|x|≤2k+1

∫|y|>2k+2

|f(y)||x− y|n dy

q

ω1(x) dx

1/q

≤

208 R. A. Bandaliev

≤ C1

∑k∈Z

∫2k<|x|≤2k+1

ω1(x)

∫|y|≥|x|

|f(y)||y|n dy

q

dx

1/q

=

= C2

∫Rn

ω1(x)

∫|y|≥|x|

|f(y)||y|n dy

q

dx

1/q

=

= C2

∞∫0

tn−1

∫|η|=1

ω1(tη) dη

∞∫t

s−1

∫|ξ|=1

|f(sξ)| dξ

ds

q

dt

1/q

.

Further, using the Theorem 2.3 (part two) by condition 2) of Theorem 3.4 we get

A3 ≤ C3

∫Rn

|f(x)|p ω(x) dx

1/p

.

Finally, we estimate A2. By Theorem 1.5 and by condition 3) of Theorem 3.4 we get

A2 =

∑k∈Z

∫2k<|x|≤2k+1

∣∣∣T (f · χ2k−1<|y|≤2k+2)

(x)∣∣∣q ω1(x) dx

1/q

≤

≤

∑k∈Z

sup2k<|x|≤2k+1

ω1(x)

∫Rn

∣∣∣T (f · χ2k−1<|y|≤2k+2)

(x)∣∣∣q ω1(x) dx

1/q

≤

≤ C

∑k∈Z

sup2k−1<|x|≤2k+2

ω1(x)

∫2k−1<|x|≤2k+2

|f(x)|q ω1(x) dx

1/q

≤

≤ C

∫Rn

|f(x)|p ω(x) dx

1/p

.

This completes the proof of Theorem 3.4.

3.5. Corollary. Let 1 < q < p < ∞ and the kernel of convolution operator (1.3)satisfies the conditions (a)-(d). Let ω(t) and ω1(t) be positive increasing functions on(0,∞) satisfying condition 1) of Theorem 3.4. Then the inequality (3.2) holds.

3.6. Corollary. Let 1 < q < p < ∞ and the kernel of convolution operator (1.3)satisfies the conditions (a)-(d). Let ω(t) and ω1(t) be positive decreasing functions on(0,∞) satisfying condition 2) of Theorem 3.4. Then the inequality (3.2) holds.

3.7. Example. Let

ω1(t) =

tq−1 lnβ 1t

for t < e− pp−q

ep(λ−q+1)p−q

(pp−q

)βtλ for t ≥ e−

pp−q ,

ω(t) =

tp−1 lnγ 1t

for t < e− pp−q

ep(µ−p+1)p−q

(pp−q

)γtµ for t ≥ e−

pp−q ,


where p− 1 < γ <p(p− 1)

p− q , β <q

p(γ + 1)− q − 1, β 6= −1, 0 ≤ λ < q

p(µ+ 1)− 1 and

q

p− 1 < µ < p− 1. Then the pair (ω, ω1) satisfies the condition of Theorem 3.4 for n = 1.

The sufficient condition for pair of general weights guaranteeing the two-weight in-equalities of a weak (p, 1) type for convolution operator (1.3) are formulate in the followingTheorem.

3.8. Theorem. Let 1 < p <∞ and the kernel of convolution operator (1.3) satisfies theconditions (a)-(d). Let ω and ω1 be positive functions on Rn. Suppose that the weightpair (ω1, ω) satisfy the following conditions:

1′)

∫Rn

∫|y|>|x|

ω1(y)

|y|n dy

p′

ω1−p′(x) dx <∞;

2′)

∫Rn

∫|y|<|x|

ω1(y) dy

p′

ω1−p′(x)

|x|np′dx <∞.

3′) there exists a constant d1 > 0 such that for any f ∈ Lp, ω (Rn) the inequality

∑k∈Z

sup2k−1<|x|≤2k+2

ω1(x)

∫2k−1<|x|≤2k+2

|f(x)| dx ≤ d1

∫Rn

|f(x)|p ω(x) dx

1/p

holds. Then there exists a constant C > 0 such that for any f ∈ Lp, ω (Rn) and λ > 0the inequality

(3.3)

∫x∈Rn: |Tf(x)|>λ

ω1(x) dx ≤ C

λ

∫Rn

|f(x)|p ω(x) dx

1/p

is valid.

3.9. Corollary. Let 1 < q < p <∞ and the kernel of convolution operator (1.3) satisfiesthe conditions (a)-(d). Let ω1(t) be increasing and ω(t) be forall positive functions on(0,∞) satisfying condition 1’) of Theorem 3.8. Then the inequality (3.3) holds.

3.10. Corollary. Let 1 < q < p <∞ and the kernel of convolution operator (1.3) satis-fies the conditions (a)-(d). Let ω1(t) be decreasing and ω(t) be forall positive functionson (0,∞) satisfying condition 2’) of Theorem 3.8. Then the inequality (3.3) holds.

The Theorems 3.4 and 3.8 are pioneering results in the case 1 ≤ q p(λ+ 1)− 1, −1 < λ < 0, β < −1 and γ > p(β+ 2) + 1. Then the pair (ω, ω1)satisfies the condition of Theorem 3.8.

3.12. Remark. Note that for p = q of the special weights the Theorem 3.4 was provedin [1] (see also [2, 11]). Some others results for p = q was proved in [8].

210 R. A. Bandaliev

Acknowledgements

The author was supported by the Science Development Foundation under the Pres-ident of the Republic of Azerbaijan EIF-2010-1(1)-40/06-1. The author would like toexpress their gratitude to the referee for his very valuable comments and suggestions.

References

[1] Bandaliev, R. A. Two-weight inequalities for convolution operators in Lebesgue spaces,

Mat.Zametki, 80 (1), 3–10, 2006 (in Russian). English translation: in Math. Notes 80(1), 3–10, 2006.

[2] Bandaliev, R. A. and Omarova, K. K. Two-weight norm inequalities for certain singular

integrals, Taiwanese Jour. of Math., N2, 713–732, 2012.[3] Calderon, A. P. and Zygmund, A. On the existence of certain singular integrals, Acta Math.,

88, 85–139, 1952.

[4] Calderon, A. P. and Zygmund, A. On singular integrals, American Math.J., 78 (2), 289–309,1956.

[5] Davis, K. and Chang, Y. Lectures on Bochner-Riesz means, London Math. Soc., LectureNotes ser. 114, Cambridge Univ. Press., 1987.

[6] Garsia-Cuerva, J. and Rubio de Francia, J. L. Weighted norm inequalities and related topics,

North-Holland Math.Studies, 1985.[7] Grubb, D. J. and Moore, C. N. A variant of Hormander’s condition for singular integrals,

Colloq. Math., 73 (2), 165–172, 1997.

[8] Guliyev, V. S. Two-weight inequalities for singular integrals satisfying a variant of Horman-der condition, Journal of Function Spaces and Appl., 7 (1), 43–54, 2009.

[9] Hormander, L. Estimates for translation invariant operators in Lp spaces, Acta Math., 104,

93–140, 1960.[10] Kabaila, V. P. On the embedding Lp(µ) into Lq(ν). (in Russian) Litovsky Mat.Sb., 21,

143–148, 1981.

[11] Kokilashvili, V. and Meskhi, A. Two-weight inequalities for singular integrals defined onhomogeneous groups, Proc. A.Razmadze Math.Inst., 112, 57–90, 1997.

[12] Maz’ya, V. G. Sobolev spaces, Springer-Verlag, Berlin, 1985.[13] Stein, E. M. Note on singular integral, Proc. Amer. Math. Soc., 8(2), 250–254, 1957.


SOME NEW IDENTITIES CONCERNING

GENERALIZED FIBONACCI

AND LUCAS NUMBERS

Zafer Siar∗, Refik Keskin†


Abstract

In this paper we obtain some identities containing generalized Fibonacciand Lucas numbers. Some of them are new and some are well known.By using some of these identities we give some congruences concerninggeneralized Fibonacci and Lucas numbers such as

V2mn+r ≡ (−(−t)m)nVr (mod Vm),

U2mn+r ≡ (− (−t)m)nUr (mod Vm),

and

V2mn+r ≡ (−t)mn Vr (mod Um),

U2mn+r ≡ (−t)mn Ur (mod Um).

Keywords: Generalized Fibonacci numbers; Generalized Lucas numbers

2000 AMS Classification: 11B37, 11B39, 40C05

1. Introduction

Let k and t be nonzero real numbers. Generalized Fibonacci sequence Un is definedby U0 = 0, U1 = 1, and Un+1 = kUn + tUn−1 for n ≥ 1 and generalized Lucas sequenceVn is defined by V0 = 2, V1 = k, and Vn+1 = kVn + tVn−1 for n ≥ 1. Un and Vn arecalled generalized Fibonacci numbers and generalized Lucas numbers respectively.

For k = t = 1, we have classical Fibonacci and Lucas sequences Fn and Ln .For k = 2 and t = 1, we have Pell and Pell-Lucas sequences Pn and Qn . For more

∗Bilecik Seyh Edebali University, Faculty of Science and Arts, Department of Mathematics,Bilecik, Turkey.

E-Mail (Corresponding author): [email protected]†Sakarya University, Faculty of Science and Arts, Department of Mathematics, 54187

Sakarya, Turkey.E-Mail: [email protected]

212 Z. Siar, R. Keskin

information about generalized Fibonacci and Lucas numbers one can consult [1], [2], [3],and [4]. For t = 1, the sequence Un has been investigated in [5] and [6].

Generalized Fibonacci and Lucas numbers for negative subscript are defined as

(1.1) U−n =−Un

(−t)n and V−n =Vn

(−t)n

respectively.Now assume that k2 + 4t > 0. Then it is well known that

(1.2) Un =αn − βn

α− β and Vn = αn + βn

where α = (k +√k2 + 4t)/2 and β = (k −

√k2 + 4t)/2. The above identities are known

as Binet formulae. Let α and β be the roots of the equations x2 − kx − t = 0. Clearlyα+ β = k, α− β =

√k2 + 4t, and αβ = −t. Moreover, it can be seen that

(1.3) Vn = Un+1 + tUn−1 = kUn + 2tUn−1

and

(1.4) (k2 + 4t)Un = Vn+1 + tVn−1

for every n ∈ ZFor t = 1, ∓(Un, Vn) are all the integer solutions of the equation x2− (k2 + 4)y2 = ∓4

and for t = −1, ∓(Un, Vn) are all the integer solutions of the equation x2−(k2−4)y2 = 4.Also, for t = 1, ∓(Un, Un−1) are all the integer solutions of the equation x2−kxy−y2 =∓1and for t = −1, ∓(Un, Un−1) are all the integer solutions of the equation x2−kxy+y2 =1 (see[7],[8], and [9]).

Many identities concerning generalized Fibonacci and Lucas numbers can be provedby using Binet formulae, induction and matrices. In the literature, the matrices[

0 1t k

]and

[k t1 0

]are used in order to produce identities (see[4],[10]). Since[

k t1 0

]and

[0 1t k

]are similar matrices, they give the same identities.

In this study we will characterize all the 2 × 2 matrices X satisfying the relationX2 = kX + tI. Then we will obtain different identities by using this property. In factthe matrices[

k t1 0

]and

[0 1t k

]are special cases of the 2× 2 matrices X satisfying X2 = kX + tI.

2. Main Theorems

2.1. Theorem. If X is a square matrix with X2 = kX + tI, then Xn = UnX + tUn−1Ifor every n ∈ Z.

Some new identities concerning generalized Fibonacci and Lucas numbers 213

Proof. If n = 0, then the proof is obvious. It can be shown by induction that Xn =UnX + tUn−1I for every n ∈ N. We now show that X−n = U−nX + tU−n−1I for everyn ∈ N. Let Y = kI −X = −tX−1. Then

Y 2 = (kI −X)2 = k2I − 2kX +X2

= k2I − 2kX + kX + tI = k(kI −X) + tI = kY + tI.

Thus Y n = UnY + tUn−1I and this shows that

(−t)nX−n = UnY + tUn−1I = Un(kI −X) + tUn−1I

= (kUn + tUn−1)I − UnX = −UnX + Un+1I.

Then we get X−n =−UnX

(−t)n +Un+1I

(−t)n . This implies that X−n = U−nX + tU−n−1I by

(1.1). This completes the proof.

2.2. Theorem. Let X be an arbitrary 2× 2 matrix. Then X2 = kX + tI if and only ifX is of the form

X =

[a bc k − a

]with detX = −t

or X = λI where λ ∈ α, β , where α = (k +√k2 + 4t)/2 and β = (k −

√k2 + 4t)/2.

Proof. Assume that X2 = kX + tI. Then the minimum polynomial of X must dividesx2−kx− t. Therefore it must be x−α or x−β or x2−kx− t. In the first case X = αI, inthe second case X = βI, and in the third case, since X is 2× 2 matrix, its characteristicpolynomial must be x2−kx− t, so its trace is k and its determinant is −t. The argumentreverses.

2.3. Corollary. If X =

[a bc k − a

]is a matrix with detX = −t, then Xn =[

aUn + tUn−1 bUn

cUn Un+1 − aUn

].

Proof. Since X2 = kX + tI, the result follows from Theorem 2.1.

2.4. Corollary. αn = αUn + tUn−1 and βn = βUn + tUn−1 for every n ∈ Z.

Proof. Take X =

[α 00 β

]with detX = αβ = −t. Then by Theorem 2.1, it follows

that

Xn =

[αn 00 βn

]=

[αUn + tUn−1 0

0 βUn + tUn−1

].

This implies that αn = αUn + tUn−1 and βn = βUn + tUn−1.

2.5. Corollary. Un =αn − βn

α− β and Vn = αn + βn for every n ∈ Z.

Proof. The result follows from Corollary 2.4.

2.6. Corollary. Let S =

[k/2 (k2 + 4t)/21/2 k/2

]. Then Sn =

[Vn/2 (k2 + 4t)Un/2Un/2 Vn/2

]for every n ∈ Z.

Proof. Since S2 = kS + tI, the proof follows from Corollary 2.3.

2.7. Corollary. Let X =

[k t1 0

]. Then Xn =

[Un+1 tUn

Un tUn−1

].


Proof. Since X2 = kX + tI, the proof follows from Corollary 2.3.

2.8. Lemma. Let a, b, and ka + b be nonzero real numbers and let k2 + 4t be not aperfect square. Then

n∑j=0

(n

j

)ajbn−jUj+r = −(−t)r

n∑j=0

(n

j

)(−a)j(ka+ b)n−jUj−r

and

n∑j=0

(n

j

)ajbn−jVj+r = (−t)r

n∑j=0

(n

j

)(−a)j(ka+ b)n−jVj−r.

Proof. Let Z [α] = aα+ b | a, b ∈ Z . Define ϕ : Z [α]→ Z [α] by ϕ(aα+ b) = aβ + b =a(k − α) + b = −aα + ka + b. Then it can be shown that ϕ is ring homomorphism.Moreover, it can be shown that ϕ is injective. On the other hand, we get

−αUn + Un+1 = −αUn + kUn + tUn−1 = ϕ(αUn + tUn−1)

= ϕ(αn) = βn = (−t)nα−n.

Then it is seen that

ϕ((aα+ b)nαr) = ϕ((aα+ b)n)ϕ(αr) = (−aα+ ka+ b)n(−t)rα−r

= (−t)rn∑

j=0

(n

j

)(−aα)j(ka+ b)n−jα−r

= (−t)rn∑

j=0

(n

j

)(−a)j(ka+ b)n−jαj−r

= (−t)rn∑

j=0

(n

j

)(−a)j(ka+ b)n−j(αUj−r + tUj−r−1)

= α

((−t)r

n∑j=0

(n

j

)(−a)j(ka+ b)n−jUj−r

)

+

(−(−t)r+1

n∑j=0

(n

j

)(−a)j(ka+ b)n−jUj−r−1

)


On the other hand, we have

ϕ((aα+ b)nαr) = ϕ

(n∑

j=0

(n

j

)ajbn−jαj+r

)

= ϕ

(n∑

j=0

(n

j

)ajbn−j(αUj+r + tUj+r−1)

)

= α

(−

n∑j=0

(n

j

)ajbn−jUj+r

)

+

(n∑

j=0

(n

j

)ajbn−j(kUj+r + tUj+r−1)

)

= α

(−

n∑j=0

(n

j

)ajbn−jUj+r

)+

(n∑

j=0

(n

j

)ajbn−jUj+r+1

).

Then the proof follows.

2.9. Theorem. Let m, r ∈ Z with m 6= 0 and m 6= 1. Then

Umn+r =

n∑j=0

(n

j

)U j

mUn−jm−1Uj+rt

n−j

and

Vmn+r =

n∑j=0

(n

j

)U j

mUn−jm−1Vj+rt

n−j .

Proof. From Corollary 2.6, it follows that

Smn+r =

Vmn+r

2

(k2 + 4t)Umn+r

2Umn+r

2

Vmn+r

2

.On the other hand, Sm = UmS + tUm−1I and therefore

Smn+r = (Sm)nSr = (UmS + tUm−1I)nSr =

n∑j=0

(n

j

)U j

mUn−jm−1t

n−jSj+r

=

12

n∑j=0

(nj

)U j

mUn−jm−1t

n−jVj+r(k2+4t)

2

n∑j=0

(nj

)U j

mUn−jm−1t

n−jUj+r

12

n∑j=0

(nj

)U j

mUn−jm−1t

n−jUj+r12

n∑j=0

(nj

)U j

mUn−jm−1t

n−jVj+r

.Then the proof follows.

2.10. Corollary. Let m, r ∈ Z with m 6= 0 and m 6= 1. If k2 + 4t is not a perfect square,then

Umn+r = −(−t)rn∑

j=0

(n

j

)(−Um)jUn−j

m+1Uj−r

and

Vmn+r = (−t)rn∑

j=0

(n

j

)(−Um)jUn−j

m+1Vj−r.


Proof. The proof follows from Lemma 2.8 and Theorem 2.9 by taking a = Um andb = tUm−1

2.11. Corollary. V 2n − (k2 + 4t)U2

n = 4(−t)n for every n ∈ Z.

Proof. From Theorem 2.9, it follows that

detSn = (detS)n = (−t)n

and

detSn =V 2n − (k2 + 4t)U2

n

4.


2.12. Theorem. Let n ∈ N and m be a nonzero integer. Then

(2.1) 2nVmn+r =

bn2 c∑j=0

(n

2j

)U2j

m V n−2jm (k2 + 4t)jVr+

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m V n−2j−1m (k2 + 4t)j+1Ur

and

(2.2) 2nUmn+r =1

2n

bn2 c∑j=0

(n

2j

)U2j

m V n−2jm (k2 + 4t)jUr+

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m V n−2j−1m (k2 + 4t)jVr

Proof. Let K = S + tS−1 =

[0 k2 + 4t1 0

]. Then K2j = (k2 + 4t)jI and K2j+1 =

(k2 + 4t)jK. Since

Sm =1

2(VmI + UmK),

it follows that

Smn+r = (Sm)nSr = (1

2(VmI + UmK))nSr =

1

2n

(n∑

j=0

(n

j

)U j

mKjV n−j

m

)Sr

and therefore

2nSmn+r =

bn2 c∑j=0

(n

2j

)U2j

m V n−2jm K2jSr +

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m V n−2j−1m K2j+1Sr

=

bn2 c∑j=0

(n

2j

)U2j

m V n−2jm (k2 + 4t)jSr

+

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m V n−2j−1m (k2 + 4t)jKSr


Since

KSr =

(k2 + 4t)Ur

2

(k2 + 4t)Vr

2Vr

2

(k2 + 4t)Ur

2

and

Smn+r =

Vmn+r

2

(k2 + 4t)Umn+r

2Umn+r

2

Vmn+r

2

,the proof follows.

2.13. Theorem.

(2.3) Um+n = UmUn+1 + tUm−1Un

and

(2.4) (−t)n−1Um−n = Um−1Un − UmUn−1

for every m, n ∈ Z.

Proof. Let X =

[k t1 0

]. Then from Corollary 2.7, it follows that

Xm+n = XmXn =

[Um+1 tUm

Um tUm−1

] [Un+1 tUn

Un tUn−1

]and

Xm−n = Xm(Xn)−1 =

[Um+1 tUm

Um tUm−1

] [Un+1 tUn

Un tUn−1

]−1

=

[Um+1 tUm

Um tUm−1

]1

(−t)n

[tUn−1 −tUn

−Un Un+1

].


Now we give some identities, which we will use later. All the given identities can beshown by using the previously obtained formulae for Sn and Xn.

(2.5) UnVm+1 + tUn−1Vm = Vn+m

(2.6) VmVn − (k2 + 4t)UmUn = 2(−t)nVm−n

(2.7) UmVn − UnVm = 2(−t)nUm−n

(2.8) VmVn = Vm+n + (−t)nVm−n

(2.9) (k2 + 4t)UmUn = Vm+n − (−t)nVm−n

(2.10) UmVn = Um+n + (−t)nUm−n

(2.11) (−t)nVm−n = Um+1Vn − Vn+1Um

(2.12) VrVr+2 − V 2r+1 = (−t)r(k2 + 4t)

2.14. Theorem. Let m, n, r ∈ Z with r 6= 0. Then

UrUm+n+r = Um+rUn+r − (−t)rUmUn,

UrUm+n−r = UmUn − (−t)rUm−rUn−r,

and

UrUm+n = UmUn+r − (−t)rUm−rUn.


Proof. Take a =Ur+1

Urand consider A =

[a bc k − a

]with detA = −t. Then by

Corollary 2.3, we get

An =

[aUn + tUn−1 bUn

cUn Un+1 − aUn

]=

Ur+1

UrUn + tUn−1 bUn

cUn Un+1 −Ur+1

UrUn

.

Using (2.3) and (2.4) we see that

An =

Un+r

UrbUn

cUn−(−t)rUn−r

Ur

.Since detA = −t and a =

Ur+1

Ur, it follows that

bc =kUrUr+1 + tU2

r − U2r+1

U2r

=Ur(kUr+1 + tUr)− U2

r+1

U2r

=UrUr+2 − U2

r+1

U2r

=−(−t)r

U2r

by (2.4). If we consider the matrix multiplication AnAm = Am+n, then we get theresult.

2.15. Corollary. Un+rUn−r − U2n = −(−t)n−rU2

r for all n, r ∈ Z.

Proof. Since detA = −t, detAn = (detA)n = (−t)n. Moreover, since

detAn = −(−t)r Un+r

Ur

Un−r

Ur− bcU2

n = −(−t)r(Un+rUn−r − U2

n

U2r

)= (−t)n,

it can be seen that Un+rUn−r − U2n = −(−t)n−rU2

r .

2.16. Theorem. Let m, n, r ∈ Z. Then

VrVm+n+r = Vm+rVn+r + (−t)r(k2 + 4t)UmUn,

VrVm+n−r = (k2 + 4t)UmUn + (−t)rVm−rVn−r,

and

VrUm+n = UnVm+r + (−t)rVn−rUm.

Proof. Take a =Vr+1

Vrand consider B =

[a bc k − a

]with detB = −t. Then by

Corollary 2.3, we get

Bn =

[aUn + tUn−1 bUn

cUn Un+1 − aUn

]=

Vr+1

VrUn + tUn−1 bUn

cUn Un+1 −Vr+1

VrUn

.Using (2.5) and (2.11) we see that

Bn =

Vn+r

VrbUn

cUn(−t)rVn−r

Vr

.Since detB = −t and a =

Vr+1

Vr, it follows that


bc =kVrVr+1 + tV 2

r − V 2r+1

V 2r

=Vr(kVr+1 + tVr)− V 2

r+1

V 2r

=VrVr+2 − V 2

r+1

V 2r

=(−t)r(k2 + 4t)

V 2r

by (2.12). If we consider the matrix multiplication BnBm = Bm+n, then we get theresult.

2.17. Corollary. Vn+rVn−r − (k2 + 4t)U2n = (−t)n−rV 2

r for all n, r ∈ Z.

Proof. Since detB = −t, detBn = (detB)n = (−t)n. Moreover, since

detBn = (−t)r Vn+r

Vr

Vn−r

Vr−bcU2

n = (−t)r(Vn+rVn−r

V 2r

− (k2 + 4t)U2n

V 2r

)= (−t)n,

it can be seen that Vn+rVn−r − (k2 + 4t)U2n = (−t)n−rV 2

r .

3. Sums and Congruences

Now we will give some sums containing generalized Fibonacci and Lucas numbers.Then we will give some congruences concerning generalized Fibonacci and Lucas numbers.Firstly, we will prove a lemma to use in the following theorems. It can be seen that

(3.1) α2n = αnVn − (−t)n

and

(3.2) α2n = αnUn

√k2 + 4t+ (−t)n

by (1.2). Now we can give the following lemma.

3.1. Lemma.

(3.3) S2n = SnVn − (−t)nIand

(3.4) S2n = UnKSn + (−t)nI

for every n ∈ N, where K is as in Theorem 2.12.

Proof. Let Z [α] = aα+ b | a, b ∈ Z and Z [S] = aS + b | a, b ∈ Z . We define afunction ϕ : Z [α]→ Z [S] , given by ϕ(aα+ b) = aS+ bI. Then ϕ is ring homomorphism.Moreover it is clear that ϕ(α) = S and therefore we get ϕ(αn) = (ϕ(α))n = Sn. Thusfrom (3.1), we get

S2n = (ϕ(α))2n = ϕ(α2n) = ϕ(αnVn − (−t)n) = SnVn − (−t)nI.That is, S2n = SnVn − (−t)nI. Also from (3.2), we get

S2n = (ϕ(α))2n = ϕ(α2n) = ϕ(Un

√k2 + 4tαn + (−t)n) =

Unϕ(√

k2 + 4t)Sn + (−t)nI.

Since

ϕ(√

k2 + 4t)

= ϕ(2α− k) = 2S − kI =

[0 k2 + 4t1 0

]= K,

we get S2n = UnKSn + (−t)nI.


3.2. Theorem. Let m, r ∈ Z . Then

U2mn+r = (−(−t)m)nn∑

j=0

(n

j

)V jmUmj+r(−(−t)m)−j

and

V2mn+r = (−(−t)m)nn∑

j=0

(n

j

)V jmVmj+r(−(−t)m)−j

for every n ∈ N.

Proof. It is known that

(3.5) S2m = SmVm − (−t)mI

by (3.3). Taking the n-th power of (3.5), we get

S2mn = (SmVm − (−t)mI)n =

n∑j=0

(n

j

)V jm(−(−t)m)n−jSmj .

Multiplying both sides of this equation by Sr, we obtain

S2mn+r = (−(−t)m)nn∑

j=0

(n

j

)V jm(−(−t)m)−jSmj+r.

Thus it follows that

U2mn+r = (−(−t)m)nn∑

j=0

(n

j

)V jmUmj+r(−(−t)m)−j

and

V2mn+r = (−(−t)m)nn∑

j=0

(n

j

)V jmVmj+r(−(−t)m)−j

by Corollary 2.6.

3.3. Corollary. Let k and t be integers. Then for all n,m ∈ N ∪ 0 and r ∈ Z suchthat mn+ r ≥ 0 if t 6= ±1, we get

U2mn+r ≡ (− (−t)m)nUr (mod Vm)

and

V2mn+r ≡ (− (−t)m)nVr (mod Vm).

3.4. Theorem. Let m, r ∈ Z and m be nonzero integer. Then

U2mn+r = (−t)mn

bn2 c∑j=0

(n

2j

)U2j

m U2mj+rDjt−2mj

+ (−t)mn

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m V2mj+m+rDj(−t)m(−2j−1)


and

V2mn+r = (−t)mn

bn2 c∑j=0

(n

2j

)U2j

m V2mj+rDjt−2mj

+ (−t)mn

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m U2mj+m+rDj+1(−t)m(−2j−1)

for every n ∈ N, where D = k2 + 4t.

Proof. It is known that

S2m = UmKSm + (−t)mI

by (3.4). It is clear that

S2mn+r = (UmKSm + (−t)mI)n Sr =

n∑j=0

(n

j

)U j

mKj((−t)m)n−jSmj+r.

On the other hand, it can be seen that K2j = DjI and K2j+1 = DjK. Therefore, we get

S2mn+r = (−t)mn

bn2 c∑j=0

(n

2j

)U2j

mK2jt−2mjS2mj+r

+ (−t)mn

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m K2j+1(−t)m(−2j−1)S2mj+m+r

= (−t)mn

bn2 c∑j=0

(n

2j

)U2j

mDjt−2mjS2mj+r

+ (−t)mn

bn−12 c∑

j=0

(n

2j + 1

)U2j+1

m Dj(−t)m(−2j−1)KS2mj+m+r.

The proof follows from Corollary 2.6.

3.5. Corollary. Let k and t be integers. Then for all n,m ∈ N and r ∈ Z such thatmn+ r ≥ 0 if t 6= ±1, we get

U2mn+r ≡ (−t)mn Ur (mod Um)

and

V2mn+r ≡ (−t)mn Vr (mod Um).

References

[1] P. Ribenboim, My numbers, My Friends, (Springer-Verlag New York, Inc., 2000).

[2] S. Rabinowitz, Algorithmic Manipulation of Fibonacci Identities, Applications of FibonacciNumbers 6, Kluwer Academic Pub., Dordrect, The Netherlands, 389–408, 1996.

[3] J. B. Muskat, Generalized Fibonacci and Lucas Sequences and Rootfinding Methods, Math-ematics of Computation, 61, Number 203, 365–372, July 1993.

[4] D. Kalman and R. Mena, The Fibonacci Numbers-Exposed, Mathematics Magazine 76,

167–181, 2003.

[5] S. Falcon and A. Plaza, The k-Fibonacci Sequence and The Pascal 2-Triangle, Chaos,Solitions and Fractals 33 , 38–49, 2007.


[6] S. Falcon and A. Plaza, On The Fibonacci k-numbers, Chaos, Solitions and Fractals 32,

1615–1624, 2007.

[7] R. Keskin and B. Demirturk, Solutions of Some Diophantine Equations Using GeneralizedFibonacci and Lucas Sequences, Ars Combinatoria (in press).

[8] J. P. Jones, Representation of Solutions of Pell equations Using Lucas Sequences, Acta

Academia Pead. Agr., Sectio Mathematicae 30, 75–86, 2003.[9] M. E. H. Ismail, One Parameter Generalizations of the Fibonacci and Lucas Numbers, The

Fibonacci Quarterly 46-47, 167–180, 2009.

[10] T.-X. He and P. J. S. Shiue, On Sequences of Numbers and Polynomials Defined By LinearRecurrence Relations of Order 2, International Journal of Mathematics and Mathematical

Sciences 2009 , 21 page, 2009.

[11] R. Keskin and B. Demirturk, Some New Fibonacci and Lucas Identities by Matrix Methods,International Journal of Mathematical Education in Science and Technology, 1-9, 2009.


ON EDELSTEIN TYPE

MULTIVALUED RANDOM

OPERATORS

Akbar Azam ∗, Muhammad Arshad † and Pasquale Vetro ‡


Abstract

The purpose of this paper is to provide stochastic versions of severalresults on fixed point theorems in the literature.

Keywords: Common fixed point, Multivalued mappings, Random operator, Metricspace

2000 AMS Classification: 47H09, 47H10, 47H40, 54H25, 60H25

1. Introduction and Preliminaries

Random operator theory is needed for the study of various classes of random operatorequations in probabilistic functional analysis. During the last three decades several results(e.g., see, [3, 4, 6, 8, 10, 11, 13, 14, 15] and references therein) regarding random fixedpoints of various types of random operators have been established and a number oftheir applications have been obtained after a survey article of Bharucha Reid [5]. Infact, random fixed point theorems are stochastic generalizations of deterministic/classicalfixed point theorems and have important applications in random operator equations,random differential equations and differential inclusions [5, 6, 7, 10]. In the presentpaper we derive common random fixed point theorems for a sequence of multivaluedrandom operators satisfying Edelstein type contractive condition. We give, also a resultof a common random fixed point for a sequence of multivalued random operators thathave a common deterministic fixed point. Our paper establish stochastic versions ofmany Banach type fixed point theorems e.g., see, [2] and references therein.

∗Department of Mathematics, COMSATS Institute of Information Technology, ChakShahzad, Islamabad, 44000, Pakistan. E-mail: (A. Azam) [email protected]†Department of Mathematics, International Islamic University, H-10, Islamabad,44000, Pak-

istan. E-mail: (M. Arshad) marshad [email protected]‡Universita degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi

34, 90123 Palermo, Italy E-mail: (P. Vetro) [email protected]

224 A. Azam, M. Arshad, P. Vetro

For a metric space (X, d), we denote by 2X the family of all nonempty subsets of X,CB (X) the family of all nonempty closed and bounded subsets of X, we define Hausdorffmetric H on CB (X) as follows:

H (A,B) = max

supα∈A

d (a,B) , supb∈B

d (A, b)

for A,B ∈ CB (X) , where

d (x,E) = inf d (x, y) : y ∈ E .

Let (Ω,Σ) be a measurable space (i.e., Σ is a σ−algebra of subsets of Ω). A functionξ : Ω −→ X is said to measurable if for any open subset C of X, ξ−1 (C) ∈ Σ. Amultivalued mapping T : Ω −→ 2X is called measurable if for any open subset C of X.

T−1 (C) = w ∈ Ω : T (w) ∩ C 6= ∅ ∈ Σ.

This type of measurability is usually called weakly measurability (cf. Himmelberg [9]),but as in this paper we always use this type of measurability, thus we omit the term“weakly” for simplicity. A mapping ξ : Ω −→ X is said to be measurable selector of ameasurable mapping T : Ω→ 2X if ξ is measurable and for each w ∈ Ω, ξ (w) ∈ T (w) .A mapping T : Ω ×X −→ 2X is called multivalued random operator if for any x ∈ X,T (·, x) is measurable.

A measurable mapping ξ : Ω −→ X is said to be a random fixed point of multivaluedrandom operator T : Ω ×X −→ 2X if for each w ∈ Ω, ξ (w) ∈ T (w, ξ (w)). A mappingξ : Ω −→ X is said to be a deterministic fixed point of multivalued random operatorT : Ω×X −→ 2X if for each w ∈ Ω, ξ (w) ∈ T (w, ξ (w)).

In [8] Fierro et al. introduced a condition, named condition (P), and we prove somerandom fixed points theorems. A mapping T : X → 2X is said to satisfy condition (P)if, for every closed ball B of X with radius r ≥ 0 and any sequence xn ⊂ X for whichd(xn, B) → 0 and d(xn, Txn) → 0 as n → ∞, there exists x0 ∈ B such that x0 ∈ Tx0.The operator T : Ω × X → 2X satisfies condition (P) if, for each ω ∈ Ω, the mappingT (ω, ·) : X → 2X satisfy condition (P).

1.1. Lemma. ([1, Lemma 2]) Let An be a sequence in CB(X) and there exists A ∈CB(X) such that

limn→∞

H(An, A)→ 0.

If xn ∈ An (n = 1, 2, 3, ..) and there exists x ∈ X such that

limn→∞

d(xn, x)→ 0

then x ∈ A.

1.2. Lemma. [9, Theorems 3.2(i), 3.3] Let (X, d) be a Polish space and T : Ω −→ 2X

is a closed valued mapping. Consider the following statements:

(a) for any closed subset C of X

T−1 (C) = w ∈ Ω : T (w) ∩ C 6= ∅ ∈ Σ.

(b) T is measurable.(c) w 7−→ d (x, T (w)) is measurable function of w for each x ∈ X.

Then

(a)⇒ (b)⇔ (c)

On Edelstein type multivalued random operators 225

2. Main Results

Let (X, d) be a metric space, ε > 0 and x, y ∈ X. An ε-chain from x to y is afinite set of points x1, x2, x3, . . . , xn such that x = x1, xn = y, and d(xj−1, xj) < ε forall j = 2, 3, . . . , n. A metric space (X, d) is said to be ε-chainable if and only if givenx, y ∈ X, there exists an ε-chain from x to y. For every ε ∈ (0,∞], let Kε the family ofmappings K : [0, ε) → [0, 1) that satisfy the condition: for t > 0, there exist δ(t) > 0and s(t) < 1 such that

0 ≤ r − t < δ(t) =⇒ K(r) ≤ s(t).The following theorem is proved in [2].

2.1. Theorem. Let (X, d) be a complete ε-chainable metric space and Tn∞n=1be a se-quence of multivalued mapping from X to CB(X) satisfying the following condition: x, y ∈X and 0 < d(x, y) < ε implies

H(Tnx, Tmy) ≤ K(d(x, y))d(x, y)

for n,m = 1, 2..., where K ∈ Kε. Then there exists a point y∗ ∈ X such that y∗ ∈∞⋂n=1

Tny∗.

The following theorem is the stochastic versions of the above result.

2.2. Theorem. Let (X, d) be an ε-chainable Polish space and Tn∞n=1be a sequence ofrandom operators from Ω×X to CB(X) satisfying the following condition: x, y ∈ X and0 < d(x, y) < ε implies

H(Tn (w, x) , Tm (w, y)) ≤ K(w, d(x, y))d(x, y)

for n,m = 1, 2..., where K : Ω × [0, ε) → [0, 1) is a mapping such that K (w, ·) ∈ Kε

and it is measurable for each w ∈ Ω. If Tj enjoys condition (P) for every j ∈ N, thenthere exists a common random fixed point of Tn∞n=1, that is, there exists a measurablemapping ξ : Ω −→ X such that for all w ∈ Ω

ξ (w) ∈∞⋂n=1

Tn (w, ξ (w)) .

Proof. We note, that for every w ∈ Ω, the sequence Tn(w, ·)∞n=1 of multivalued map-pings on X satisfy the hypothesis of Theorem 2.1, so there exists a point x∗ ∈ X suchthat

x∗ ∈∞⋂n=1

Tn (w, x∗) .

Now we see that each Tn(w, ·) is continuous, for all w ∈ Ω. Let β > 0 and assume thatxm → x∗, then there exists an integer Mβ > 0 such that m ≥ Mβ implies d(xm, x

∗) <minβ, ε. From inequality (2), we deduce that

H(Tn (w, xm) , Tn (w, x∗)) ≤ K(w, d(xm, x∗))d(xm, x

∗) =

Kw(d(xm, x∗))d(xm, x

∗) < d(xm, x∗) < minβ, ε ≤ β,

whenever, m ≥Mβ . Consequently, Tn(w, ·) is continuous.We consider the multivalued mapping F : Ω→ 2X defined by

F (w) =

x ∈ X : x ∈

∞⋂n=1

Tn (w, x)

.

In the view of inequality (3), it follows that F (w) is nonempty for each w ∈ Ω.


To see that F (·) is closed valued, let u be a limit point of F (w), this implies that there

exists a sequence u1, u2, u3, ... ⊂ F (w) such that ui → u. Then ui ∈∞⋂n=1

Tn (w, ui), for

every i = 1, 2, 3, ... . Since ui ∈ Tn (w, ui), for every n, the continuity of Tn (w, ·) implies

that Tn (w, ui)→ Tn (w, u) . By Lemma 1.1, it follows that u ∈∞⋂n=1

Tn (w, u), hence F (·)

is closed valued.Now, for every j = 1, 2, 3, ...., we consider the multivalued mapping Fj : Ω → 2X

defined by

Fj (w) = x ∈ X : x ∈ Tj (w, x) .To see that Fj(·) is a measurable mapping, let B = B(z, r) := y ∈ X : d(z, y) ≤ rbe a closed ball of X and we prove that F−1

j (B) ∈ Σ. Take a countable dense subset

S = x1, x2, ... of X and let

L (B) =

∞⋂n=1

⋃xi∈Sn

w ∈ Ω : d (xi, Tj (w, xi)) <

2ε

n

,

where

Sn =x ∈ S : d (x,B) <

ε

n

.

We show that F−1j (B) = L (B) . If w ∈ F−1

j (B) then Fj (w) ∩ B 6= ∅. Let x ∈ B such

that x ∈ Tj (w, x), then S ∩ z : d(x, z) < εn 6= ∅. It follows that for each n there exists

xi(n) ∈ Sn such that d(x, xi(n)

)< ε

n≤ ε. This implies that

H(Tj (w, x) , Tj

(w, xi(n)

))≤ K(w, d(x, xi(n)))d(x, xi(n)) < d(x, xi(n)) <

ε

n.

We obtain

d(xi(n), Tj

(w, xi(n)

))≤ d

(xi(n), x

)+ d

(x, Tj

(w, xi(n)

))≤

d(xi(n), x

)+ H

(Tj (w, x) , Tj

(w, xi(n)

))<ε

n+ε

n.

As xi(n) ∈ Sn, it follows that w ∈ L (B) and F−1j (B) ⊆ L (B).

Conversely, if ω ∈ L(B), then for each n, we can take xi(n) ∈ Sn for which

d(xi(n), Tj(ω, xi(n))) < 2ε/n.

We have

d(xi(n), B)→ 0 i.e. d(xi(n), Tj(ω, xi(n))→ 0,

since the mapping Tj(ω, ·) satisfies condition (P), there exists x ∈ B such that x ∈Tj(ω, x) and hence ω ∈ F−1

j (B). Therefore L(B) = F−1j (B).

Now for any x ∈ X define a mapping Gj(x) : Ω→ R as Gj(x) (·) = d (x, Tj (·, x)) , since(by hypotheses) Tj (·, x) is closed valued and measurable. By Lemma 1.2, the mappingGj(x) (·) is measurable. It follows that

w ∈ Ω : d (xi, Tj (w, xi)) <2ε

n

∈ Σ.

Hence F−1j (B) = L (B) ∈ Σ. To complete the proof, let G be an arbitrary open subset

of X, by the separability of X there exists a sequence of closed ball Bn such that

G =

∞⋃n=1

Bn.

Since F−1j (G) =

⋃∞n=1 F

−1j (Bn), we conclude that Fj is measurable.


Hence, F (·) is measurable. The Kuratowski-Ryll-Nardzewski theorem [12] furtherimplies that there exists a measurable mapping ξ : Ω −→ X such that for all w ∈ Ω

ξ (w) ∈∞⋂n=1

Tn (w, ξ (w)) .

This completes the proof.

The following results are direct consequences of the above theorem.

2.3. Theorem. Let (X, d) be an ε-chainable Polish space and T : Ω ×X → CB(X) bea multivalued random operator satisfying the following condition: x, y ∈ X and 0 <d(x, y) < ε implies

H(T (w, x) , T (w, y)) ≤ K(w, d(x, y))d(x, y),

where K : Ω × [0, ε) → [0, 1) is a mapping such that K (w, ·) ∈ Kε and it is measurablefor each w ∈ Ω. If T enjoys condition (P), then T has a random fixed point.

2.4. Corollary. Let (X, d) be a Polish space and T : Ω×X → CB(X) be a multivaluedrandom operator satisfying the following condition: x, y ∈ X

H(T (w, x) , T (w, y)) ≤ K(w, d(x, y))d(x, y),

where K : Ω× [0,∞)→ [0, 1) is a mapping such that K (w, ·) ∈ K∞ and it is measurablefor each w ∈ Ω. If T enjoys condition (P), then T has a random fixed point.

3. A result for a sequence with a common deterministic fixedpoint

In this section we consider sequences of multivalued random operators with a commondeterministic fixed point and we deduce the existence of a common random fixed point.

Let (X, d) be a metric space and ε > 0. A sequence Tn∞n=1 of multivalued randomoperators from Ω×X to CB(X) is ε-locally nonexpansive if for every x, y ∈ X such that0 < d(x, y) < ε holds

H(T (ω, x), T (ω, y)) ≤ d(x, y).

3.1. Theorem. Let (X, d) be a Polish space and Tn∞n=1 be a sequence of ε-locallynonexpansive multivalued random operators from Ω×X to CB(X). Assume that the Tjenjoys condition (P) for every j ∈ N. If the sequence Tn∞n=1 of random operators, hasa common deterministic fixed point, then there exists a common random fixed point ofTn∞n=1, that is, there exists a measurable mapping ξ : Ω −→ X such that for all w ∈ Ω

ξ (w) ∈∞⋂n=1

Tn (w, ξ (w)) .

Proof. Let F : Ω→ 2X be defined for every ω ∈ Ω from

F (ω) := x ∈ X : x ∈+∞⋂n=1

Tn(ω, x).

Since the random operators Tn (n = 1, 2, ...) have a common deterministic fixed point wededuce that F (ω) 6= ∅ for all ω ∈ Ω. We note that Tj(ω, ·) : X → CB(X) is continuousfor all ω ∈ Ω and j ∈ N. In fact if 0 < d(x, y) < ε, we have

H(Tj(ω, x), Tj(ω, y)) ≤ d(x, y).

The set

Aj(ω) := x ∈ X : x ∈ Tj(ω, x)


is closed for all j and it implies that F (ω) is closed for all ω ∈ Ω. Proceeding as inthe proof of Theorem 2.2, we obtain that there exists a common random fixed point ofTn∞n=1.

3.2. Proposition. Let (X, d) be a locally compact metric space and T be a ε-locally non-expansive multivalued random operator from Ω×X to CB(X). Then T enjoys condition(P).

Proof. Let B be a compact ball of X and xn a sequence such that d(xn, B) → 0,it is not restrictive to suppose that xn → x0 ∈ B. If 0 < d(xn, x0) < ε, thenH(T (ω, x0), T (ω, xn)) ≤ d(x0, xn), consequently

H(T (ω, x0), T (ω, xn)→ 0.

By Lemma 1.1, we deduce that x0 ∈ T (ω, x0) and thus T satisfies condition (P).

The following results are direct consequences of the Theorem 3.1 and Proposition 3.2.

3.3. Corollary. Let (X, d) be a locally compact separable complete metric space andTn∞n=1 be a sequence of ε-locally nonexpansive multivalued random operators from Ω×Xto CB(X). If the random operators Tn∞n=1 have a common deterministic fixed point,then there exists a common random fixed point of Tn∞n=1.

3.4. Corollary. Let (X, d) be a locally compact separable complete metric space and Tbe a ε-locally nonexpansive multivalued random operator from Ω ×X to CB(X). If therandom operator T has a deterministic fixed point, then there exists a random fixed pointof T .

References

[1] Assad N. A. and Kirk, W. A. Fixed point theorems for set-valued mappings of contractivetype, Pacific J. Math., 43, 553–562, 1972.

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ANTI-INVARIANT ξ⊥-RIEMANNIAN

SUBMERSIONS FROM ALMOST

CONTACT MANIFOLDS

Jae Won Lee ∗


Abstract

We introduce anti-invariant ξ⊥-Riemannian submersions from almostcontact manifolds onto Riemannian manifolds. We give an example,investigate the geometry of foliations which are arisen from the defi-nition of a Riemannian submersion and check the harmonicity of suchsubmersions. We also find necessary and sufficient conditions for a spe-cial anti-invariant ξ⊥-Riemannian submersion to be totally geodesic.Moreover, we obtain decomposition theorems for the total manifold ofsuch submersions.

Keywords: Riemannian submersion, Sasakian manifold, Anti-invariant ξ⊥-Riemannainsubmersion

2000 AMS Classification: 53C25, 53C40, 53C50

1. Introduction

Riemannian submersions between Riemannian manifolds were studied by O’Neil [9]and Gray [7]. In [13], Waston defined almost Hermitian submersions between almostHermitian manifolds and he showed that the base manifold and each fiber has the samekind of structure as the total space, in most cases. He also showed that the verti-cal and horizontal distributions are invariant. On the other hand, the geometry ofanti-invariant Riemannian submersions is quite different from the geometry of almostHermitian submersions. For example, since every holomorphic map between Kahlermanifolds is harmonic [5], it follows that any holomorphic submersion between Kahlermanifolds is harmonic. However, this result is not valid for anti-invariant Riemanniansubmersions, which was first studied by Sahin in [11]. Simlarly, Ianus and Pastore [8]shows φ-holomorphic maps between contact manifolds are harmonic. This implies that

∗University of Missouri, Department of Mathematics, 202 Mathematical Sciences Building,

Columbia, MO 65211E-mail: [email protected]

232 J.W. Lee

any contact submersion is harmonic. However, this result is not valid for anti-invariantξ⊥-Riemannian submersions.

We also note that Riemannian submersions have applications in Klauza-Klein thoeryand the theory of robotics. Indeed, in Klauza-Klein theory, the general solution of a recentmodel can be expressed in harmonic maps which satisfies Einstein equations. However,a very general class of solution is given by Riemannian submersions from the extradimensional space onto the space in which the scalar fields take values ( see [6] for details). On the other hand, Altafini [1] used the Riemannian submersion in redundant robots,it means that the robotic chain has more than six joints, and showed that the forwardkinematic map from joint space to the workspace of the end effector is a Riemanniansubmersion. He also showed that there is a close relationship between inverse kinematicin robotics and the horizontal lift of vector fields in Riemannian submersions.

In [4], Chinea defined almost contact Riemannian submersions between almost contactmetric manifolds and examined the differential geometric properties of Riemannian sub-mersions between almost contact metric manifolds. More precisely, let (M, gM , ϕ, ξ, η)and (N, gN , ϕ

′, ξ′, η′) be almost contact manifolds with dimM = 2m + 1 and dimN =2n + 1. A Riemannian submersion F : M −→ N is called th almost contact metricsubmersion if F is an almost contact mapping, i.e., ϕ′F∗ = F∗ϕ. The main result of thisnotion is that the vertical and horizontal distributions are ϕ-invariant. Moreover, thecharacteristic vector field ξ is horizontal. We note that only ϕ-holomorphic submersionshave been considered on almost contact manifolds [6].

In this paper, we consider a Riemannian submersion from an almost contact manifoldunder the assumption that the fibers are anti-invariant with respect to the tensor field oftype (1, 1) of the almost contact manifold. This assumption implies that the horizontaldistribution is not invariant under the action of the tensor field of the total manifoldof such submersions. Roughly speaking, almost contact submersions are useful for de-scribing the geometry of base manifolds, anti-invariant submersions are however servedto determine the geometry of total manifolds.

The paper is organized as follows: In Section 2, we present the basic information,needed for this paper. In Section 3, we give the definition of anti-invariant ξ⊥-Riemanniansubmersions, provide an example and investigate the geometry of leaves of the distri-butions. We also introduce a special anti-invariant ξ⊥-Riemannian submersions andobtain necessary and sufficient conditions for such submersions to be totally geodesicor harmonic. In Section 4, we give decomposition theorems by using the existence ofanti-invariant ξ⊥-Riemannian submersions and observe that such submersions put somerestrictions on the geometry of the total manifold.

2. Preliminaries

In this section, we define almost contact manifolds, recall the notion of Riemanniansubmersions between Riemannian manifolds and give a brief review of basic facts of Rie-mannian submersions.

Let M be an almost contact metric manifold with structure tensors (ϕ, ξ, η, gM ) whereϕ is a tensor field of type (1, 1), ξ a vector field, η a 1-form and gM is the Riemannianmetric on M . Then these tensors satisfy [2]

(2.1) ϕξ = 0, η ϕ = 0, η(ξ) = 1

(2.2) ϕ2 = −I + η ⊗ ξ, and gM (ϕX,ϕY ) = gM (X,Y )− η(X)η(Y ),

Anti-invariant ξ⊥-Riemannian submersions from almost contact manifolds 233

where I denotes the identity endomorphism of TM and X,Y are any vector fields on M .Moreover, if M is Sasakian [12], then we have

(2.3) (∇Xϕ)Y = −g(X,Y )ξ + η(Y )X and ∇Xξ = ϕX,

where ∇ is the connection of Levi-Civita covariant differentiation.

Let (Mm, gM ) and (Nn, gN ) be Riemannian manifolds, where dimM = m, dimN = nand m > n. A Riemannian submersion F : M −→ N is a map from M onto N satisfyingthe following axioms:

(S1) F has the maximal rank.(S2) The differential F∗ preserves the lengths of horizontal vectors.

For each q ∈ N , F−1(q) is an (m − n)-dimensional submanifold of M . The sub-manifolds F−1(q) are called fibers. A vector field on M is called vertical if it is alwaystangent to fibers. A vector field on M is called horizontal if it is always orthogonalto fibers. A vector field X on M is called basic if X is horizontal and F -related toa vector field X∗ on N , i.e., F∗Xp = X∗F (p) for all p ∈ M . Note that we denote the

projection morphisms on the distributions kerF∗ and (kerF∗)⊥ by V and H, respectively.

We recall the following lemma from O’Neil [9].

2.1. Lemma. Let F : M −→ N be a Riemannian submersion between Riemannianmanifolds and X,Y be basic vector fields of M . Then

(a) gM (X,Y ) = gN (X∗, Y∗) F .

(b) the horizontal part [X,Y ]H of [X,Y ] is a basic vector field and corresponds to[X∗, Y∗], i.e., F∗([X,Y ]) = [X∗, Y∗].

(c) [V,X] is vertical for any vector field V of kerF∗.

(d) (∇MX Y )H is the basic vector field corresponding to ∇N

X∗Y∗.

The geometry of Riemannian submersions is characterized by O’Neil’s tensors T andA defined for vector fields E, F on M by

AEF = H∇HEVF + V∇HEHF(2.4)

TEF = H∇VEVF + V∇VEHF ,(2.5)

where ∇ is the Levi-Civita connection of gM . It is easy to see that a Riemannian sub-mersion F : M −→ N has totally geodesic fibers if and only if T vanishes identically. Forany E ∈ Γ(TM), TC = TVE and A is horizontal, A = AHE . We note that the tensor T

and A satisfy

TUW = TWU, U,W ∈ Γ(kerF∗)(2.6)

AXY = −AYX =1

2V[X,Y ], X, Y ∈ Γ((kerF∗)

⊥).(2.7)

On the other hand, from (2.4) and (2.5), we have

∇VW = TVW + ∇VW(2.8)

∇VX = H∇VX + TVX(2.9)

∇XV = AXV + V∇XV(2.10)

∇XY = H∇XY + AXY(2.11)

for X,Y ∈ Γ((kerF∗)⊥) and V,W ∈ Γ(kerF∗), where ∇VW = V∇VW . If X is basic,

then H∇VX = AXV .

234 J.W. Lee

Finally, we recall the notion of harmonic maps betwwen Riemannian manifolds. Let(M, gM ) and (N, gN ) be Riemannian manifolds and supposed that ϕ : M −→ N isa smooth map. Then the differential ϕ∗ of ϕ can be viewed a section of the bun-dle Hom(TM,ϕ−1TN) → M , where ϕ−1TN is the pullback bundle which has fibers(ϕ−1TN)p = Tϕ(p)N , p ∈M . Hom(TM,ϕ−1TN) has a connection ∇ induced from the

Levi-Civita connection ∇M and the pullback connection ∇ϕ. Then the second funda-mental form of ϕ is given by

(2.12) (∇ϕ∗)(X,Y ) = ∇ϕXϕ∗(Y )− ϕ∗(∇M

X Y )

for X,Y ∈ Γ(TM). It is known that the second fundamental form is symmetric. Asmooth map ϕ : (M, gM ) −→ (N, gN ) is said to be harmonic if trace(∇ϕ∗) = 0. On theother hand, the tensor field of ϕ is the sectionτ(ϕ) of Γ(ϕ−1TN) defined by

(2.13) τ(ϕ) = divϕ∗ =

m∑i=1

(∇ϕ∗)(ei, ei),

where e1, · · · , em is the orthonormal frame on M . Then it follows that ϕ is harmonicif and only if τ(ϕ) = 0 (for details, see [3]).

3. Anti-invariant ξ⊥-Rieamannian submersions

In this section, we define anti-invariant ξ⊥-Riemannian submersion from an almostcontact metric manifold onto a Riemannian manifold and investigate the integrabiltiyof distributions and obtain a necessary and sufficient condition for such submersions tobe totally geodesic map. We also investigate the harmonicity of a special Riemanniansubmersions.

3.1. Definition. Let (M, gM , ϕ, ξ, η) be an almost contact metric manifold and (N, gN ) aRiemannian manifold. Suppose that there exists a Riemannian submersion F : M −→ Nsuch that ξ is normal to kerF∗ and kerF∗ is anti-invariant with respect to ϕ, i.e.,ϕ(kerF∗) ⊂ (kerF∗)

⊥. Then we say that F is an anti-invariant ξ⊥-Riemannian sub-mersion.

Now, we assume that F : (M, gM , ϕ, ξ, η) −→ (N, gN ) is an anti-invariant ξ⊥-Riemanniansubmersion. First of all, from Definition 3.1, we have (kerF∗)

⊥∩kerF∗ 6= 0. We denotethe complementary orthogonal distribution to ϕ(kerF∗) in (kerF∗)

⊥ by µ. Then we have

(3.1) (kerF∗)⊥ = ϕ(kerF∗)⊕ µ,

where ϕ(µ) ⊂ µ. Hence µ contains ξ. Thus, for X ∈ Γ((kerF∗)⊥), we have

(3.2) ϕX = BX + CX,

where BX ∈ Γ(kerF∗) and CX ∈ Γ(µ). On the other hand, since F∗((kerF∗)⊥) = TN

and F is a Riemannian submersion, using (3.2), we have gN (F∗ϕV, F∗ϕCX) = 0 for anyX ∈ Γ((kerF∗)

⊥) and V ∈ Γ(kerF∗), which implies that

TN = F∗(ϕ(kerF∗))⊕ F∗(µ)

3.2. Example. Let (R5, g, ϕ0, ξ, η) denote the manifold R5 with its usual Sasakian struc-ture given by

η =1

2

(dz − x2dx1 − x4dx3

), ξ = 2

∂

∂z,

g = η ⊗ η +1

4

(dx1 ⊗ dx1 + dx2 ⊗ dx2 + dx3 ⊗ dx3 + dx4 ⊗ dx4

),


ϕ0

(X1

∂

∂x1+X2

∂

∂x2+X3

∂

∂x3+X4

∂

∂x4+ Z

∂

∂z

)= X2

∂

∂x1−X1

∂

∂x2+X4

∂

∂x3−X3

∂

∂x4+(X2x

2 +X4x4) ∂∂z,

where(x1, x2, x3, x4, z

)are the Cartesian coordinates.

Let F : (R5, g, ϕ0, ξ, η) −→ R3 be a map defined by F(x1, x2, x3, x4, z

)=(

x1+x4√2, x2+x3√2, z)

.

Then, by the direct computations, we have

kerF∗ = span

V =

∂

∂x1− ∂

∂x4, W =

∂

∂x2− ∂

∂x3

,

and

(kerF∗)⊥ = span

X =

∂

∂x1+

∂

∂x4+ x4

∂

∂z, Y =

∂

∂x2+

∂

∂x3+ x2

∂

∂z, ξ = 2

∂

∂z

.

Then it is easy to see that F is a Riemannian submersion. Moreover, since ϕ0(V ) =−Y and ϕ0(W ) = X, ϕ0(kerF∗) ⊂ (kerF∗)

⊥. As a result, F is an anti-invariant ξ⊥-Riemannian submersion.

3.3. Lemma. Let F be an anti-invariant ξ⊥-Riemannian submersion from a Sasakianmanifold (M, gM , ϕ, ξ, η) onto a Riemannian manifold (N, gN ). Then we have

(3.3) gM (CY, ϕV ) = 0

and

(3.4) gM (∇XCY, ϕV ) = −gM (CY, ϕAXV )

for X,Y ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗).

Proof. For Y ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗), using (2.2), we have

gM (CY, ϕV ) = gM (ϕY−BY,ϕV ) = gM (ϕY, ϕV ) = gM (Y, V )+η(Y )η(V ) = gM (Y, V ) = 0

since BY ∈ Γ(kerF∗) and ϕV, ξ ∈ Γ((kerF∗)⊥). Differentiating (3.3) with respect to X,

we get

gM (∇XCY, ϕV ) = −gM (CY,∇XϕV )

= gM (CY, (∇Xϕ)V )− gM (CY, ϕ(∇XV ))

= −gM (CY, ϕ(∇XV ))

= −gM (CY, ϕAXV )− gM (CY, ϕV∇XV )

= −gM (CY, ϕAXV )

due to ϕV∇XV ∈ Γ(ϕ(kerF∗)). Our assertion is complete.

We study the ingetgrability of the distribution (kerF∗)⊥ and then we investigate the

geometry of leaves of kerF∗ and (kerF∗)⊥. We note it is known that the distribution

kerF∗ is integrable.

3.4. Theorem. Let F be an anti-invariant ξ⊥-Riemannian submersion from a Sasakianmanifold (M, gM , ϕ, ξ, η) onto a Riemannian manifold (N, gN ). The followings are equiv-alent.

(a) (kerF∗)⊥ is integrable

(b)

gN ((∇F∗)(Y,BX), F∗ϕV ) = gN ((∇F∗)(X,BY ), F∗ϕV )

+ gM (CY, ϕAXV )− gM (CX,ϕAY V )

+ η(Y )gM (X,ϕV )− η(X)gM (Y, ϕV )

236 J.W. Lee

(c)

gM (AXBY −AYBX,ϕV ) = gM (CY, ϕAXV )− gM (CX,ϕAY V )

+ η(Y )gM (X,ϕV )− η(X)gM (Y, ϕV )

for X,Y ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗).

Proof. For Y ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗), from Definition 3.1, ϕV ∈ Γ((kerF∗)

⊥)and ϕY ∈ Γ(kerF∗ ⊕ µ). Using (2.2) and (2.3), we note that for X ∈ Γ((kerF∗)

⊥),

(3.5) gM (∇XY, V ) = gM (∇XϕY, ϕV )− η(Y )gM (X,ϕV )

Therefore, from (3.5), we get

gM ([X,Y ], V ) = gM (∇XϕY, ϕV )− gM (∇Y ϕX,ϕV )

− η(Y )gM (X,ϕV ) + η(X)gM (Y, ϕV )

= gM (∇XBY,ϕV ) + gM (∇XCY, ϕV )

− gM (∇YBX,ϕV )− gM (∇Y CX,ϕV )

− η(Y )gM (X,ϕV ) + η(X)gM (Y, ϕV ).

Since F is a Riemannian submersion, we obtain

gM ([X,Y ], V ) = gN (F∗∇XBY,F∗ϕV ) + gM (∇XCY, ϕV )

− gN (F∗∇YBX,F∗ϕV )− gM (∇Y CX,ϕV )

− η(Y )gM (X,ϕV ) + η(X)gM (Y, ϕV ).

Thus, from (2.13) and (3.4), we have

gM ([X,Y ], V ) = gN (−(∇F∗)(X,BY ) + (∇F∗)(Y,BX), F∗ϕV )

− gM (CY, ϕAXV ) + gM (CX,ϕAY V )

− η(Y )gM (X,ϕV ) + η(X)gM (Y, ϕV ),

which proves (a) ⇐⇒ (b).On the other hand, using (2.12), we obtain

(∇F∗)(Y,BX)− (∇F∗)(X,BY ) = −F∗(∇YBX −∇XBY ) = −F∗(AYBX −AXBY ),

which shows that (b) ⇐⇒ (c).

3.5. Corollary. Let F be an anti-invariant ξ⊥-Riemannian submersion from a Sasakianmanifold (M, gM , ϕ, ξ, η) onto a Riemannian manifold (N, gN ) with (kerF∗)

⊥ = ϕ(kerF∗)⊕ <ξ >. Then the followings are equivalent.

(a) (kerF∗)⊥ is integrable

(b) (∇F∗)(X,ϕY ) + η(X)F∗Y = (∇F∗)(Y, ϕX) + η(Y )F∗X(c) AXϕY + η(X)Y = AY ϕX + η(Y )X, for X,Y ∈ Γ((kerF∗)

⊥).


(a) (kerF∗)⊥ defines a totally geodesic foliation on M .

(b) gM (AXBY,ϕV ) = gM (CY, ϕAXV )− η(Y )gM (X,ϕV )(c) gN ((∇F∗)(Y, ϕX), F∗ϕV ) = gM (CY, ϕAXV ) − η(Y )gM (X,ϕV ), for X,Y ∈

Γ((kerF∗)⊥) and V ∈ Γ(kerF∗).


Proof. For X,Y ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗), from (3.5), we have

gM (∇XY, V ) = gM (AXBY,ϕV ) + gM (∇XCY, ϕV )− η(Y )gM (X,ϕV )

Then from (3.4), we have

gM (∇XY, V ) = gM (AXBY,ϕV )− gM (CY, ϕAXV )− η(Y )gM (X,ϕV ),

which shows (a) ⇐⇒ (b). On the other hand, from (2.10) and (2.12), we havegM (AXBY,ϕV ) = gN (−(∇F∗)(X,BY ), F∗ϕV ), which proves (b) ⇐⇒ (c).



(a) (kerF∗)⊥ defines a totally geodesic foliation on M .

(b) AXϕY = η(Y )X(c) (∇F∗)(Y, ϕX) = η(Y )F∗X, for X,Y ∈ Γ((kerF∗)

⊥).


(a) kerF∗ defines a totally geodesic foliation on M .(b) gN ((∇F∗)(V, ϕX), F∗ϕW ) = 0(c) TVBX + ACXV ∈ Γ(µ), for X ∈ Γ((kerF∗)

⊥) and V,W ∈ Γ(kerF∗).

Proof. For X ∈ Γ((kerF∗)⊥) and V,W ∈ Γ(kerF∗), gM (W, ξ) = 0 implies that from (2.3)

gM (∇VW, ξ) = −gM (W,∇V ξ) = −g(W,ϕV ) = 0. Thus we have

gM (∇VW,X) = gM (ϕ∇VW,ϕX) + η(∇VW )η(X)

= gM (ϕ∇VW,ϕX)

= gM (∇V ϕW,ϕX)− gM ((∇V ϕ)W,ϕX)

= −gM (ϕW,∇V ϕX)

Since F is a Riemannian submersion, we have

gM (∇VW,X) = −gN (F∗ϕW,F∗∇V ϕX) = gN (F∗ϕW, (∇F∗)(V, ϕX)),

which proves (a) ⇐⇒ (b).By direct calculation, we derive

gN (F∗ϕW, (∇F∗)(V, ϕX)) = −gM (ϕW,∇V ϕX)

= −gM (ϕW,∇VBX +∇V CX)

= −gM (ϕW,∇VBX + [V,CX] +∇CXV )

Since [V,CX] ∈ Γ(kerF∗), from (2.8) and (2.10), we obtain

gN (F∗ϕW, (∇F∗)(V, ϕX)) = −gM (ϕW,TVBX + ACXV ),

which proves (b) ⇐⇒ (c).

As an analogue of a Lagrangian Riemannian submersion in [11], we have a similarresult;



(a) kerF∗ defines a totally geodesic foliation on M .(b) (∇F∗)(V, ϕX) = 0(c) TV ϕW = 0, for X ∈ Γ((kerF∗)


238 J.W. Lee

Proof. From Theorem 3.6, it is enough to show (b) ⇐⇒ (c). Using (2.12) and (2.9), wehave

gN (F∗ϕW, (∇F∗)(V, ϕX)) = gM (∇V ϕW,ϕX)

= gM (TV ϕW,ϕX)

Since TV ϕW ∈ Γ(kerF∗), the proof is complete.

We note that a differentiable map F between two Riemannian manifolds is called to-tally geodesic if ∇F∗ = 0. For the special Riemannian submersion, we have the followingcharacterization.

3.10. Theorem. Let F be an anti-invariant ξ⊥-Riemannian submersion from a Sasakianmanifold (M, gM , ϕ, ξ, η) onto a Riemannian manifold (N, gN ) with (kerF∗)

⊥ = ϕ(kerF∗)⊕ <ξ >. Then F is a totally geodesic manp if and only if

(3.6) TV ϕW = 0 V,W ∈ Γ(kerF∗)

and

(3.7) AXϕW = 0 X ∈ Γ((kerF∗)⊥)

Proof. First of all, we recall that the second fundamental form of a Riemannian submer-sion satisfies

(3.8) (∇F∗)(X,Y ) = 0 ∀X,Y ∈ Γ((kerF∗)⊥)

For V,W ∈ Γ(kerF∗), we get

(3.9) (∇F∗)(V,W ) = F∗(ϕTV ϕW ).

On the other hand, from (2.1), (2.2) and (2.12), we get

(3.10) (∇F∗)(X,W ) = F∗(ϕAXϕW ), X ∈ Γ((kerF∗)⊥)

Therefore, F is totally geodesic if and only if

(3.11) ϕ(TV ϕW ) = 0, ∀V,W ∈ Γ(kerF∗)

and

(3.12) ϕ(AXϕW ) = 0, ∀X ∈ Γ((kerF∗)⊥)

From (2.2), (2.4) and (2.5), we have

TV ϕW = 0, ∀V,W ∈ Γ(kerF∗)

and

AXϕW = 0, ∀X ∈ Γ((kerF∗)⊥)

From (2.3), F is totally geodesic if and only if the equations (3.6) and (3.7) hold.

Finally, in this section, we give a necessary and sufficient condition for a special Riemann-ian submersion to be harmonic as an analogue of a Lagrangian Riemannian submersionin [11];


⊥ = ϕ(kerF∗)⊥ <ξ >. Then F is harmonic if and only if Trace(ϕTV ) = 0 for V ∈ Γ(kerF∗).


Proof. From [5], we know that F is harmonic if and only if F has minimal fibers. ThusF is harmonic if and only if

∑m1i=1 Teiei = 0. On the other hand, from (2.3), (2.9) and

(2.8), we have

(3.13) TV ϕW = ϕTVW

due to ξ ∈ Γ((kerF∗)⊥) for any V,W ∈ Γ(kerF∗). Using (3.13), we get

m1∑i=1

gM (Teiϕei, V ) =

m1∑i=1

gM (ϕTeiei, V ) = −m1∑i=1

gM (Teiei, ϕV )

for any V ∈ Γ(kerF∗). Thus skew-symmetric T implies thatm1∑i=1

gM (ϕei,TeiV ) =

m1∑i=1

gM (Teiei, ϕV ).

Using (2.6) and (2.2), we havem1∑i=1

gM (ei, ϕTV ei) = −m1∑i=1

gM (ϕei,TV ei) = −gM (

m1∑i=1

Teiei, ϕV ),

which shows our assertion.

4. Decomposition theorems

In this section, we obtain decompostion theorems by using the existence of anti-invariant ξ⊥-Riemannian submersions. First, we recall the following.

4.1. Theorem. [10]. Let g be a Riemannian metric tensor on the manifold B = M ×Nand assume that the canonical foliations DM and DN intersect perpendicularly every-where. Then g is the metric tensor of

(i) a twisted product M ×f N if and olny if DM is a totally geodesic foliation andDN is a totally umbilical foliation.

(ii) a warped product M ×f N if and olny if DM is a totally geodesic foliation andDN is a spheric foliation, i.e., it is umbilic and its mean curvature vector fieldis parallel.

(iii) a usual product of Riemannian manifold if and only if DM and DN are totallygeodesic foliations.

Our first decomposition theorem for an anti-invariant ξ⊥-Riemannian submersioncomes from Theorem 3.4 and 3.6 in terms of the second fundamental forms of suchsubmersions.

4.2. Theorem. Let F be an anti-invariant ξ⊥-Riemannian submersion from a Sasakianmanifold (M, gM , ϕ, ξ, η) onto a Riemannian manifold (N, gN ). Then M is a locallyproduct manifold if and only if

gN ((∇F∗)(Y, ϕX), F∗ϕV ) = gM (CY, ϕAXV )− η(Y )gM (X,ϕV )

andgN ((∇F∗)(V, ϕX), F∗ϕW ) = 0

for X,Y ∈ Γ((kerF∗)⊥) and V,W ∈ Γ(kerF∗)

From Corollary 3.5 and 3.7, we have the following decomposition theorem:


⊥ = ϕ(kerF∗)⊥ <ξ >. Then M is a locally product manifold if and only if AXϕY = η(Y )X and TV ϕW = 0,for X,Y ∈ Γ((kerF∗)


240 J.W. Lee

Next we obtain a decomposition theorem which is related to the notion of a twistedproduct manifold.


⊥ = ϕ(kerF∗)⊥ <ξ >. Then M is locally twisted product manifold of the form M(kerF∗)⊥ ×f MkerF∗ if andonly if

TV ϕX = −gM (X,TV V )||V ||−2ϕV

and

AXϕY = η(Y )X

for X,Y ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗), where M(kerF∗)⊥ and MkerF∗ are integral

manifolds of the distributions (kerF∗)⊥ and kerF∗.

Proof. For X ∈ Γ((kerF∗)⊥) and V ∈ Γ(kerF∗) , from (2.3) and (2.9), we obtain

gM (∇VW,X) = gM (TV ϕW,ϕX)

= −gM (ϕW,TV ϕX)

since TV is skew-symmetric. This implies that kerF∗ is totally umbilical if and only if

TV ϕW = −X(λ)ϕV,

where λ is a function on M . By the direct computation,

TV ϕX = −gM (X,TV V )||V ||−2ϕV.

Then the proof follows from Corollary 3.5.

However, in the sequel, we show that the notion of anti-invariant ξ⊥-Riemannian sub-mersion puts some restrictions on the source manifold.

4.5. Theorem. Let (M, gM , ϕ, ξ, η) be a Sasakian manifold and (N, gN ) be a Riemannianmanifold. Then there does not exist an anti-invariant ξ⊥-Riemannian submersion fromM to N with (kerF∗)

⊥ = ϕ(kerF∗)⊥ < ξ > such that M is a locally proper twistedproduct manifold of the form MkerF∗ ×f M(kerF∗)⊥ .

Proof. Suppose that F : (M, gM , ϕ, η, ξ) −→ (N, gN ) is an anti-invariant ξ⊥-Riemanniansubmersion with (kerF∗)

⊥ = ϕ(kerF∗)⊥ < ξ > and M is a locally twisted product of thform MkerF∗ ×f M(kerF∗)⊥ . Then MkerF∗ is a totally geodesic foliation and M(kerF∗)⊥

is a totally umbilical foliation. We denote the second fundamental form of M(kerF∗)⊥ byh. Then we have

(4.1) gM (∇XY, V ) = gM (h(X,Y ), V ) X,Y ∈ Γ((kerF∗)⊥), V ∈ Γ(kerF∗).

Since M(kerF∗)⊥ is a totally umbilical foliation, we have

gM (∇XY, V ) = gM (H,V )gM (X,Y ),

where H is the mean curvature vector field of M(kerF∗)⊥ .

On the other hand, from (3.5), we derive

gM (∇XY, V ) = −gM (ϕY,∇XϕV )− η(Y )gM (X,ϕV ).(4.2)

Using (2.11), we obtain

gM (∇XY, V ) = −gM (ϕY,AXϕV )− η(Y )gM (X,ϕV )(4.3)

= −gM (Y, ϕAXϕV + gM (X,ϕV )ξ)


Therefore, from (4.1), (4.3) and (2.2), we have

AXϕV = gM (H,V )ϕX + η(AXϕV )ξ

Since AXϕV is in Γ(kerF∗), η(AXϕV ) = gM (AXϕV, ξ) = 0. Thus, we have

AXϕV = gM (H,V )ϕX.

Hence, we derive

gM (AXϕV, ϕX) = gM (H,V )||X||2 − η2(X)Then using (2.11) we have

gM (∇XϕV, ϕX) = gM (H,V )||X||2 − η2(X)Thus (3.5) implies that

gM (∇XX,V ) + η(X)gM (X,ϕV ) = gM (H,V )||X||2 − η2(X)Then using (2.7), we have AXX = 0, which implies

η(X)gM (X,ϕV ) = gM (H,V )||X||2−η2(X),∀X ∈ Γ((kerF∗)⊥), V ∈ Γ(kerF∗).

Choosing X which is orthogonal to ξ, 0 = gM (H,V )|X||2. Since gM is the Riemannianmetric and H ∈ Γ(kerF∗), we conclude that H = 0, which shows (kerF∗)

⊥ is totallygeodesic, so M is usual product of Riemannian manifolds.

AcknowledgmentsThe author is grateful to the referee for his/ her valuable comments and suggestions.

References

[1] Altafini, C. Redundant robotic chains on Riemannian submersions, IEEE Transactions onRobotics and Automation, 20(2), 335-340, 2004.

[2] Blair, D. E. Contact manifold in Riemannian geometry,(Lecture Notes in Math., 509,

Springer-Verlag, Berlin-New York, 1976).[3] Baird, P., Wood, J. C. Harmonic Morphisms Between Riemannian Manifolds, (London

Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press, Ox-

ford, 2003).[4] Chinea, C. Almost contact metric submersions, Rend. Circ. Mat. Palermo, 43(1), 89–104,

1985.[5] Eells, J., Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86,

109–160, 1964.

[6] Falcitelli, M., Ianus, S., Pastore, A. M. Riemannian Submersions and Related Topics, (WorldScientific, River Edge, NJ, 2004).

[7] Gray, A. Pseudo-Riemannian almost product manifolds and submersion, J. Math. Mech., 16,

715–737, 1967.[8] Ianus, S., Pastore A. M. Harmonic maps on contact metric manifolds, Ann. Math. Blaise

Pascal, 2(2), 43–53, 1995.

[9] O’Neill, B. The fundamental equations of a submersion, Mich. Math. J.,13, 458–469, 1966.[10] Ponge, R. Reckziegel, H. Twisted products in pseudo-Riemannian geometry, Geom. Dedi-

cata, 48(1), 15–25, 1993.

[11] Sahin, B. Anti-invariant Riemannian submersions from almost hermitian manifolds, Cent.Eur. J. Math., 8(3), 437–447, 2010.

[12] Sasaki, S., Hatakeyama, Y. On differentiable manifolds with contact metric structure, J.Math. Soc. Japan, 14, 249–271, 1961.

[13] Watson, B. Almost Hermitian submersions, J. Differential Geometry, 11(1), 147–165, 1976.


SOME HERMITE-HADAMARD TYPE

INEQUALITIES FOR DIFFERENTIABLE

CONVEX FUNCTIONS AND APPLICATIONS

Bo-Yan Xi ∗ † § and Feng Qi ‡ §


Abstract

In the paper, the authors offer some new inequalities for differentiableconvex functions, which are connected with Hermite-Hadamard integralinequality, and apply these inequalities to special means of two positivenumbers.

Keywords: Integral inequality, Hermite-Hadamard integral inequality, Convex func-tion, Mean, Application

2000 AMS Classification: 26D15, 26A51, 26D20, 26E60, 41A55.

1. Introduction

In [2], the following Hermite-Hadamard type inequalities for differentiable convexfunctions were proved.

1.1. Theorem ([2, Theorem 2.2]). Let f : I ⊂ R→ R be a differentiable mapping anda, b ∈ I with a < b. If |f ′(x)| is convex on [a, b], then

(1.1)

∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ ≤ (b− a)(|f ′(a)|+ |f ′(b)|)8

.

∗College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, InnerMongolia Autonomous Region, 028043, China. E-mail: (B.-Y. Xi) [email protected]†Corresponding author‡Department of Mathematics, School of Science, Tianjin Polytechnic University, Tian-

jin City, 300387, China; School of Mathematics and Informatics, Henan Polytechnic Univer-sity, Jiaozuo City, Henan Province, 454010, China. E-mail: (F. Qi) [email protected],

[email protected], [email protected]. URL: http://qifeng618.wordpress.com§This work was partially supported by the Foundation of Research Program of Science and

Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159and by the NSF of China under Grant No. 10962004, China

244 B.-Y. Xi, F. Qi

1.2. Theorem ([2, Theorem 2.3]). Let f : I ⊂ R → R be a differentiable mapping,

a, b ∈ I with a < b, and let p > 1. If the new mapping |f ′(x)|p/(p−1) is convex on [a, b],then

(1.2)

∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

2(p+ 1)1/p

[|f ′(a)|p/(p−1) + |f ′(b)|p/(p−1)

2

](p−1)/p

.

In [6], the above inequalities were generalized as follows.

1.3. Theorem ([6, Theorems 1 and 2]). Let f : I ⊂ R → R be differentiable on I,a, b ∈ I with a < b, and let q ≥ 1. If |f ′(x)|q is convex on [a, b], then

(1.3)

∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ ≤ b− a4

[|f ′(a)|q + |f ′(b)|q

2

]1/qand

(1.4)

∣∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ ≤ b− a4

[|f ′(a)|q + |f ′(b)|q

2

]1/q.

In [4], the above inequalities were further generalized as follows.

1.4. Theorem ([4, Theorems 2.3 and 2.4]). Let f : I ⊂ R → R be differentiable on I,

a, b ∈ I with a < b, and let p > 1. If |f ′(x)|p/(p−1) is convex on [a, b], then

(1.5)

∣∣∣∣ 1

b− a

∫ b

a

f(x) dx− f(a+ b

2

)∣∣∣∣ ≤ b− a16

(4

p+ 1

)1/p

×[|f ′(a)|p/(p−1) + 3|f ′(b)|p/(p−1)](p−1)/p

+[3|f ′(a)|p/(p−1) + |f ′(b)|p/(p−1)](p−1)/p

and

(1.6)

∣∣∣∣ 1

b− a

∫ b

a

f(x) dx− f(a+ b

2

)∣∣∣∣ ≤ b− a4

(4

p+ 1

)1/p

(|f ′(a)|+ |f ′(b)|).

In [3], an inequality similar to the above ones was given as follows.

1.5. Theorem ([3, Theorem 3]). Let f : [a, b]→ R be an absolutely continuous mappingwhose derivative belongs to Lp[a, b]. Then

(1.7)

∣∣∣∣13[f(a) + f(b)

2+ 2f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ 1

6

[2q+1 + 1

3(q + 1)

]1/q(b − a)1/q‖f ′‖p,

where 1p

+ 1q

= 1 and p > 1.

For more information on this topic, please refer to [1, 8, 9, 10, 11, 12] and plenty ofreferences cited therein.

In this paper, motivated by Theorems 1.1 to 1.5, we will establish some new Hermite-Hadamard type inequalities for differentiable functions and apply them to derive someinequalities of special means.

Hermite-Hadamard type inequalities for convex functions . . . 245

2. A Lemma

For establishing our Hermite-Hadamard type inequalities, we need the following lemma.

2.1. Lemma. Let f : I ⊂ R → R be differentiable on I, a, b ∈ I with a < b. Iff ′ ∈ L[a, b], then

(2.1)1

2

[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

=b− a

4

∫ 1

0

(1

2− t)[f ′(ta+ (1− t)a+ b

2

)+ f ′

(ta+ b

2+ (1− t)b

)]dt.

Proof. Integrating by part and changing variables of definite integrals yield∫ 1

0

(1

2− t)f ′(ta+ (1− t)a+ b

2

)dt

= − 2

b− a

[(1

2− t)f

(ta+ (1− t)a+ b

2

)∣∣∣∣10

+

∫ 1

0

f

(ta+ (1− t)a+ b

2

)dt

]=

1

b− a

[f(a) + f

(a+ b

2

)]− 4

(b− a)2

∫ (a+b)/2

a

f(x) dx

and ∫ 1

0

(1

2− t)f ′(ta+ b

2+ (1− t)b

)dt

= − 2

b− a

[(1

2− t)f

(ta+ b

2+ (1− t)b

)∣∣∣∣10

+

∫ 1

0

f

(ta+ b

2+ (1− t)b

)dt

]=

1

b− a

[f

(a+ b

2

)+ f(b)

]− 4

(b− a)2

∫ b

(a+b)/2

f(x) dx.

Combining the above two equations leads to Lemma 2.1.

2.2. Remark. Lemma 2.1 is the key point of this paper, which will lead to the newHermite-Hadamard type inequalities for differentiable convex functions in next section.We notice that choosing a new and suitable function g(t) instead of 1

2− t in integrals∫ 1

0

(1

2− t)f ′(ta+(1− t)a+ b

2

)dt and

∫ 1

0

(1

2− t)f ′(ta+ b

2+(1− t)b

)dt

will generalize Lemma 2.1 and, by this generalization, some more general and new in-equalities of Hermite-Hadamard type can be derived. Due to the limitation of length ofthis paper, we will study in this direction in subsequent papers.

3. New inequalities of Hermite-Hadamard type

Now we are in a position to establish some new Hermite-Hadamard type inequalitiesfor differentiable convex functions.

3.1. Definition. A function f(x) is said to be convex on an interval I if

(3.1) f(λx1 + (1− λ)x2) ≤ λf(x1) + (1− λ)f(x2)

holds for all x1, x2 ∈ I and 0 < λ < 1.

246 B.-Y. Xi, F. Qi

3.2. Theorem. Let f : I ⊂ R → R be a differentiable function on I, a, b ∈ I witha < b, and f ′ ∈ L[a, b]. If |f ′(x)|q for q ≥ 1 is convex on [a, b], then

(3.2)

∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ ≤ b− a16

(1

q + 1

)1/q

×[

(2q + 5)|f ′(a)|q + (2q + 3)|f ′(b)|q

4(q + 2)

]1/q+

[|f ′(a)|q + (4q + 7)|f ′(b)|q

4(q + 2)

]1/q+

[(4q + 7)|f ′(a)|q + |f ′(b)|q

4(q + 2)

]1/q+

[(2q + 3)|f ′(a)|q + (2q + 5)|f ′(b)|q

4(q + 2)

]1/q.

Proof. When q > 1, since |f ′(x)|q is convex on [a, b], by Lemma 2.1 and Holder integralinequality, we have∣∣∣∣12

[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

∫ 1/2

0

(1

2− t)[∣∣∣∣f ′(ta+ (1− t)a+ b

2

)∣∣∣∣+

∣∣∣∣f ′(ta+ b

2+ (1− t)b

)∣∣∣∣] dt

+

∫ 1

1/2

(t− 1

2

)[∣∣∣∣f ′(ta+ (1− t)a+ b

2

)∣∣∣∣+

∣∣∣∣f ′(ta+ b

2+ (1− t)b

)∣∣∣∣] dt

≤ b− a

4

(1

2

)1−1/q[(∫ 1/2

0

(1

2− t)q(

(1 + t)|f ′(a)|q

2+

(1− t)|f ′(b)|q

2

)dt

)1/q

+

(∫ 1/2

0

(1

2− t)q(

t

2|f ′(a)|q +

2− t2|f ′(b)|q

)dt

)1/q]+

(1

2

)1−1/q[(∫ 1

1/2

(t− 1

2

)q(1 + t

2|f ′(a)|q +

1− t2|f ′(b)|q

)dt

)1/q

+

(∫ 1

1/2

(t− 1

2

)q(t

2|f ′(a)|q +

2− t2|f ′(b)|q

)dt

)1/q].

A direct calculation gives∫ 1/2

0

(1

2− t)q(

1 + t

2|f ′(a)|q +

1− t2|f ′(b)|q

)dt

=1

2q+3

[(3

q + 1− 1

q + 2

)|f ′(a)|q +

(1

q + 2+

1

q + 1

)|f ′(b)|q

]=

1

2q+3(q + 1)(q + 2)

[(2q + 5)|f ′(a)|q + (2q + 3)|f ′(b)|q

],∫ 1/2

0

(1

2− t)q(

t

2|f ′(a)|q +

2− t2|f ′(b)|q

)dt

=1

2q+3

[(1

q + 1− 1

q + 2

)|f ′(a)|q +

(1

q + 2+

3

q + 1

)|f ′(b)|q

]=

1

2q+3(q + 1)(q + 2)

[|f ′(a)|q + (4q + 7)|f ′(b)|q

],∫ 1

1/2

(t− 1

2

)q(1 + t

2|f ′(a)|q +

1− t2|f ′(b)|q

)dt

=1

2q+3

[(1

q + 2+

3

q + 2

)|f ′(a)|q +

(1

q + 1− 1

q + 2

)|f ′(b)|q

]=

1

2q+3(q + 1)(q + 2)

[(4q + 7)|f ′(a)|q + |f ′(b)|q

],

Hermite-Hadamard type inequalities for convex functions . . . 247∫ 1

1/2

(t− 1

2

)q(t

2|f ′(a)|q +

2− t2|f ′(b)|q

)dt

=1

2q+3

[(1

q + 2+

1

q + 1

)|f ′(a)|q + |f ′(b)|q

(3

q + 1− 1

q + 2

)]=

1

2q+3(q + 1)(q + 2)

[(2q + 3)|f ′(a)|q + (2q + 5)|f ′(b)|q

].

Substituting the above identities into the above inequality results in∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

(1

2

)1−1/q[1

2q+3(q + 1)(q + 2)

((2q + 5)|f ′(a)|q + (2q + 3)|f ′(b)|q

)]1/q+

[1

2q+3(q + 1)(q + 2)

(|f ′(a)|q + (4q + 7)|f ′(b)|q

)]1/q+

[1

2q+3(q + 1)(q + 2)

((4q + 7)|f ′(a)|q + |f ′(b)|q

)]1/q+

[1

2q+3(q + 1)(q + 2)

((2q + 3)|f ′(a)|q + (2q + 5)|f ′(b)|q

)]1/q=b− a

16

[1

4(q + 1)(q + 2)

]1/q[((2q + 5)|f ′(a)|q + (2q + 3)|f ′(b)|q

)1/q+(|f ′(a)|q + (4q + 7)|f ′(b)|q

)1/q+((4q + 7)|f ′(a)|q + |f ′(b)|q

)1/q+((2q + 3)|f ′(a)|q + (2q + 5)|f ′(b)|q

)1/q].

Thus, the inequality (3.2) is valid for q > 1.When q = 1, by Lemma 2.1, we have∣∣∣∣12

[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

∫ 1/2

0

(1

2− t)[∣∣∣∣f ′(ta+ (1− t)a+ b

2

)∣∣∣∣+

∣∣∣∣f ′(ta+ b

2+ (1− t)b

)∣∣∣∣] dt

+

∫ 1

1/2

(t− 1

2

)[∣∣∣∣f ′(ta+ (1− t)a+ b

2

)∣∣∣∣+

∣∣∣∣f ′(ta+ b

2+ (1− t)b

)∣∣∣∣] dt

≤ b− a

4

[∫ 1/2

0

(1

2− t)(

1 + t

2|f ′(a)|+ 1− t

2|f ′(b)|

)dt

+

∫ 1/2

0

(1

2− t)(

t

2|f ′(a)|+ 2− t

2|f ′(b)|

)dt

+

∫ 1

1/2

(t− 1

2

)(1 + t

2|f ′(a)|+ 1− t

2|f ′(b)|

)dt

+

∫ 1

1/2

(t− 1

2

)(t

2|f ′(a)|+ 2− t

2|f ′(b)|

)dt

]=b− a

16(|f ′(a)|+ |f ′(b)|).

Thus, the inequality (3.2) holds for q = 1. Theorem 3.2 is proved.

From Theorem 3.2 we can derive the following two corollaries.

248 B.-Y. Xi, F. Qi

3.3. Corollary. Let f : I ⊂ R → R be differentiable on I, a, b ∈ I with a < b, andf ′ ∈ L[a, b]. If |f ′(x)|q is convex on [a, b] for q ≥ 1 and

(3.3)f(a) + f(b)

2= f

(a+ b

2

),

then

(3.4)

∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ =

∣∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

16

(1

q + 1

)1/q[(2q + 5)|f ′(a)|q + (2q + 3)|f ′(b)|q

4(q + 2)

]1/q+

[|f ′(a)|q + (4q + 7)|f ′(b)|q

4(q + 2)

]1/q+

[(4q + 7)|f ′(a)|q + |f ′(b)|q

4(q + 2)

]1/q+

[(2q + 3)|f ′(a)|q + (2q + 5)|f ′(b)|q

4(q + 2)

]1/q.

3.4. Corollary. Let f : I ⊂ R → R be differentiable on I, a, b ∈ I with a < b, and letf ′ ∈ L[a, b]. If |f ′(x)| is convex on [a, b], then

(3.5)

∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ ≤ b− a16

(|f ′(a)|+ |f ′(b)|).

Furthermore, if the equality (3.3) is valid, then

(3.6)

∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ =

∣∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

16(|f ′(a)|+ |f ′(b)|).

3.5. Remark. Somebody may ask that whether the condition (3.3) is significant. Inother words, can one find an example satisfying conditions of Corollary 3.3? This questioncan be affirmatively answered by the functions

f(x) = ±1

3x(x2 − 9x+ 27

)on the interval [1, 5]. Therefore, Corollary 3.3 and the inequality (3.6) in Corollary 3.4are significant.

3.6. Definition. A function f : I ⊆ R → (0,∞) is said to be logarithmically convex ifthe function ln f is convex on I, that is,

(3.7) f(λx1 + (1− λ)x2) ≤ [f(x1)]λ[f(x2)]1−λ

for any two points x1 and x2 on I and any λ meeting 0 < λ < 1.

3.7. Theorem. Let f : I ⊂ R → R be differentiable on I, a, b ∈ I with a < b, andf ′ ∈ L[a, b]. If |f ′(x)|q is logarithmically convex on [a, b] for q ≥ 1, then

(3.8)

∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

(1

8

)1−1/q|f ′(a)f ′(b)|1/2[g1(µ)]1/q + |f ′(b)|[g1(µ)]1/q

+ |f ′(a)|[g2(µ)]1/q + |f ′(a)f ′(b)|1/2[g2(µ)]1/q,


where

(3.9)

µ =

∣∣∣∣f ′(a)

f ′(b)

∣∣∣∣q/2,g1(u) =

u1/2 − 1

(lnu)2− 1

2 lnu, u 6= 1,

1

8, u = 1,

and

(3.10) g2(u) =

u−1/2 − 1

(lnu)2+

1

2 lnu, u 6= 1,

1

8, u = 1.

Proof. When q > 1, from the logarithmic convexity of |f ′(x)|q on [a, b] and (3.7),Lemma 2.1, and Holder integral inequality, it follows that

∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

∫ 1/2

0

(1

2− t)[∣∣∣∣f ′(ta+ (1− t)a+ b

2

)∣∣∣∣+

∣∣∣∣f ′(ta+ b

2+ (1− t)b

)∣∣∣∣] dt

+

∫ 1

1/2

(t− 1

2

)[∣∣∣∣f ′(ta+ (1− t)a+ b

2

)∣∣∣∣+

∣∣∣∣f ′(ta+ b

2+ (1− t)b

)∣∣∣∣] dt

≤ b− a

4

(1

8

)1−1/q([∫ 1/2

0

(1

2− t)|f ′(a)|q(1+t)/2|f ′(b)|q(1−t)/2 dt

]1/q+

[∫ 1/2

0

(1

2− t)|f ′(a)|qt/2|f ′(b)|q(2−t)/2 dt

]1/q)+

(1

8

)1−1/q([∫ 1

1/2

(t− 1

2

)|f ′(a)|q(1+t)/2|f ′(b)|q(1−t)/2 dt

]1/q+

[∫ 1

1/2

(t− 1

2

)|f ′(a)|qt/2|f ′(b)|q(2−t)/2 dt

]1/q).

If µ = 1, then

∫ 1/2

0

(1

2− t)µt dt =

∫ 1

1/2

(t− 1

2

)µt dt =

1

8.

If µ 6= 1, we have

∫ 1/2

0

(1

2− t)µt dt =

1

lnµ

(µ1/2 − 1

lnµ− 1

2

)= g1(µ),∫ 1

1/2

(t− 1

2

)µt dt = µ

(1

2 lnµ+µ−1/2 − 1

(lnµ)2

)= µg2(µ).

250 B.-Y. Xi, F. Qi

Hence, ∫ 1/2

0

(1

2− t)|f ′(a)|q(1+t)/2|f ′(b)|q(1−t)/2 dt =|f ′(a)f ′(b)|q/2

∫ 1/2

0

(1

2− t)µt dt

=|f ′(a)f ′(b)|q/2g1(µ),∫ 1/2

0

(1

2− t)|f ′(a)|qt/2|f ′(b)|q(2−t)/2 dt =|f ′(b)|q

∫ 1/2

0

(1

2− t)µt dt

=|f ′(b)|qg1(µ),∫ 1

1/2

(t− 1

2

)|f ′(a)|q(1+t)/2|f ′(b)|q(1−t)/2 dt =|f ′(a)f ′(b)|q/2

∫ 1

1/2

(t− 1

2

)µt dt

=|f ′(a)|qg2(µ),∫ 1

1/2

(t− 1

2

)|f ′(a)|qt/2|f ′(b)|q(2−t)/2 dt =|f ′(b)|q

∫ 1

1/2

(t− 1

2

)µt dt

=|f ′(a)f ′(b)|q/2g2(µ).

Substituting these equalities into the first inequality above and rearranging yield theinequality (3.8).

If q = 1, by the same argument as above, we have∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

[∫ 1/2

0

(1

2− t)|f ′(a)|(1+t)/2|f ′(b)|(1−t)/2 dt

+

∫ 1

1/2

(t− 1

2

)|f ′(a)|t/2|f ′(b)|(2−t)/2 dt

+

∫ 1/2

0

(1

2− t)|f ′(a)|t/2|f ′(b)|(2−t)/2 dt

+

∫ 1

1/2

(t− 1

2

)|f ′(a)|(1+t)/2|f ′(b)|(1−t)/2 dt

]=b− a

4

|f ′(a)f ′(b)|1/2[g1(µ) + g2(µ)] + |f ′(a)|g2(µ) + |f ′(b)|g1(µ)

.

Thus, Theorem 3.7 is proved.

From Theorem 3.7 we can deduce the following corollaries.

3.8. Corollary. Let f : I ⊂ R → R be differentiable on I, a, b ∈ I with a < b, andf ′ ∈ L[a, b]. If |f ′(x)| is logarithmically convex on [a, b], then

(3.11)

∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

[|f ′(a)f ′(b)|1/2[g1(µ) + g2(µ)] + |f ′(b)|g1(µ) + |f ′(a)|g2(µ)

].

In particular, if the identity (3.3) is also valid, then

(3.12)

∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣ =

∣∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣≤ b− a

4

[|f ′(a)f ′(b)|1/2[g1(µ) + g2(µ)] + |f ′(b)|g1(µ) + |f ′(a)|g2(µ)

].


3.9. Remark. Similar to Remark 3.5, we can give an example

f(x) = ln(secx+ tanx)

on the symmetric interval [−a, a], where 0 < a < π2

, meeting conditions of the inequal-ity (3.12). An alternative example is

f(x) = ±(1 + x2k

)x

for k ∈ N, which satisfy conditions of the inequality (3.12) on the interval [−a, a] for0 < a < 1. These show us that the inequality (3.12) in Corollary 3.8 is significant.

3.10. Theorem. Under conditions of Theorem 3.7, the inequality (3.8) is better than (3.2).Consequently, under conditions of Corollary 3.8, the inequality (3.11) is sharper than (3.5);if the equality (3.3) is also valid, then the inequality (3.12) is stronger than (3.6).

Proof. This follows from considering the following two facts in the proofs of Theorems 3.2and 3.7.

(1) Any logarithmically convex function must be convex. See [7, Remarks 1.2and 1.9] and related references therein.

(2) The inequality between the arithmetic and geometric weighted means (see [5,p. 49, Remark 1]) implies

|f ′(a)|q(1+t)/2|f ′(b)|q(1−t)/2 ≤ 1 + t

2|f ′(a)|q +

1− t2|f ′(b)|q,

and

|f ′(a)|qt/2|f ′(b)|q(2−t)/2 ≤ t

2|f ′(a)|q +

2− t2|f ′(b)|q.

Theorem 3.10 is thus proved.

3.11. Remark. The term∣∣∣∣12[f(a) + f(b)

2+ f

(a+ b

2

)]− 1

b− a

∫ b

a

f(x) dx

∣∣∣∣in (3.2) and (3.8) is different from the left hand sides of inequalities (1.1), (1.2), (1.3),(1.4), (1.5), (1.6), and (1.7), so Theorems 3.2 and 3.7 can not be compared with Theo-rems 1.1, 1.2, 1.3, 1.4, and 1.5 mentioned in the first section.

4. Applications to means

For two positive numbers a > 0 and b > 0, define

A(a, b) =a+ b

2, G(a, b) =

√ab , H(a, b) =

2ab

a+ b,

252 B.-Y. Xi, F. Qi

and

I(a, b) =

1

e

(bb

aa

)1/(b−a)

, a 6= b,

a, a = b,

Ls(a, b) =

[bs+1 − as+1

(s+ 1)(b− a)

]1/s, s 6= 0,−1 and a 6= b,

b− aln b− ln a

, s = −1 and a 6= b,

I(a, b), s = 0 and a 6= b,

a, a = b,

Hω,s(a, b) =

[as + ω(ab)s/2 + bs

ω + 2

]1/s, s 6= 0,

√ab , s = 0

for 0 ≤ ω < ∞. It is well known that A, G, H, L = L−1, I = L0, Ls, and Hω,sare respectively called the arithmetic, geometric, harmonic, logarithmic, exponential,generalized logarithmic, and generalized Heronian means of two positive number a andb.

In what follows we will apply theorems and corollaries in the above section to establishinequalities for some special mean values.

4.1. Theorem. Let b > a > 0, q ≥ 1, and s ∈ R.

(1) If s > 1 and (s− 1)q ≥ 1, then

(4.1)

∣∣∣∣A(as, bs) +As(a, b)

2− [Ls(a, b)]

s

∣∣∣∣ ≤ (b− a)s

16

(1

q + 1

)1/q

×[

(2q + 5)a(s−1)q + (2q + 3)b(s−1)q

4(q + 2)

]1/q+

[a(s−1)q + (4q + 7)b(s−1)q

4(q + 2)

]1/q+

[(4q + 7)a(s−1)q + b(s−1)q

4(q + 2)

]1/q+

[(2q + 3)a(s−1)q + (2q + 5)b(s−1)q

4(q + 2)

]1/q.

(2) If s < 1 and s 6= 0,−1, then∣∣∣∣A(as, bs) +As(a, b)

2− [Ls(a, b)]

s

∣∣∣∣ ≤ (b− a)1−1/q|s|16

[8

(1− s)q

]1/q×A

(a(s−1)/2, b(s−1)/2)[L(a, b)]1/q

×b(s−1)/2

[a(s−1)q/4(L(1−s)q/4−1(a, b)

)(1−s)q/4−1L(a, b)− 1

]1/q+ a(s−1)/2

[1− b(s−1)q/4(L(1−s)q/4−1(a, b)

)(1−s)q/4−1L(a, b)

]1/q≤ b− a

16|s|(

1

q + 1

)1/q[(2q + 5)a(s−1)q + (2q + 3)b(s−1)q

4(q + 2)

]1/q+

[a(s−1)q + (4q + 7)b(s−1)q

4(q + 2)

]1/q+

[(4q + 7)a(s−1)q + b(s−1)q

4(q + 2)

]1/q+

[(2q + 3)a(s−1)q + (2q + 5)b(s−1)q

4(q + 2)

]1/q.


(3) If s = −1, then∣∣∣∣12[

1

H(a, b)+

1

A(a, b)

]− 1

L(a, b)

∣∣∣∣ ≤ (b− a)1−1/q

16

(4

q

)1/q

× [L(a, b)]1/q

H(a, b)

1

b

[(Lq/2−1(a, b))q/2−1L(a, b)

aq/2− 1

]1/q+

1

a

[1−

(Lq/2−1(a, b))q/2−1L(a, b)

bq/2

]1/q≤ b− a

16

(1

q + 1

)1/q[(2q + 5)/a2q + (2q + 3)/b2q

4(q + 2)

]1/q+

[1/a2q + (4q + 7)/b2q

4(q + 2)

]1/q+

[(4q + 7)/a2q + 1/b2q

4(q + 2)

]1/q+

[(2q + 3)/a2q + (2q + 5)/b2q

4(q + 2)

]1/q.

Proof. Let f(x) = xs for x > 0 and s 6= 0, 1. Then f ′(x) = sxs−1 and |f ′(x)|q =

|s|x(s−1)q. Further, it follows that(|f ′(x)|q

)′′= |s|q(s− 1)q[(s− 1)q − 1]x(s−1)q−2

and (ln |f ′(x)|q

)′′=

(1− s)qx2

.

If s > 1 and (s − 1)q ≥ 1, the function |f ′(x)|q = |s|qx(s−1)q is convex on [a, b]. ByTheorem 3.2, the inequality (4.1) follows.

If s < 1 and s 6= 0, the function |f ′(x)|q = |s|qx(s−1)q is logarithmically convex on[a, b]. Meanwhile, the formulas (3.9) and (3.10) become

µ =

(b

a

)(1−s)q/2

,

g1(µ) =L(a, b)

q(1− s)(b− a)

a(s−1)q/4[L(1−s)q/4−1(a, b)](1−s)q/4−1L(a, b)− 1

,

g2(µ) =L(a, b)

q(1− s)(b− a)

1− b(s−1)q/4[L(1−s)q/4−1(a, b)](1−s)q/4−1L(a, b)

.

Substituting these scalars into Theorem 3.10 yields the required results.

4.2. Corollary. Let b > a > 0 and s ∈ R.

(1) If s ≥ 2, we have

(4.2)

∣∣∣∣A(as, bs) + [A(a, b)]s

2− [Ls(a, b)]

s

∣∣∣∣ ≤ b− a8|s|A

(as−1, bs−1).

(2) If s < 1 and s 6= 0,−1, we have

(4.3)

∣∣∣∣A(as, bs) + [A(a, b)]s

2− [Ls(a, b)]

s

∣∣∣∣ ≤ b− a8|s|A

(a(s−1)/2, b(s−1)/2)

× [G(a, b)]s−1L(a, b)

2

[L−(1+s)/2(a, b)](1+s)/2− L(a, b)

[L−(3+s)/2(a, b)](3+s)/2

≤ b− a

8|s|A

(as−1, bs−1).

254 B.-Y. Xi, F. Qi

(3) If s = −1, we have∣∣∣∣12[

1

H(a, b)+

1

A(a, b)

]− 1

L(a, b)

∣∣∣∣ ≤ b− a8

L(a, b)

H(a, b)[G(a, b)]2≤ b− a

8

1

H(a2, b2

) .4.3. Theorem. For b > a > 0 and q ≥ 1, we have∣∣∣∣ lnG(a, b) + lnA(a, b)

2− ln I(a, b)

∣∣∣∣ ≤ (b− a)1−1/q

16

(8

q

)1/q[L(a, b)]1/q

H(a1/2, b1/2)

×

1

b1/2

[(Lq/4−1(a, b))q/4−1L(a, b)

aq/4− 1

]1/q+

1

a1/2

[1−

(Lq/4−1(a, b))q/4−1L(a, b)

bq/4+

]1/q≤ b− a

16

1

(q + 1)1/q

[(2q + 5)/aq + (2q + 3)/bq

4(q + 2)

]1/q+

[1/aq + (4q + 7)/bq

4(q + 2)

]1/q+

[(4q + 7)/aq + 1/bq

4(q + 2)

]1/q+

[(2q + 3)/aq + (2q + 5)/bq

4(q + 2)

]1/q.

Proof. Let f(x) = lnx for x > 0. It is clear that f ′(x) = 1x

and |f ′(x)|q = 1xq

. Further,

we have(ln |f ′(x)|q

)′′= q

x2. This shows that the function |f ′(x)|q = 1

xqis logarithmically

convex. On the other hand, we have

µ =

(b

a

)q/2,

g1(µ) =L(a, b)

q(b− a)

[Lq/4−1(a, b)]q/4−1L(a, b)

aq/4− 1

,

g2(µ) =L(a, b)

q(b− a)

1−

[Lq/4−1(a, b)]q/4−1L(a, b)

bq/4

.

Substituting these equations into Theorem 3.10 leads to our required results.

4.4. Corollary. For b > a > 0 and q ≥ 1, we have

(4.4)

∣∣∣∣ lnG(a, b) + lnA(a, b)

2− ln I(a, b)

∣∣∣∣≤ b− a

8

L(a, b)

H(a1/2, b1/2)G(a, b)

2

[L−1/2(a, b)]1/2− L(a, b)

[L−3/4(a, b)]3/2

≤ b− a

8

1

H(a, b).

4.5. Theorem. Let b > a > 0 and ω ≥ 0.

(1) If s ≥ 4 or s < 1 with s 6= −1,−2, then

(4.5)

∣∣∣∣12

[Hω,s(a, b)]s

H(as, bs)+

[Hω,s

(ab, a2 + b2

)]s[G(a, b)]2s

− 1

ω + 2

[L2s+1(a, b)]2s+1

A(a, b)[G(a, b)]2s+ ω

[Ls+1(a, b)]s+1

A(a, b)[G(a, b)]s+ 1

∣∣∣∣≤ b− a

8

|s|ω + 2

A(a, b)

2A(a2(s−1), b2(s−1)

)[G(a, b)]2s

+ ωA(as−2, bs−2

)[G(a, b)]s

.


(2) If s = −1, then

(4.6)

∣∣∣∣12Hω,1(a, b)

H(a, b)+Hω,1

(ab, a2 + b2

)2[A(a, b)]2

− 1

ω + 2

[1

H(a, b)L(a, b)+ ω

G(a, b)

A(a, b)+ 1

]∣∣∣∣≤ b− a

8

1

ω + 2A(a, b)

2A(a4, b4

)[G(a, b)]6

+ ωA(a3, b3

)[G(a, b)]5

.

(3) If s = −2, we have

(4.7)

∣∣∣∣12

[Hω,2(a, b)]2

H(a2, b2

) +1

4

[Hω,2

(ab, a2 + b2

)A(a2, b2

) ]2− 1

ω + 2

[ω

H(a, b)L(a, b)+ 2

]∣∣∣∣≤ b− a

4

1

ω + 2A(a, b)

2A(a6, b6

)[G(a, b)]8

+ ωA(a4, b4

)[G(a, b)]4

.

Proof. Let

f(x) =xs + ωxs/2 + 1

ω + 2

for x > 0 and s 6= 0. Then

f ′(x) =s

ω + 2

(xs−1 +

ω

2xs/2−1

),

(|f ′(x)|)′′ =|s|

ω + 2xs/2−3

[(s− 1)(s− 2)xs/2 +

ω

8(s− 2)(s− 4)

].

This means that when s ≥ 4 or s < 1 with s 6= 0 the function |f ′(x)| is convex on [a, b].It is easy to see that

1

2

[f(b/a) + f(a/b)

2+f

(b/a+ a/b

2

)]=

1

2

[Hω,s(a, b)]

s

H(as, bs

) +

[Hω,s

(ab, a2 + b2

)]s[G(a, b)]2

.

When s ≥ 4 or s < 1 with s 6= 0,−1,−2, we have

1

b/a+ a/b

∫ b/a

a/b

f(x) dx =1

b/a+ a/b

∫ b/a

a/b

xs + ωxs/2 + 1

ω + 2dx

= − 1

ω + 2

[L2s+1(a, b)]2s+1

A(a, b)[G(a, b)]2s+ ω

[Ls+1(a, b)]s+1

A(a, b)[G(a, b)]s+ 1

and

b/a+ a/b

16

∣∣∣∣f ′(ab)∣∣∣∣+

∣∣∣∣f ′( ba)∣∣∣∣ =

b− a8

|s|ω + 2

A(a, b)

×

2A(a2(s−1), b2(s−1)

)[G(a, b)]2s

+ ωA(as−2, bs−2

)[G(a, b)]s

.

Substituting these equations into Corollary 3.4 leads to the inequality (4.5).

256 B.-Y. Xi, F. Qi

When s = −1, we have

1

b/a+ a/b

∫ b/a

a/b

f(x) dx =1

b/a+ a/b

∫ b/a

a/b

1/x+ ω/x1/2 + 1

ω + 2dx

=1

ω + 2

[1

H(a, b)L(a, b)+ ω

G(a, b)

A(a, b)+ 1

],

1

2

[f(b/a) + f(a/b)

2+f

(b/a+ a/b

2

)]=

1

2

Hω,1(a, b)

H(a, b) +

1

2

Hω,1(ab, a2 + b2

)[A(a, b)]2

,

and

b/a+ a/b

16

∣∣∣∣f ′(ab)∣∣∣∣+

∣∣∣∣f ′( ba)∣∣∣∣ =

b− a8

A(a, b)

ω + 2

2A(a4, b4

)[G(a, b)]6

+ ωA(a3, b3

)[G(a, b)]5

.

Substituting these equations into Corollary 3.4 leads to the inequality (4.6).Finally, when s = −2, it is not difficult to obtain that

1

b/a+ a/b

∫ b/a

a/b

f(x) dx =1

b/a+ a/b

∫ b/a

a/b

1/x2 + ω/x+ 1

ω + 2dx

=1

ω + 2

[ω

H(a, b)L(a, b)+ 2

],

b/a+ a/b

16

∣∣∣∣f ′(ab)∣∣∣∣+

∣∣∣∣f ′( ba)∣∣∣∣ =

b− a4

A(a, b)

ω + 2

2A(a6, b6

)[G(a, b)]8

+ ωA(a4, b4

)[G(a, b)]4

,

and

1

2

[f(b/a) + f(a/b)

2+ f

(b/a+ a/b

2

)]=

1

2

[Hω,2(a, b)]2

H(a2, b2

) +1

4

[Hω,2

(ab, a2 + b2

)A(a2, b2

) ]2.

Substituting these equations into Corollary 3.4 leads to the inequality (4.7). The proofof Theorem 4.5 is complete.

Acknowledgements

The authors appreciate the anonymous referee for his/her helpful suggestions andvaluable comments on this manuscript.

References

[1] Bai, R.-F., Qi, F. and Xi, B.-Y. Hermite-Hadamard type inequalities for the m- and (α,m)-

logarithmically convex functions, Filomat 27 (1), 1–7, 2013.[2] Dragomir, S. S. and Agarwal, R. P. Two inequalities for differentiable mappings and appli-

cations to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11(5), 91–95, 1998.

[3] Dragomir, S. S., Agarwal, R. P. and Cerone, P. On Simpson’s inequality and applications,J. Inequal. Appl. 5 (6), 533–579, 2000.

[4] Kirmaci, U. S. Inequalities for differentiable mappings and applications to special means of

real numbers to midpoint formula, Appl. Math. Comput. 147 (1), 137–146, 2004.

[5] Mitrinovic, D. S. Analytic Inequalities, Springer-Verlag, Berlin, 1970.[6] Pearce, C. E. M. and Pecaric, J. Inequalities for differentiable mappings with application to

special means and quadrature formulae, Appl. Math. Lett. 13 (2), 51–55, 2000.[7] Qi, F. Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010, Article ID

493058, 84 pages, 2010.


[8] Qi, F., Wei, Z.-L. and Yang, Q. Generalizations and refinements of Hermite-Hadamard’s

inequality, Rocky Mountain J. Math. 35 (1), 235–251, 2005.

[9] Wang, S.-H., Xi, B.-Y. and Qi, F. On Hermite-Hadamard type inequalities for (α,m)-convexfunctions, Int. J. Open Probl. Comput. Sci. Math. 5 (4), 47–56, 2012.

[10] Wang, S.-H., Xi, B.-Y. and Qi, F. Some new inequalities of Hermite-Hadamard type for

n-time differentiable functions which are m-convex, Analysis (Munich) 32 (3), 247–262,2012.

[11] Xi, B.-Y., Bai, R.-F. and Qi, F. Hermite-Hadamard type inequalities for the m- and (α,m)-

geometrically convex functions, Aequationes Math. 84 (3), 261–269, 2012.[12] Xi, B.-Y. and Qi, F. Some integral inequalities of Hermite-Hadamard type for convex func-

tions with applications to means, J. Funct. Spaces Appl. 2012, Article ID 980438, 14 pages,

2012.


APPROXIMATION BY FEJER SUMS

OF FOURIER TRIGONOMETRIC SERIES

IN WEIGHTED ORLICZ SPACES

Sadulla Z. Jafarov ∗


Abstract

In this work we investigate the approximation problems of the functionsby Fejer sums of Fourier series in the reflexive weighted Orlicz spaceswith Muckenhoupt weights and of the functions by Fejer sums of Faberseries in weighted Smirnov-Orlicz classes defined on simply connecteddomains with a Dini-smooth boundary of the complex plane.

Keywords: Orlicz space, weighted Orlicz space, Boyd indices, Muckenhoupt weight,Fejer sums, weighted Smirnov-Orlicz class, Dini-smooth curve, Faber series.

2000 AMS Classification: 41A10, 42A10, 41A25, 46A25

1. Introduction, main results and some auxiliary results

A convex and continuous function M : [0,∞)→ [0,∞) for which M(0) = 0, M(x) > 0for x > 0 and

limx→0

M(x)

x= 0, lim

x→∞

M(x)

x=∞

is called a Young function.Let T := [−π, π], and let M be a Young function. We denote by LM (T ) the linear

space Lebesgue measurable functions f : T → R satisfying the condition∫T

M (α|f(t)|) dt <∞

∗Department of Mathematics, Faculty of Art and Sciences, Pamukkale University, 20017,Denizli, Turkey and Institute of Mathematics and Mechanics of NAS of Azerbaijan 9,

B.Vahabzade St., Baku, AZ1141 AzerbaijanE-mail: [email protected]

260 S. Z. Jafarov

for some α > 0. Equipped with the norm

‖f‖LM (T ) := inf

α > 0 :

∫T

M

(|f(t)|α

)dt < 1

,

the space LM (T ) become a Banach space [43, pp.52-68].The norm ‖·‖LM (T ) is called Orlicz norm and the Banach space LM (T ) is called

Orlicz space. It is known [43, p.50] that every function in LM (T ) is integrable on T , i.e.LM (T ) ⊂ L1(T ).

Let M−1 : [0,∞) → [0,∞) be the inverse of the Young function M . The lower andupper indices αM , βM

αM := limx→0

log h(x)

log x, βM := lim

x→∞

log h(x)

log x

of the function

h : [0,∞)→ [0,∞), h(x) := limt→∞

supM−1(t)

M−1(tx

) , x > 0

first considered by Matuszewska and Orlicz [38] are called the Boyd indices of the Orliczspace LM (T ). It is known that 0 ≤ αM ≤ 1. The Boyd indices αM , βM said to benontrivial if 0 < αM and βM < 1. The Orlicz space LM (T ) is reflexive if and only if0 < αM ≤ βM < 1, i.e. if the Boyd indices are nontrivial. The detailed informationabout Orlicz spaces and the Boyd indices can be found in [29] and [6], respectively.

A function ω is called a weight on T if ω : T → [0,∞] is measurable and ω−1 (0,∞)has measure zero (with respect to Lebesgue measure). With any given weight ω weassociate the ω-weighted Orlicz space LM (T, ω) consisting of all measurable functions fon T such that

‖f‖LM (T,ω) := ‖fω‖LM (T ) .

Let 1 < p <∞, 1/p+ 1/p′ = 1 and let ω be a weight function on T . ω is said to satisfyMuckenhoupt’s Ap-condition on T if

supJ

1

|J |

∫J

ωp (t) dt

1/p 1

|J |

∫J

ω−p′ (t) dt

1/p′

<∞,

where J is any subinterval of T and |J | denotes its length.Let us denote by Ap (T ) the set of all weight functions satisfying Muckenhoupt’s Ap-

condition on T .Note that by [34, Lemma 3.3], [ 35, Theorem 4.5 ] and [ 33, Section 2.3] if LM (T )

is reflexive and ω weight function satisfying the condition ω ∈ A1/αM (T ) ∩ A1/βM (T ) ,then the space LM (T, ω) is also reflexive.

Let LM (T, ω) be a weighted Orlicz space, let αM and βM be nontrivial, and letω ∈ A 1

αM

(T ) ∩A 1βM

(T ). For f ∈ LM (T, ω) we set

(νhf) (x) :=1

2h

h∫−h

f (x+ t) dt, 0 < h < π, x ∈ T.

By reference [20, Lemma 1] the shift operator νh is a bounded linear operator onLM (T, ω):

‖νh (f)‖LM (T,ω) ≤ C ‖f‖LM (T,ω) .

Approximation by Fejer sums of Fourier trigonometric series . . . 261

The function

ΩkM,ω (δ, f) := sup0<hi≤δ1≤i≤k

∥∥∥∥∥k∏i=1

(I − νhi) f

∥∥∥∥∥LM (T,ω)

, δ > 0, k = 1, 2, ...

is called k-th modulus of smoothness of f ∈ LM (T, ω), where I is the identity operator.It can easily be shown that ΩkM,ω (·, f) is a continuous, nonnegative and nondecreasing

function satisfying the conditions

limδ→0

ΩkM,ω (δ, f) = 0, ΩkM,ω (δ, f + g) ≤ ΩkM,ω (δ, f) + ΩkM,ω (δ, g)

for f, g ∈ LM (T, ω).Let

(1.1)a0

2+

∞∑k=1

(ak (f) cos kx+ bk (f) sin kx)

be the Fourier series of the function f ∈ L1(T ), where αk(f) are βk(f) the Fouriercoefficients of the function f .

The n-th partial sums and Fejer sums of series (1.1) are defined, respectively as

Sn (x, f) =a0

2+

n∑k=1

(ak (f) cos kx+ bk (f) sin kx) ,

σn (x, f) =1

n+ 1

n∑k=0

Sk (x, f) .

Note that Fejer sums were introduced by Fejer [9].The best approximation to f ∈ LM (T, ω) in the class

∏n of trigonometric polynomials

of degree not exceeding n is defined by

En (f)M,ω := inf‖f − Tn‖LM (T,ω) : Tn ∈

∏n

.

Note that the existence of T ∗n ∈ Πn such that

En (f)M,ω = ‖f − T ∗n‖LM (T,ω)

follows, for example, from Theorem 1.1 in [17, p.59].We put

ρn,M (f) = ‖f − σn−1 (., f)‖LM (T,ω) .

We use c, c1, c2, ... to denote constants (which may, in general, differ in different relations)depending only on numbers that are not important for the questions of interest.

The problems of approximation theory in weighted, non-weighted Lebesgue spacesand weighted, non-weighted Orlicz spaces have been investigated by several authors (see,for example, [1-5, 8, 11-14,18-28, 30, 31, 36, 37, 39, 40, 42, 44, 45 ]). Note that theapproximation problems by trigonometric polynomials in weighted Lebesgue spaces withweights belonging to the Muckenhoupt class Ap(T ) were investigated in [11], [36] and[37].

Detailed information on weighted polynomial approximation can be found in the books[15] and [40].

In this work we obtain the general estimate for the deviation ρn,M (f) of the functionf from its Fejer sums σn(f) in weighted Orlicz spaces LM (T, ω). Note that the estimateobtained in this study depends on sequence of the best approximation En(f)M,ω. Thisresult was applied to estimate of approximation of Fejer sums of Faber series in weightedSmirnov-Orlicz classes defined on simply connected domains of the complex plane interms of the modulus of smoothness.

262 S. Z. Jafarov

The following results hold.

1.1. Theorem. Let LM (T, ω) be a weighted Orlicz space with Boyd indices 0 < αM ≤βM < 1, and let ω ∈ A1/αM (T ) ∩A1/βM (T ). Then for f ∈ LM (T, ω), the inequality

ρn,M (f) = ‖f − σn−1 (., f)‖LM (T,ω) ≤c1n

n∑m=1

Em (f)M,ω , (n = 1, 2, ...)

holds with a positive constant c1, not depend on n.

1.2. Corollary. Let LM (T, ω) be a weighted Orlicz space with Boyd indices 0 < αM ≤βM < 1, and let ω ∈ A1/αM (T )∩A1/βM (T ). Then for every f ∈ LM (T, ω), the estimate

(1.2) ‖f − σn−1 (., f)‖LM (T,ω) ≤c2n

n∑m=1

ΩkM,ω

(1

m+ 1, f

),

holds with a c2 > 0 independent of n.

Now, we obtain the analogs of the above results in the weighted Smirnov-Orlicz classes,defined on the finite simple connected domains of the complex plane.

Let G be a finite domain in the complex plane C, bounded by a rectifiable Jordancurve Γ, and let G− := extΓ. Further let

T := w ∈ C : |w| = 1 , D := int T and D− := ext T.

Let w = ϕ(z) be the conformal mapping of G− onto D− normalized by

ϕ(∞) =∞, limz→∞

ϕ(z)

z> 0,

and let ψ denote the inverse of ϕ.Let w = ϕ1(z) denote a function that maps the domain G conformally onto the disk

|w| < 1.The inverse mapping of ϕ1 will be denoted by ψ1. Let Γr denote circular images in

the domain G, that is, curves in G corresponding to circle |ϕ1(z)| = r under the mappingz = ψ1(w).

Let us denote by Ep, where p > 0, the class of all functions f(z) 6= 0 that are analyticin G and have the property that the integral∫

Γr

|f(z)|p |dz|

is bounded for 0 < r < 1. We shall call the Ep-class the Smirnov class. If the functionf(z) belongs to Ep, then f(x) has definite limiting values f(z′) almost every where on Γ,over all nontangential paths; |f(z′)| is summable on Γ; and

limr→1

∫Γr

|f(z)|p |dz| =∫Γ

∣∣f(z′)∣∣p |dz| .

It is known that ϕ′ = E1(G−) and ψ′ ∈ E1(D−). Note that the general informationabout Smirnov classes can be found in the books [10, pp. 438-453] and [16, pp. 168-185].

Let LM (T, ω) is a weighted Orlicz space defined on Γ. We define also the ω-weightedSmirnov-Orlicz class EM (G,ω) as

EM (G,ω) := f ∈ E1 (G) : f ∈ LM (Γ, ω) .

With every weight function ω on Γ, we associate another weight ω0 on T defined by

ω0 (t) := ω (ψ (t)) , t ∈ T.


For f ∈ LM (Γ, ω) we define the function

f0(t) := f (ψ(t)) , t ∈ T.

Let h be a continuous function on [0, 2π]. Its modulus of continuity is defined by

ω (t, h) := sup |h (t1)− h (t2)| : t1, t2 ∈ [0, 2π] , |t1 − t2| ≤ t , t ≥ 0.

The curve Γ is called Dini-smooth if it has a parameterization

Γ : ϕ0(s), 0 ≤ s ≤ 2π

such that ϕ′0(s) is Dini-continuous, i.e.

π∫0

ω (t, ϕ′0)

tdt <∞

and ϕ′0 (s) 6= 0 [ 41, p. 48].If Γ Dini-smooth curve, then there exist [46] the constants c3 and c4 such that

(1.3) 0 ≤ c3 ≤∣∣ψ′ (t)∣∣ ≤ c4 <∞, |t| > 1.

Note that if Γ is a Dini-smooth curve, then by (1.3) we have f0 ∈ LM (Γ, ω0) andf ∈ LM (Γ, ω).

Let 1 0

1

r

∫Γ∩D(z,r)

|ω (τ)|p |dτ |

1/p1

r

∫Γ∩D(z,r)

[ω (τ)]−p′|dτ |

1/p′

<∞,

where D(z, r) is an open disk with radius r and centered z.Let us denote by Ap(Γ) the set of all weight functions satisfying Muckenhoupt’s Ap

-condition on Γ. For a detailed discussion of Muckenhoupt weights on curves, see, e.g.[7].

Let Γ be a rectifiable Jordan curve and f ∈ L1(Γ). Then the function f+ defined by

f+(z) :=1

2πi

∫Γ

f(s)ds

s− z , z ∈ G

is analytic in G. Note that if 0 < αM ≤ βM < 1, ω ∈ A1/αM (Γ) ∩ A1/βM (Γ) and

f ∈ LM (Γ, ω), then by Lemma1 in [25] f+ ∈ EM (G,ω).Let ϕk(z), k = 0, 1, 2, ... be the Faber polynomials for G. The Faber polynomials

ϕk(z), associated with G ∪ Γ, are defined through the expansion

(1.4)ψ′ (t)

ψ (t)− z =

∞∑k=0

ϕk (z)

tk+1, z ∈ G, t ∈ D−

and the equalities

(1.5) ϕk (z) =1

2πi

∫T

tkψ′ (t)

ψ (t)− z dt z ∈ G,

(1.6) ϕk (z) = ϕk (z) +1

2πi

∫Γ

ϕk (s)

s− z ds, z ∈ G−

hold [45, p.33-48].

264 S. Z. Jafarov

Let f ∈ EM (G,ω). Since f ∈ E1(G), we have

f(z) =1

2πi

∫Γ

f (s)

s− z ds =1

2πi

∫T

f (ψ (t))ψ′ (t)

ψ (t)− z dt,

for every z ∈ G. Considering this formula and expansion (1.4), we can associate with fthe formal series

(1.7) f (z) ∼∞∑k=0

ak (f)ϕk (z) ,

where

ak (f) :=1

2πi

∫T

f (ψ (t))

tk+1dt.

This series is called the Faber series expansion of f , and the coefficients ak(f) are saidto be the Faber coefficients of f .

The n-th partial sums and Fejer sums of the series (1.7) are defined, respectively, as

Sn (z, f) =

n∑k=0

ak (f)ϕk (z) ,

σn (z, f) =1

n+ 1

n∑k=0

Sk (z, f) .

Let Γ be a Dini-smooth curve. Using the nontangential boundary values of f+0 on T

we define the r−th modulus of smoothness of f ∈ LM (Γ, ω) as

ΩkΓ,M,ω (δ, f) := ΩkM,ω0

(δ, f+

0

), δ > 0,

for k = 1, 2, 3, ...The following theorem holds.

1.3. Theorem. Let Γ be a Dini-smooth curve, LM (Γ, ω) be a weighted Orlicz spacewith Boyd indices 0 < αM ≤ βM < 1, and ω ∈ A1/αM (Γ) ∩ A1/βM (Γ). Then forf ∈ EM (G,ω) the inequality

‖f − σn−1 (., f)‖LM (Γ,ω) ≤c5n

n∑m=1

ΩkΓ,M,ω

(1

m+ 1, f

)holds with a constant c5 > 0 independent of k.

Let P := all polynomials (with no restriction on the degree), and let P (D) be theset of traces of members of P on D. We define the operator

T : P (D)→ EM (G,ω)

as

T (P )(z) :=1

2πi

∫T

P (w)ψ′ (w)

ψ (w)− z dw, z ∈ G.

Then using (1.5) and (1.6) we get

T

(n∑k=0

akwk

)=

n∑k=0

ak (f)ϕk (z) , z ∈ G.

The following theorems hold for the linear operator T [25].

1.4. Theorem. Let Γ be a Dini-smooth curve and LM (Γ) be a reflexive Orlicz space. Ifω ∈ A1/αM (Γ) ∩A1/βM (Γ), then the linear operator T : P (D)→ EM (G,ω) is bounded.


1.5. Theorem. If Γ is a Dini-smooth curve, 0 < αM ≤ βM < 1 and ω ∈ A1/αM (Γ) ∩A1/βM (Γ), then the operator

T : EM (D,ω0)→ EM (G,ω)

is one-to-one and onto.

The following theorem was proved in [20, Theorem 2].

1.6. Theorem. Let LM (T, ω) be a weighted Orlicz space with Boyd indices 0 < αM ≤βM < 1, and let ω ∈ A1/αM (T )∩A1/βM (T ). Then for every f ∈ LM (T, ω) the estimate

En (f)M,ω ≤ c6ΩkM,ω

(1

n+ 1, f

), k = 1, 2, ...

holds with a constant c6 > 0 independent of n.

2. Proofs of the Main ResultsProof of Theorem 1.1 We can write the following equality:

(1.8) σn−1 (f) =1

n

n−1∑m=0

Sm (f) =

1

n

S0 (f) +

j−1∑i=1

2i−1∑m=2i−1

Sm (f) +

n−1∑m=2j−1

Sm (f)

.

Using (1.8) we get

(1.9) f − σn−1 (f) =

1

n

(f − S0 (f)) +

j−1∑i=1

2i−1∑m=2i−1

(f − Sm (f)) +

n−1∑m=2j−1

(f − Sm (f))

By using ineguality

‖f − Sn (., f)‖LM (T,ω) ≤ c7En (f)M,ω

given [20] and (1.9) we obtain

(1.10)

∥∥f − σn−1 (f)∥∥LM (T,ω)

≤

≤c8

n

[E1 (f)M,ω +

j−1∑i=1

(2i + 2i−1 − 1

)E

2i−1 (f)M,ω +(2n− 2j−1 − 1

)En−2j−1 (f)M,ω

]

≤c9

n

[E1 (f)M,ω + E1 (f)M,ω +

j−1∑i=2

2i−1E2i−1 (f)M,ω +

(2n− 2j−1

)En−2j−1 (f)M,ω

].

By [20 ] the following inequality holds:

(1.11) 2i−1E2i−1 (f)M,ω ≤ 2

2i−1∑m=2i−2+1

Em (f)M,ω .

Selecting j such that 2j ≤ n < 2j+1, from (1.11) we get

(1.12)(

2n− 2j−1)En−2j−1 (f)M,ω ≤

2n− 2j−1

n− 2j−1 − 2j−2

n−2j−1∑m=2i−2+1

Em (f)M,ω

=

(2 +

2j

n− 2j−1 − 2j−2

) n−2j−1∑m=2i−2+1

Em (f)M,ω ≤ c10

n∑m=2j−2+1

Em (f)M,ω

266 S. Z. Jafarov

By (1.10), (1.11) and (1.12) we obtain

(1.13) ‖f − σn−1 (f)‖LM (T,ω) ≤c11

n

E1 (f)M,ω +

j−1∑i=2

2i−1∑m=2i−2+1

Em (f)M,ω +

n∑m=2j−2+1

Em (f)M,ω

≤ c12

n

n∑m=1

Em (f)M,ω ,

which completes the proof of Theorem 1.1.Proof of Corollary 1.2. By Theorem 1.6 the following inequality holds

(1.14) En (f)M,ω ≤ c13ΩkM,ω

(1

n+ 1, f

)k = 1, 2, ...

Then using (1.13) and (1.14) we obtain inequality (1.2).Proof of Theorem 1.3. Let f ∈ EM (G,ω). Then by Theorem 1.5 the operator

T : EM (D,ω0) → EM (G,ω) is bounded, one-to-one and onto and T(f+

0

)= f . The

function f has the following Faber series

f(z) ∼∞∑k=0

ak(f)ϕk(z).

Since ω0 ∈ A1/αM (T ) ∩ A1/βM (T ), by Lemma 1 in [25, p. 760] we have f+0 ∈

EM (D,ω0). For the function f+0 the following Taylor expansion holds:

f+0 (w) =

∞∑k=0

ak(f)wk.

It is known that f+0 ∈ E1 (D) and boundary function f+

0 ∈ LM (T, ω0). Then using[16, Th. 3.4] for the function f+

0 we have Fourier expansion

f+0 (w) ∼

∞∑k=0

ak(f)wikt.

Using the boundedness of the operator T Theorem 1.1 and Corollary 1.2 we get

‖f − σn−1 (·, f)‖LM (Γ,ω) =∥∥T (f+

0

)− T

(σn−1

(·, f+

0

))∥∥LM (Γ,ω)

≤

≤ c14

∥∥f+0 − σn−1

(·, f+

0

)∥∥LM (T,ω0)

≤ c15

n

n∑m=1

Em(f+

0

)M,ω≤

≤ c16

n

n∑m=1

ΩkM,ω0

(1

m+ 1, f+

0

)=c17

n

n∑m=1

ΩkΓ,M,ω

(1

m+ 1, f

).

The proof of Theorem 1.3 is completed.

Acknowledgements

The author wishes to express deep gratitude to the referee for valuable suggestions.

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OPERATOR VALUED DIRICHLET

PROBLEM IN THE PLANE

Nihat Gokhan Gogus ∗


Abstract

We consider an operator valued Dirichlet problem for harmonic map-pings and prove the existence of a Perron-like solution. To formulatethe Perron’s construction we make use of Olson’s notion of spectralorder. We introduce a class of operator valued subharmonic mappingsand establish some of their elementary properties.

Keywords: Operator theory, Harmonic mappings, Perron method, Spectral order.

2000 AMS Classification: 47A56, 47A63, 47B15

1. Introduction

With this paper we would like to initiate a research on potential theory of harmonicand subharmonic functions in the plane with values in the class of bounded linear oper-ators on a Hilbert space. The main purpose is to describe the solution of the Dirichletproblem using a Perron’s method. Making use of the Olson’s notion of spectral order weshow that there is a Perron-like solution whenever the boundary values are commutingor the Poisson integral of the boundary mapping is a projection (Theorem 4.10).

We recall that inequality in spectral order implies the inequality in the usual order onthe class of self-adjoint operators. By Olson S is a conditionally complete lattice underthe partial order . Making use of functional calculus and spectral order we obtain somegeneralizations of well known properties of subharmonic functions to the operator valuedcase. One of the main results of the paper is the maximum principle for subharmonicmappings in terms of the spectral order.

Recently quite a few papers are written about harmonic mappings which take theirvalues in infinite dimensional spaces. We consider in this paper a Dirichlet problemfor operator valued harmonic mappings of the complex plane. To formulate the Perronsolution we introduce a class of operator valued subharmonic mappings and establishsome of their elementary properties which will be used for our purpose. In section 4we provide an example of a mapping of the form F (z) = T + zS, where T and S are

∗Sabanci University, Orhanli, Tuzla 34956, Istanbul, TurkeyEmail: [email protected]

270 N. G. Gogus

Hermitian 2× 2 matrices such that F (z)F ∗(z) fails to be subharmonic. On the contraryan example of subharmonic mappings is given by the following result.

Theorem. The mapping Log (F (z)F ∗(z)) is harmonic and F (z)F ∗(z) is subharmonicwhenever the mapping F : G → A is holomorphic, F (z) is normal, and the spectrum ofF (z) does not contain any element from the set

x+ iy ∈ C : y = 0, x ≤ 0.

The type of the Dirichlet problem is determined according the WOT, SOT or normconvergence of the solution on the boundary of the domain. It turns out that for a generaldomain G in the complex plane there is always a SOT continuous up to the boundaryof harmonic mapping on G for a given SOT continuous function on the boundary pro-vided that the domain G is regular for the Dirichlet problem for real valued harmonicfunctions. The theory of operator-valued holomorphic mappings has found many appli-cations in functional analysis. These studies provide a better understanding and a wayof formulation of different phenomena about vector-valued function spaces. The ideasand methods of this theory is used in our time not only in mathematical physics, butalso in function theory, functional analysis, probability theory, approximation theory andharmonic analysis.

We start with investigating the class of harmonic mappings from an open subset of thecomplex plane which take values in a von Neumann algebra. Most of the classical resultsare carried in a natural way to the setting of operator-valued harmonic mappings. Wegather several well known information on harmonic mappings in Section 3: A mappingis harmonic if and only if it is weakly harmonic for example. This result has severaluseful applications. We show that for self-adjoint harmonic mappings the norm is alwaysa subharmonic function and for complex combinations of harmonic mappings the squareof its numerical radius is also subharmonic. We prove the main results in section 4.

2. Preliminaries

Basic notation. A will denote a von Neumann algebra. There exists a Hilbert spaceH so that A is a sub-algebra of B(H) and we will make use of this Hilbert space relatedto A throughout. The class of self-adjoint elements in A is denoted by S. More generally,we will denote by A a C∗-algebra. We will denote by S or SA the class of self-adjointelements in A. For arbitrary elements a, b ∈ B(H) we set [a, b] := ab − ba. We willdenote by D(w, r) the open disk in the complex plane with center w and radius r andby T(w, r) its boundary. We denote the open unit disk and unit circle in C by D and Trespectively. Let C∞ denote the Riemann sphere.

Suppose A is a sub-algebra of B(H) for some Hilbert space H. If ψ : X → A is amapping from a set X, and h ∈ H, k ∈ H, we denote by ψh, respectively by ψh,k thecomplex-valued function on X defined by

ψh(x) = 〈ψ(x)h, h〉

and respectively,

ψh,k(x) = 〈ψ(x)h, k〉

for any x ∈ X.Order. Let A be a von Neumann algebra of operators on a Hilbert space H. Let S

be the real vector space of self-adjoint elements of A. For two operators a and b in S,a ≤ b means 〈ah, h〉 ≤ 〈bh, h〉 for all h ∈ H. With this order, S is a partially orderedvector space but not a vector lattice.

Operator valued Dirichlet problem in the plane 271

Another order, so called the spectral order on S is defined by Olson in [8]. Let a and bbe self-adjoint elements of the von Neumann algebra A. Let Ea and Eb be the resolutionsof the identity of a and b, respectively. We write a b if

Ea(t) ≥ Eb(t), t ∈ R.The order ”” is called spectral order. It was proved in [8] that S is a conditionallycomplete lattice under the partial order . Some of the basic properties of the spectralorder which will be used frequently in the text are collected in the next proposition.

2.1. Proposition. Let a, b ∈ S.

i. a 0 if and only if a ≥ 0. The two orders, therefore, have the same positivecones.

ii. If a b, then a ≤ b.iii. If a and b commute, then a b if and only if a ≤ b.iv. Suppose a and b are also bounded positive. Then a b if and only if ak ≤ bk for

k = 1, 2, . . ..v. a b if and only if −a −b.vi. Let aj, bj ∈ S so that kI ≤ aj bj, j ≥ 1, for some k ∈ R, and

strong limj aj = a, strong limj bj = b.

Then a b.vii. Let a1, . . . ak ∈ S be nonnegative elements. Then

strong limr→∞(ar1 + . . .+ ark)1/r = sup1≤j≤k

aj

and the limit is increasing in usual order. Likewise it follows that

strong limr→∞(a−r1 + . . .+ a−rk )−1/r = inf1≤j≤k

aj

and the limit is decreasing in usual order.viii. Let S ≥ 0 and T be a projection. Then S T if and only if S ≤ T .

Proof. The first four properties are in [8]. Property v. although appears in several paperswe could find a proof in [9]. To prove part vi. we may assume without loss of generalitythat aj ≥ 0 and bj ≥ 0 since for any two operators a and b it is true that a b if andonly if a−kI b−kI. The condition aj bj implies akj ≤ bkj for every k ≥ 1 by part iv.

Hence ak ≤ bk for every k ≥ 1 and the result follows again using part iv. Property vii.above is proved for positive bounded operators in [8]. It is proved in full generality in[6]. If 0 ≤ S ≤ I, then Sk ≤ S for every k ≥ 1. Property viii. then follows from iv.

The following version of Jensen’s inequality is proved in [2].

2.2. Theorem. Let A be a C∗-algebra and B be a von Neumann algebra. Let a = a∗ ∈ A,f be a monotone convex real-valued function on an open interval containing the spectrumof a, and ϕ be a unital positive map from A to B. Then f(ϕ(a)) ϕ(f(a)).

A real-valued function defined on an interval O of R is said to be spectral orderpreserving on O if a b implies that f(a) f(b) for every a, b ∈ S whenever thespectra of a and b are contained in O. The next result helps us to generate spectral orderpreserving functions.

2.3. Proposition. a. Let f : O → R and g : J → R be spectral order preservingfunctions on intervals O and J , respectively, so that g(J) ⊂ O. Then f g is spectralorder preserving on J .

b. Let g(t) be a non-decreasing continuous function on an open interval J so thatg(J) ⊂ (0,∞), and let p ≥ 0 be a number. Then gp(t) is spectral order preserving on J .

272 N. G. Gogus

Proof. Part a. is rather apparent. Let g(t) be a function as in b., and let p ≥ 0 be anumber. Then gkp > 0 and non-decreasing on J for every integer k ≥ 1. To prove thesecond part take operators a b with their spectra lying in J . Then 0 ≤ gkp(a) ≤ gkp(b)for every k by Proposition 2.1. Again by Proposition 2.1, this implies that gp(a) gp(b).

We will borrow a result from [1] which will be used later in the next sections. LetN ⊂M be von Neumann algebras, Φ : M→ N a faithful conditional expectation (that is,a projection of norm 1 so that Φ(a) > 0 whenever a > 0), and a ∈M a positive element.We let

a+ := infb ∈ SN : a b,the spectral order majorant of a in N.

2.4. Theorem. [1, Theorem 9] The sequence [Φ(an)]1/n converges in SOT to a+.

For last we will state a result from [10] for commuting n-tuples of operators.

2.5. Theorem. [10] Let F : Rn → R be a continuous function which is increasingin each variable. Let ak, bk ∈ S, k = 1, . . . , n, be operators such that ak bk, and[aj , ak] = [bj , bk] = 0 for every j, k = 1, . . . n. Then

F (a1, . . . , an) F (b1, . . . , bn).

Taking F (t) = t1 + . . .+ tn, t ∈ Rn in Theorem 2.5 we obtain the following result.

2.6. Corollary. Let ak, bk ∈ S, k = 1, . . . , n, be operators such that ak bk, and[aj , ak] = [bj , bk] = 0 for every j, k = 1, . . . n. Then

a1 + · · ·+ an b1 + · · ·+ bn.

3. Harmonic mappings on the plane

In this section we will introduce the harmonic mappings on the complex plane. Wewill present the Dirichlet problem for operator valued harmonic mappings on the plane.The main theorem of this section is that if a domain is regular for the classical real valuedDirichlet problem, then it is regular for the operator valued Dirichlet problem.

First we shall gather the well-known properties of harmonic mappings. Let G be anopen set in C. The target space of the harmonic mappings which will be defined in thissection is always the class of self-adjoint elements S of a von Neumann algebra A.

3.1. Definition. A mapping u : G→ S is called harmonic if ∆u = 0 in G.

We denote the class of harmonic mappings by Har[G, S]. Let us denote by Har[G,A]the complex linear combination of all self-adjoint harmonic mappings. There are naturalexamples of harmonic mappings. Let F : G→ A be a holomorphic mapping, then ReFand ImF belong to Har[G, S]. Note that the class of all holomorphic mappings from Gto A belongs to Har[G,A].

σ-Dirichlet problem on the disk. We denote by σ the weak opeartor topology(WOT), the strong operator topology (SOT) or the norm topology on the class S ofselfadjoint elements in A. Let G be a domain in C. Given a function ϕ : ∂G → S

continuous in σ find a harmonic mapping u ∈ Har[G, S] so that

limz→ζ u(z) = ϕ(ζ)

in σ for every ζ ∈ ∂G. This problem is the generalization of the classical Dirichletproblem to the operator-valued potential theory. A domain G is said to be regular for


the σ-Dirichlet problem or simply σ-regular if for every σ-continuous S-valued functionϕ from ∂G the σ-Dirichlet problem has a solution. Although the statements and proofsin this section are quite standard, we could not find a good reference for the materialpresented here. We refer to [11] for the proof in the classical case.

3.2. Theorem. a. There is at most one solution to the Dirichlet problem when G is abounded domain in C.

b. The Poisson transform P [ϕ,D] is harmonic on D for any mapping ϕ ∈ L1[T, S].c. If ϕ is continuous at a point ζ0 ∈ T, then limz→ζ0 P [ϕ,D](z) = ϕ(ζ0) in norm.

For general domains the solution of the Dirichlet problem is again quite similar tothe real valued case. Let G be a domain in C∞ such that ∂G is non-polar. It is well-known that there is a unique harmonic measure ωG for G (see [11, Theorem 4.3.2]). Letϕ : ∂G→ S be a bounded mapping so that the complex-valued function ϕh,k is Borel forevery h, k ∈ H. We set

P [ϕ,G](z) :=

∫∂G

ϕ(ζ) dωG(z, ζ), z ∈ G.

Note that

〈P [ϕ,G](z)h, h〉 = P [ϕh, G](z)

and the real-valued function P [ϕh, G](z) is harmonic in G for every h ∈ H. It turns outthat the mapping P [ϕ,G](z) is harmonic. The big part of the next theorem is obtainedin [3] as a consequence of Theorem 17 there.

3.3. Theorem. Let G be an open set in C. Let u : G→ S be a mapping which is locallyintegrable on G. The following statements are equivalent:

(1) u is harmonic on G.(2) u(z) = P [u,∆](z) for any disk ∆ compactly belonging to G and for every z ∈ ∆.(3) u(z) = P [u,Ω](z) for any open set Ω compactly belonging to G and for every

z ∈ Ω.(4) The complex valued function uh,k is harmonic for every h, k ∈ H.(5) The real valued function uh is harmonic on G for every h ∈ H.(6) For every open disk ∆ in G, u is the real part of a holomorphic function from

∆ to A on ∆.(7) ψ u : G→ R is harmonic for every continuous functional ψ : S→ R.(8) If ∆ is an open disk which is relatively compact in G, then u = ReF for some

holomorphic mapping F : ∆→ S.

Thus all the natural topologies, norm topology, SOT and the WOT in B(H) give riseto the same class of harmonic mappings. We state a corollary of Theorem 3.3 (see also[3] and [5]).

3.4. Corollary. If (un)n≥1 is a sequence of harmonic mappings on an open set G ⊂ Cthat converge locally uniformly in WOT to a mapping u, then u is harmonic on G.

Theorem 3.3 has several other consequences. The next one is trivial.

3.5. Corollary. Suppose G is a bounded domain in C. Let u, v ∈ Har[G, S]. If u ≤ von ∂G, then u ≤ v on G.

Using Theorem 2.2 we prove the integral version of Jensen’s inequality.

3.6. Corollary. Let G be an open set in C, A and C be C∗-algebras so that A ⊂L1[∂G, SC], and u ∈ A. Let f be a monotone convex real-valued function on an openinterval containing the spectrum of u(z) for every z ∈ ∂G. Then

f(P [u,G])(z) P [f(u), G](z)

274 N. G. Gogus

for every z ∈ G.

Proof. Take a point z ∈ G. Let B = C, and ϕ(v) = P [v,G](z) for any v ∈ A. Clearly ϕis a unital positive map from A to B. Hence Theorem 2.2 can be applied.

The next result of this section shows that the norm of a harmonic mapping is subhar-monic.

3.7. Theorem. Let G be an open subset of C, and let u ∈ Har[G, S]. Then ‖u‖ belongsto SH[G].

Proof. We have

‖u(z)‖ = sup|ψ u(z)| : ψ ∈ S∗, ‖ψ‖ = 1

for every z ∈ G. Since ψ u ∈ Har[G] by Theorem 3.3 the conclusion follows by notingthat norm is continuous.

Recall that for any operator T ∈ B(H) we define the numerical range W [T ] of T asthe set

W [T ] := 〈Tx, x〉 : x ∈ H.The numerical radius w[T ] of T is defined as the number

w[T ] := sup‖x‖=1

|〈Tx, x〉|.

If T is self-adjoint, then the numerical radius w(T ) coincides with the spectral radiusρ(T ) which is the same as the norm ‖T‖ of T . We prove in the next result that thesquare of the numerical radius of a harmonic mapping is subharmonic.

3.8. Theorem. Let u ∈ Har[G,B(H)]. Then the function v(z) defined by

v(z) := (w[u(z)])2 , z ∈ Gis continuous subharmonic on G.

Proof. It is sufficient to note that we have

v(z) = sup‖x‖=1

|〈u(z)x, x〉|2,

the functions |〈u(z)x, x〉|2 are subharmonic on G for every x ∈ H and that v(z) iscontinuous. This proves that v is subharmonic on G.

4. Subharmonic mappings and Perron method

In the previous section we gathered some necessary information on harmonic map-pings. In this section we describe the solution of the operator valued Dirichlet problemusing a Perron-like method. We introduce a notion of subharmonic mappings to describePerron’s construction. The following example is imortant. There by Log z we denote theprincipal branch of the complex logarithm.

4.1. Theorem. The mapping

Log (F (z)F ∗(z))

belongs to Har[G, S] whenever F : G → A is holomorphic, F (z) is normal, and thespectrum of F (z) does not contain any element from the set

x+ iy ∈ C : y = 0, x ≤ 0for every z ∈ G.


Proof. By functional calculus for normal operators LogF ∗(z) = (LogF (z))∗ and LogF (z)F ∗(z) = LogF (z) + LogF ∗(z) for every z ∈ G. Since the mapping LogF (z) is holomor-phic, we see that the mapping LogF (z)F ∗(z) is harmonic in G.

We recall that for any φ : ∂G → R the associated Perron function HGφ : G → R isdefined by

HGφ(z) := supU[φ]

u(z),

where U[φ] is the class of all subharmonic functions u on G so that lim supz→ζ u(z) ≤ φ(ζ)for every ζ ∈ ∂G. It is clear that when there is a harmonic solution of the Dirichletproblem for φ in G, then HGφ(z) is the solution. An application of the classical potentialtheory gives the following result.

4.2. Theorem. Let G be a domain in C∞ such that ∂G is non-polar. Let ϕ : ∂G→ S bea bounded mapping so that the complex-valued function ϕh,k is Borel for every h, k ∈ H.The following statements hold:

a. For every h ∈ H the function HGϕh coincides with the function P [ϕh, G] on G.b. If for every h, k ∈ H the function ϕh,k is continuous at nearly every point in

∂G, then

SOT − limz∈G,z→ζ P [ϕ,G](z) = ϕ(ζ)

for nearly every point ζ ∈ ∂G.c. Let ζ0 be a regular boundary point of G. If ϕ is SOT-continuous at ζ0, then

SOT − limz∈G,z→ζ0 P [ϕ,G](z) = ϕ(ζ0).

d. If G is a regular domain and ϕ : ∂G→ S is SOT-continuous, then P [ϕ,G](z) isthe unique harmonic mapping on G such that

SOT − limz∈G,z→ζ P [ϕ,G](z) = ϕ(ζ)

for all ζ ∈ ∂G.

Proof. The proofs follow immediately from [11] Theorem 4.3.3 for part a., Corollary 4.2.6for part b., Theorem 4.1.5 for part c., and from Corollary 4.1.8 for part d.

Theorem 4.2 shows that if a domain in the complex plane is regular with respect to theclassical Dirichlet problem, then it is regular for the operator-valued Dirichlet problem.We will now define a class of subharmonic mappings. There is no exactly one way of adescription of subharmonic mappings in the operator theoretic setup. Various classes ofsubharmonic mappings and their relations within them will be studied in another project.So let us give the definition of a subharmonic mapping that fits to our purpose.

4.3. Definition. Let u : G → S be a SOT-continuous function. We say that u issubharmonic in G if

(SH) for every open domain Ω compactly belonging to G we have the inequality

u(z) P [u,Ω](z)

for every z ∈ Ω.

We say that u is superharmonic in G if −u is subharmonic in G.

We will denote the class of subharmonic and superharmonic mappings inG by SH[G, S,] and SPH[G, S,], respectively. In view of Theorem 3.3 the class Har[G, S] ⊂ SH[G, S,].The following observation follows immediately from Proposition 2.1 part v.

4.4. Proposition. Let u : G→ S be continuous. Then u ∈ Har[G, S] if and only if bothu and −u belong to SH[G, S,].

276 N. G. Gogus

A subharmonic real-valued function composed with a convex function is again sub-harmonic. We have similar results for operator-valued mappings.

4.5. Theorem. Let A be a von Neumann algebra. Let u ∈ SH[G, S,] and f be a mono-tone convex spectral order preserving real-valued function on an open interval containingthe spectrum of every u(z), z ∈ G. Then f(u) ∈ SH[G, S,].

Proof. Let ∆ be a disk compactly belonging to G. If u ∈ SH[G, S,] and f is spec-tral order preserving, then f(u(z)) f(P [u,∆](z)) for every z ∈ G. By Theorem 2.2f(P [u,∆](z)) P [f(u),∆](z). Then f(u(z)) P [f(u),∆](z). Hence f(u) ∈ SH[G, S,].

As an application of these observations we present the following corollary.

4.6. Corollary. Let A be a von Neumann algebra and let u ∈ SH[G, S,]. Thena. eu ∈ SH[G, S,];b. if u(z) ≥ 0 for all z ∈ G, then up ∈ SH[G, S,] for every number p ≥ 1.

Proof. The functions tp and et are spectral order preserving on the positive real line byProposition 2.3. Therefore if u(z) ≥ 0 for all z ∈ G, then Theorem 4.5 can be applied andhence a. and b. hold. For part a. suppose that u is an arbitrary mapping in SH[G, S,].

Take a disk ∆ compactly belonging to G and let m = 2 max‖u(z)‖ : z ∈ ∆ so that

u + mI > 0 on ∆. Observe that eu ∈ SH[∆, S,] if and only if eu+mI ∈ SH[∆, S,].Then use the first argument to conclude the proof.

4.7. Corollary. Let F : G → A be holomorphic, F (z) be normal, and suppose that thespectrum of F (z) does not contain any element from the set

x+ iy ∈ C : y = 0, x ≤ 0for every z ∈ G. Then the mapping F (z)F ∗(z) belongs to SH[G, S,].

The next result is the formulation of the maximum principle for subharmonic map-pings.

4.8. Theorem. Let G be a domain in C and u ∈ SH[G, S,]. For every relativelycompact open subset Ω in G and for every z ∈ Ω we have

u(z) supw∈∂Ω

u(w).

Proof. Let Ω in G be a relatively compact open subset and z ∈ Ω. Since u is subharmonicin G we have

u(z) P [u,Ω](z).

Hence by Corollary 3.6

un(z) (P [u,Ω](z))n P [un,Ω](z)

for every integer n ≥ 1. Note that the map v 7→ P [u,Ω](z) is a faithful conditionalexpectation from M := L∞[∂Ω,A] onto N := set of constant operator valued mappings

from ∂Ω to A. We identify N with A. By Theorem 2.4 the sequence (P [un,Ω](z))1/n

converges in strong operator topology to the operator

u+ = infa ∈ S : u(ζ) a for every ζ ∈ ∂Ω.We will show that u+ equals supw∈∂Ω u(w). Since u(ζ) supw∈∂Ω u(w) for every ζ ∈ ∂Ωwe have u+ supw∈∂Ω u(w). On the other hand since u(ζ) u+ for every ζ ∈ ∂Ω wehave supw∈∂Ω u(w) u+. Hence u+ = supw∈∂Ω u(w). To finish the proof we have

u(z) (P [un,Ω](z))1/n


for every n ≥ 1. Taking limit and using vi. of Proposition 2.1 we get

u(z) supw∈∂Ω

u(w).

This completes the proof.

In the next result we prove that uniform limit of subharmonic mappings is againsubharmonic. An analog result holds for superharmonic mappings.

4.9. Theorem. Let G be a domain in C and un ∈ SH[G, S,]. If un converges in strongoperator topology to a mapping u locally uniformly in G, then u belongs to SH[G, S,].

Proof. Let Ω be a relatively open subset of G and z ∈ Ω. Then

un(z) P [un, G](z)

for every n. By vi. of Proposition 2.1 we get that u(z) P [u,G](z). Thus u ∈ SH[G, S,].

Perron method. To describe the solution of the Dirichlet problem in classical po-tential theory one uses the Perron method. It is quite useful in applications. In thissection we extend the Perron method to certain operator valued settings. In these re-sults supremum is with respect to the spectral order. The purpose of the next result isto extend this classical result to the case for which the boundary data is commutative.

4.10. Theorem. Let G be a bounded domain in C which is regular for the classicalreal-valued Dirichlet problem. Let ϕ be a norm continuous S-valued function from ∂Gwhich has a commutative range. Let Uc[ϕ] be the class of all mappings u ∈ SH[G, S,]∩C[G, SSOT ] so that u(ζ) ϕ(ζ) for every ζ ∈ ∂G and the range of u|∂G is commutative.Then the mapping

Hc[ϕ,G](z) := sup u(z) : u ∈ Uc[ϕ] , z ∈ Gis harmonic on G, SOT-continuous on G, and P [ϕ,G](z) = Hcϕ(z) for every z ∈ G.Moreover,

limz→ζ P [ϕ,G](z) = ϕ(ζ)

in SOT for every ζ ∈ ∂G.

Proof. We only need to prove the equality of Hc[ϕ,G](z) and P [ϕ,G](z), z ∈ G, sinceall the other statements about P [ϕ,G](z) are already proved in Theorem 4.2. ClearlyP [ϕ,G](z) Hc[ϕ,G](z) for z ∈ G and P [ϕ,G](z) = Hc[ϕ,G](z) = ϕ(z) when z ∈ ∂G.To prove the reverse inequality take any z ∈ G and u ∈ Uc[ϕ]. We set a := u|∂G. Let ωGbe the harmonic measure for G. Given n ≥ 1 we can find z1, . . . , zn ∈ ∂G and relativelyopen subsets I1, . . . , In ⊂ ∂G with zk ∈ Ik, k = 1, . . . , n so that

limn

∥∥∥∥∥P [ϕ,G](z)−n∑k=1

ϕ(zk)ωG(z, Ik)

∥∥∥∥∥ =

= limn

∥∥∥∥∥P [a,G](z)−n∑k=1

a(zk)ωG(z, Ik)

∥∥∥∥∥ = 0.

By Corollary 2.6 we haven∑k=1

a(zk)ωG(z, Ik) n∑k=1

ϕ(zk)ωG(z, Ik)

for every n ≥ 1. By vi. of Proposition 2.1 we see that

u(z) P [a,G](z) P [ϕ,G](z).

278 N. G. Gogus

Taking supremum over all mappings u ∈ Uc[ϕ] we get that Hc[ϕ,G](z) P [ϕ,G](z).Thus Hc[ϕ,G](z) = P [ϕ,G](z).

In the next result we consider the case for which P [a,G](z) is a projection.

4.11. Theorem. Let G be a domain in C which is regular for the classical real-valuedDirichlet problem. Let z ∈ G and ϕ be a norm continuous S-valued function from ∂Gso that P [ϕ,G](z) is a projection. Let U+[ϕ] be the class of all nonnegative mappings

u ∈ SH[G, S,] ∩ C[G, SSOT ] so that u(ζ) ϕ(ζ) for every ζ ∈ ∂G. Then the operator

H+[ϕ,G](z) := sup u(z) : u ∈ U+[ϕ]

coincides with P [ϕ,G](z).

Proof. Let u ∈ U+[ϕ]. Then u ≤ P [ϕ,G] in G, in particular, u(z) ≤ P [ϕ,G](z). Fromviii. of Proposition 2.1 we have u(z) P [ϕ,G](z). Taking supremum over all suchu ∈ U+[ϕ] we get H+[ϕ,G](z) P [ϕ,G](z). The reverse inequality is already true.Hence H+[ϕ,G](z) coincides with P [ϕ,G](z).

In the example below we construct continuous mappings a, b on ∂D with the followingproperties:

i. a(ζ) b(ζ) for every ζ ∈ ∂D.ii. P [b,D](0) is a projection.iii. P [a,D](0) 6 P [b,D](0).

Hence, as in this example, it may happen that P [b,D](z) 6= H[b,D](z) for some z ∈ D,where

H[b,D](z) := supu(z) : u ∈ SH[D, S] ∩ C[D, S], u b in ∂D.

4.12. Example. In [8, page 543] 2 × 2 matrices P and A were constructed so that0 ≤ P −A, P is a projection and A 6 P . More precisely,

P =

[1 00 0

], A =

[−1 −

√2

−√

2 −1

], P −A =

[2√

2√2 1

].

Let 0 < ε < π. For any 2× 2 matrix X we set

gX(eit) :=

X if 0 ≤ t ≤ π−ε2

or 3π+ε2≤ t ≤ 2π,

2ε

[(π2− t)X +

(t− π−ε

2

)I]

if π−ε2≤ t ≤ π

2,

2ε

[(π+ε

2− t)I +

(t− π

2

)A]

if π2≤ t ≤ π+ε

2,

A if π+ε2≤ t ≤ 3π−ε

2,

2ε

[(3π2− t)A+

(t− 3π−ε

2

)I]

if 3π−ε2≤ t ≤ 3π

2,

2ε

[(3π+ε

2− t)I +

(t− 3π

2

)X]

if 3π2≤ t ≤ 3π+ε

2.

It can be checked easily that

1

2π

∫ 2π

0

gX(eit) dt =2π − ε

4π(X +A) +

ε

2πI.

Now we let

a(eit) :=4π

2π − εg0(eit)− 2ε

2π − ε , and b(eit) :=4π

2π − εgP−A(eit)− 2ε

2π − εfor any 0 ≤ t ≤ 2π. From the construction it is readily seen that a(eit) b(eit) for any0 ≤ t ≤ 2π,

P [a,D](0) =1

2π

∫ 2π

0

a(eit) dt = A,


and

P [b,D](0) =1

2π

∫ 2π

0

b(eit) dt = P.

Since A 6 P we have P [a,D](0) 6 P [b,D](0). In this example P [b,D](0) ≺ H[b,D](0).

Following Example 4.12 and in contrast to Corollary 4.7 we construct a linear map-ping F (z) with positive self-adjoint coefficients so that the mapping lnF (z)F ∗(z) is notharmonic.

4.13. Example. With the same notation as in Example 4.12 let S := (A + 100I)1/2,

and T := (P −A)1/2. We set

F (z) := S + zT

for every z ∈ C. Note that the spectrum of F (z) does not contain any element from theset

z = x+ iy ∈ C : y = 0, x ≤ 0

whenever |z| ≤ 2. Let ∆ := D(0, 2). We claim that the mapping u(z) := lnF (z)F ∗(z) isnot harmonic in ∆. Suppose on the contrary that it is harmonic. By Corollary 4.6 thenthe mapping F (z)F ∗(z) belongs to SH[∆, S2×2], where S2×2 is the class of self-adjoint2× 2 matrices. Let us show that this is not the case. We compute

1

2π

∫ 2π

0

F (eit)F ∗(eit) dt = S2 + T 2 = P + 100I.

Thus

F (0)F ∗(0) = A+ 100I 6 P + 100I =1

2π

∫ 2π

0

F (eit)F ∗(eit) dt.

Hence F (z)F ∗(z) 6∈ SH[∆, S2×2]. Therefore lnF (z)F ∗(z) is not harmonic in ∆.

Acknowledgements

We would like to thank Mohan Ravichandran for introducing us with the notionof spectral order and to Artur Planeta who brought the result in Theorem 2.5 to ourattention.

References

[1] Akemann, C. A. and Weaver, N. Minimal upper bounds of commuting operators, Proc.Amer. Math. Soc. 124 (11) , 3469–3476, 1996.

[2] Antezana, J. and Massey, P. and Stojanoff, D. Jensen’s inequality for spectral order and

submajorization, J. Math. Anal. Appl. 331, 297–307, 2007.[3] Bonet, J. and Frerick, L. and Jord, E. Extension of vector-valued holomorphic and harmonic

functions, Studia Math. 183 (3), 225–248, 2007.[4] Conway, J. B. A Course in operator theory, Grad. Texts in Math. 21, Amer. Math. Soc.,

1999.

[5] Enflo, P. and Smithies, L. Harnack’s theorem for harmonic compact operator-valued func-

tions, Linear Algebra and its Applications 336, 21–27, 2001.[6] Fujii, M. and Kasahara, I. A remark on the spectral order of operators, Proc. Japan Acad.

47, 986–988, 1971.[7] Jorda, E. Vitali’s and Harnack’s type results for vector-valued functions, J. Math. Anal.

Appl. 327, 739–743, 2007.

[8] Olson, M. P. The selfadjoint operators of a Von Neumann algebra form a conditionallycomplete lattice, Proc. Amer. Soc. 28, 537–544, 1971.

280 N. G. Gogus

[9] Planeta, A. and Stochel, J. Spectral order for unbounded operators, J. Math. Anal. Appl.,

10.1016/j.jmaa.2011.12.042.

[10] Planeta, A. and Stochel, J. Multidimensional spectral order, preprint.[11] Ransford, T. Potential theory in the complex plane, London Mathematical Society Student

Texts 28, Cambridge University Press, Cambridge, 1995.


SOME SUMMATION FORMULAS

FOR THE HYPERGEOMETRIC

SERIES r+2Fr+1(12)

Y. S. Kim∗, A. K. Rathie†, U. Pandey‡ and R. B. Paris§ ¶


Abstract

The aim of this paper is to obtain explicit expressions of the generalizedhypergeometric function

r+2Fr+1

[a, b,

12(a+ b+ j + 1),

(fr +mr)(fr)

; 12

]for j = 0,±1, . . . ,±5, where r pairs of numeratorial and denominatorialparameters differ by positive integers mr. The results are derived withthe help of an expansion in terms of a finite sum of 2F1( 1

2) functions and

a generalization of Gauss’ second summation theorem due to Lavoie etal. [J. Comput. Appl. Math. 72, 293–300 (1996)]. Some special andlimiting cases are also given.

Keywords: Generalized hypergeometric series, Generalized Gauss summation theorem

2000 AMS Classification: Primary 33C20, Secondary 33C15

∗Department of Mathematics Education, Wonkwang University, Iksan, Korea.E-Mail: [email protected]†Department of Mathematics, Central University of Kerala, Kasaragad 671328, Kerala, India.

E-Mail: [email protected]‡Department of Mathematics, Marudhar Engineering College, Raisar, Bikaner, Rajasthan

State, India. E-mail: [email protected]§University of Abertay Dundee, Dundee DD1 1HG, UK. E-Mail: [email protected]¶Corresponding author

282 Y. S. Kim, A. K. Rathie, U. Pandey, R. B. Paris

1. Introduction

The generalized hypergeometric function with p numeratorial and q denominatorialparameters is defined by the series [16, p. 41]

(1.1) pFq

[a1, a2, . . . , apb1, b2, . . . , bq

;x

]=

∞∑n=0

(a1)n(a2)n . . . (ap)n(b1)n(b2)n . . . (bq)n

xn

n!,

where (a)n = Γ(a+ n)/Γ(a) is the Pochhammer symbol (or ascending factorial). Whenq = p this series converges for |x| < ∞, but when q = p − 1 convergence occurs when|x| < 1 (unless the series terminates). In what follows we shall adopt the convention ofwriting the finite sequence of parameters (a1, . . . , ap) simply by (ap).

It is well known that whenever hypergeometric functions reduce to gamma functions,the results are very important from the applications point of view. Thus, the classicaltheorems of Gauss, Kummer and Bailey for the series 2F1, and of Watson, Dixon, Whip-ple and Saalschutz for the series 3F2, and others, play an important role in the theory ofhypergeometric and generalized hypergeometric series. In [3, 4, 5], Lavoie et al. consid-ered generalizations of some of the above-mentioned classical summations. In particular,they obtained a generalization of Gauss’ second summation theorem, given by

2F1

[a, b

12a+ 1

2b+ 1

2

; 12

]= π

12

Γ( 12a+ 1

2b+ 1

2)

Γ( 12a+ 1

2)Γ( 1

2b+ 1

2)

provided 12a+ 1

2b+ 1

26= 0,−1,−2, . . . , in the following form

1.1. Theorem. [5] Provided 12(a+ b+ j + 1) 6= 0,−1,−2, . . ., we have the summation

(1.2) 2F1

[a, b

12(a+ b+ j + 1)

; 12

]= π

12

Γ( 12a+ 1

2b+ 1

2j + 1

2)Γ( 1

2a− 1

2b− 1

2j + 1

2)

Γ( 12a− 1

2b+ 1

2+ 1

2|j|)

×

Aj(a, b)

Γ( 12a+ 1

2)Γ( 1

2b+ δj+1)

+Bj(a, b)

Γ( 12a)Γ( 1

2b+ δj)

for integer j. As usual, [x] denotes the greatest integer less than or equal to x, its modulusis denoted by |x| and we have defined

(1.3) δj ≡ 12j − [ 1

2j].

The coefficients Aj(a, b) and Bj(a, b) are displayed in Table 1 for 0 ≤ j ≤ 5, where thecoefficients for j < 0 satisfy

(1.4) A−j(a, b) = (−1)jAj(a, b), B−j(a, b) = (−1)j+1Bj(a, b).

Table 1. The coefficients Aj and Bj for 0 ≤ j ≤ 5.

j Aj(a, b) Bj(a, b)

0 1 0

1 −1 1

2 12(a+ b− 1) −2

3 − 12(3a+ b− 2) 1

2(3b+ a− 2)

4 14(a+ b− 1)(a+ b− 3) + ab −2(a+ b− 1)

5 − 14(a+ b− 2)(5a+ b− 4) + a(b+ 1) 1

4(a+ b− 2)(5b+ a− 4) + b(a+ 1)

Some summation formulas for the hypergeometric series r+2Fr+1(12 ) 283

Recently Miller [7], Miller and Paris [8, 9] and Miller and Srivastava [12] studiedthe generalized hypergeometric series (including integer parameter differences) and ob-tained numerous transformation and summation formulas for the series r+2Fr+1(z) and

r+1Fr+1(z). In our present investigation, we shall require the following result establishedin [8] expressing r+2Fr+1(z), with r numeratorial and denominatorial parameters dif-fering by the positive integers (mr), in terms of a finite sum of Gauss hypergeometricfunctions.

1.2. Theorem. [8] Let (mr) denote a set of positive integers with m = m1 + · · ·+mr.Then, when |z| < 1, we have

(1.5) r+2Fr+1

[a, b,c,

(fr +mr)(fr)

; z

]=

m∑k=0

(a)k(b)k(c)k

zkCk(r) 2F1

[a+ k, b+ k

c+ k; z

],

where the coefficients Ck(r) are defined by [10]

(1.6) Ck(r) =(−1)k

k!r+1Fr

[−k, (fr +mr)

(fr); 1

].

The expansion also holds when z = 1 provided Re (c− a− b) > m.

Alternatively, the coefficients can be expressed in the form [8]

(1.7) Ck(r) ≡ 1

Λ

m∑j=k

jk

σm−j , Λ = (f1)m1 . . . (fr)mr ,

with C0(r) = 1, Cm(r) = 1/Λ, where jk denotes the Stirling number of the second kindand the σj (0 ≤ j ≤ m) are generated by the relation

(1.8) (f1 + x)m1 · · · (fr + x)mr =

m∑j=0

σm−jxj .

The result (1.5) and (1.6) can also be deduced as a particular case of the more generalexpansion given by Luke in [6, Eq. (5.10.2(4))] combined with the fact that Ck(r) = 0for k > m [2, 10].

1.3. Remark. If we set z = 1 in (1.5), we immediately obtain the generalization of theKarlsson-Minton summation theorem [11, 12]

r+2Fr+1

[a, b,c,

(fr +mr)(fr)

; 1

]=

Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)

m∑k=0

(−)k(a)k(b)kCk(r)

(1 + a+ b− c)k

provided Re (c−a−b) > m. If c = b+1 and use is made of the alternative representationof the coefficients Ck(r) in (1.7), this reduces to the Karlsson-Minton summation formulagiven by [11, 13]

r+2Fr+1

[a, b,b+ 1,

(fr +mr)(fr)

; 1

]=

Γ(1 + b)Γ(1− a)

Γ(1 + b− a)

(f1 − b)m1 . . . (fr − b)mr

(f1)m1 . . . (fr)mr

when Re (−a) > m− 1.

Our aim in this paper is to obtain explicit expressions of

r+2Fr+1

[a, b,

12(a+ b+ j + 1),

(fr +mr)(fr)

; 12

]for complex parameters a, b and (fr) and for j = 0,±1, . . . ,±5, where r pairs of numera-torial and denominatorial parameters differ by positive integers (mr). For this purpose weshall make use of the expansion (1.5) combined with the generalization in (1.2) of Gauss’


second summation theorem. Several special and limiting cases of our main findings arealso presented.

2. Summation formulas

Our main result is given by the following theorem.

2.1. Theorem. Let (mr) be a set of positive integers with m = m1 + · · · + mr and letj be an integer. Then, the generalized hypergeometric function of argument 1

2and with

r pairs of numeratorial and denominatorial parameters differing by positive integers hasthe summation

(2.1) r+2Fr+1

[a, b,

12(a+ b+ j + 1),

(fr +mr)(fr)

; 12

]= π

12

Γ( 12a+ 1

2b+ 1

2j + 1

2)Γ( 1

2a− 1

2b− 1

2j + 1

2)

Γ( 12a− 1

2b+ 1

2|j|+ 1

2)

m∑k=0

(a)k(b)kCk(r)( 12)k

×

Aj(a+ k, b+ k)

Γ( 12a+ 1

2k + 1

2)Γ( 1

2b+ 1

2k + δj+1)

+Bj(a+ k, b+ k)

Γ( 12a+ 1

2k)Γ( 1

2b+ 1

2k + δj)

,

where the coefficients Aj and Bj for −5 ≤ j ≤ 5 can be obtained from the Aj(a, b) andBj(a, b) in Table 1 and use of (1.4) simply by putting a 7→ a + k and b 7→ b + k, and δjis defined in (1.3).

Proof. If we set z = 12

and c = 12(a+ b+ j + 1), where j is an integer, in (1.5) we obtain

r+2Fr+1

[a, b,

12(a+ b+ j + 1),

(fr +mr)(fr)

; 12

]=

m∑k=0

(a)k(b)kCk(r)

2k( 12a+ 1

2b+ 1

2j + 1

2)k

2F1

[a+ k, b+ k

12a+ 1

2b+ 1

2j + 1

2+ k

; 12

].

Application of (1.2) then leads to the result stated in (2.1). 2

To simplify the presentation of specific cases, we define the quantities D(1)± (k) and

D(2)± (k) by

D(1)± (k) ≡ 1

a± b

(Γ( 1

2a+ 1

2+ 1

2k)Γ( 1

2b+ 1

2k)± Γ( 1

2a+ 1

2k)Γ( 1

2b+ 1

2+ 1

2k)

),

D(2)± (k) ≡ 1

(a± b)2 − 1

((a+b+2k−1)Γ( 1

2a+ 1

2k)Γ( 1

2b+ 1

2k)

± 4Γ( 12a+ 1

2+ 1

2k)Γ( 1

2b+ 1

2+ 1

2k)

).

Then, upon use of the duplication formula for the gamma function

(2.2)√π Γ(2z) = 22z−1Γ(z)Γ(z + 1

2),

we have the following summations.

2.2. Corollary. The summations in (2.1) for j = 0,±1,±2, respectively, take the form

(2.3)

r+2Fr+1

[a, b,

12(a+b+1),

(fr +mr)(fr)

; 12

]=

2a+b−2

√π

Γ( 12a+ 1

2b+ 1

2)

Γ(a)Γ(b)

m∑k=0

2kCk(r) Γ( 12a+ 1

2k)Γ( 1

2b+ 1

2k),


(2.4)

r+2Fr+1

[a, b,

12(a+b+1)± 1

2,

(fr +mr)(fr)

; 12

]

=2a+b−1

√π

Γ( 12a+ 1

2b+1)

Γ(a)Γ(b)

m∑k=0

2kCk(r)D(1)∓ (k),

(2.5)

r+2Fr+1

[a, b,

12(a+b+1)± 1,

(fr +mr)(fr)

; 12

]

=2a+b−1

√π

Γ( 12a+ 1

2b+ 3

2)

Γ(a)Γ(b)

m∑k=0

2kCk(r)D(2)∓ (k).

When r = 0, the sequences (fr) and (fr +mr) are empty so that m = 0 and (2.3)–(2.5)reduce to (1.2).

2.3. Remark. The results for j = −2, −1, 0 and 1 have been obtained by Miller andParis [8]. However, the results for j = −2 and j = −1 are given here in corrected forms.

3. Specific examples

As specific examples, we let r = 1, m1 = 1 and f1 = f , so that C0(1) = 1, C1(1) = 1/f ,and thus obtain

(3.1) 3F2

[a, b,

12(a+ b+ 1)

f + 1f

; 12

]

= π12 Γ( 1

2a+ 1

2b+ 1

2)

1

Γ( 12a+ 1

2)Γ( 1

2b+ 1

2)

+2/f

Γ( 12a)Γ( 1

2b)

,

(3.2) 3F2

[a, b,

12(a+ b+ 1)± 1

2,f + 1f

; 12

]

= 2π12

Γ( 12a+ 1

2b+ 1)

a∓ b

(a∓ f)/f

Γ( 12a+ 1

2)Γ( 1

2b)∓ (b∓ f)/f

Γ( 12a)Γ( 1

2b+ 1

2)

,

(3.3) 3F2

[a, b,

12(a+ b+ 1)± 1,

f + 1f

; 12

]

= 2π12

Γ( 12a+ 1

2b+ 3

2)

(a∓ b)2 − 1

a+ b− 1∓ (2ab)/f

Γ( 12a+ 1

2)Γ( 1

2b+ 1

2)∓ 4∓ 2(a+ b+ 1)/f

Γ( 12a)Γ( 1

2b)

.

3.1. Remark. The summation in (3.2) with the upper signs has been obtained previouslyin [10] and [15].

The summations in (2.1) corresponding to j ≥ 1 are not valid when a−b = 0,±1, . . . ,±( 12j − 1

2) for odd j, and when a − b = ± 1

2, . . . ,±( 1

2j − 1

2) for even j. In such cases we

may use l’Hopital’s rule and the duplication formula (2.2) to obtain the limiting casesgiven below.


3.2. Corollary. When a = b, for example, we find the following summations correspond-ing to j = 1, 3 and 5 respectively:

(3.4) r+2Fr+1

[a, a,a+ 1,

(fr +mr)(fr)

; 12

]

(3.5) = 2a−1a

m∑k=0

Ck(r)(a)k

ψ( 1

2a+ 1

2k + 1

2)− ψ( 1

2a+ 1

2k)

,

(3.6) r+2Fr+1

[a, a,a+ 2,

(fr +mr)(fr)

; 12

]= 2a−1a(a+ 1)

m∑k=0

Ck(r)(a)k(2a + 2k − 1)ψ( 1

2a + 1

2k)− ψ( 1

2a + 1

2k + 1

2)+ 2

and

r+2Fr+1

[a, a,a+ 3,

(fr +mr)(fr)

; 12

]= 2a−1a(a+ 1)(a+ 2)

m∑k=0

Ck(r)(a)k(a+ k − 1

2)2 + 1

4ψ( 1

2a+ 1

2k + 1

2)− ψ( 1

2a+ 1

2k) − (a+ k − 1

2)

,

where ψ is the digamma or psi function.

3.3. Corollary. Similarly, if a− b = 1 when j = 2 and 4 we find the summations:

(3.7) r+2Fr+1

[a, a+ 1,a+ 2,

(fr +mr)(fr)

; 12

]

= 2aa(a+ 1)

m∑k=0

Ck(r)(a+ 1)k

ψ( 1

2a+ 1

2k + 1

2)− ψ( 1

2a+ 1

2k)− 1

a+ k

and

(3.8) r+2Fr+1

[a, a+ 1,a+ 3,

(fr +mr)(fr)

; 12

]= 2aa(a+ 1)(a+ 2)

m∑k=0

Ck(r)(a+ 1)k(a+ k)ψ( 1

2a+ 1

2k)− ψ( 1

2a+ 1

2k + 1

2)+ 1 +

1

2(a+ k)

.

Finally, it is worthy of mention that when r = 0 we retrieve respectively from (3.4)–(3.8), or from (1.2), the following results for the series 2F1( 1

2):

2F1

[a, aa+ 1

; 12

]= 2a−1a

ψ( 1

2a+ 1

2)− ψ( 1

2a),

2F1

[a, aa+ 2

; 12

]= 2a−1a(a+ 1)

(2a− 1)ψ( 1

2a)− ψ( 1

2a+ 1

2)+ 2

,

2F1

[a, aa+ 3

; 12

]= 2a−1a(a+ 1)(a+ 2)

(a− 12)2 + 1

4ψ( 1

2a+ 1

2)− ψ( 1

2a) − (a− 1

2),

2F1

[a, a+ 1a+ 2

; 12

]= 2aa(a+ 1)

ψ( 1

2a+ 1

2)− ψ( 1

2a)− 1

a


and

2F1

[a, a+ 1a+ 3

; 12

]= 2aa(a+ 1)(a+ 2)

aψ( 1

2a)− ψ( 1

2a+ 1

2)+ 1 +

1

2a

.

3.4. Remark. The first of the above 2F1( 12) summations is found in [1, p. 557] and in

[14, p. 492] written in slightly different form, but the remaining results are believed tobe new.

Acknowledgments

One of the authors (YSK) acknowledges the support of the Wonkwang UniversityResearch Fund (2012).

References

[1] Abramowitz, M. and Stegun, I. A. (Eds.) Handbook of Mathematical Functions (Dover, New

York, 1965).[2] Karlsson, P. W. Hypergeometric functions with integral parameter differences, J. Math.

Phys. 12, 270–271, 1971.

[3] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson’s theorem on thesum of a 3F2, Indian J. Math. 34, 23–32, 1992.

[4] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Dixon’s theorem on the sum

of a 3F2, Math. Comp. 62, 267–276, 1994.[5] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple’s theorem on the

sum of a 3F2, J. Comput. Appl. Math. 72, 293–300, 1996.[6] Luke, Y. L. Mathematical Functions and Their Approximations (Academic Press, New York,

1975).

[7] Miller, A. R. Certain summation and transformation formulas for generalized hypergeomet-ric series, J. Comput. Appl. Math. 231, 964–972, 2009.

[8] Miller, A. R. and Paris, R. B. Certain transformations and summations for the generalized

hypergeometric series with integral parameter differences, Integral Transforms and SpecialFunctions, 22, 67–77, 2011.

[9] Miller, A. R. and Paris, R. B. Euler-type transformations for the generalized hypergeometric

function r+2Fr+1(x), Zeitschrift angew. Math. Phys., 62, 31–45, 2011.[10] Miller, A. R. and Paris, R. B. On a result related to transformations and summations of

generalized hypergeometric series, Math. Communications, 17, 205–210, 2012.

[11] Miller, A. R. and Paris, R. B. Transformation formulas for the generalized hypergeometricfunction with integral parameter differences, Rocky Mountain J. Math., 43, 291-327, 2013.

[12] Miller, A. R. and Srivastava, H. M. Karlsson–Minton summation theorems for the gener-

alized hypergeometric series of unit argument, Integral Transforms and Special Functions,21, 603–612, 2010.

[13] Minton, B. M. Generalized hypergeometric function of unit argument, J. Math. Phys., 11,1375–1376, 1970.

[14] Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. Integrals and Series: More Special

Functions, vol. 3 (Gordon and Breach, New York, 1990).

[15] Rathie, A. K. and Pogany, T. K. New summation formula for 3F2( 12

) and a Kummer-type

II transformation of 2F2(x), Mathematical Communications, 13, 63–66, 2008.

[16] Slater, L. J. Generalized Hypergeometric Functions (Cambridge University Press, Cam-

bridge, 1966).


NEW INTEGRAL INEQUALITIES

VIA (α,m)-CONVEXITY AND

QUASI-CONVEXITY

Wenjun Liu∗


Abstract

In this paper, we establish some new integral inequalities involvingBeta function via (α,m)-convexity and quasi-convexity, respectively.Our results in special cases recapture known results.

Keywords: Hermite’s inequality, Euler Beta function, Holder’s inequality, (α,m)-convexity, quasi-convexity

2000 AMS Classification: 26D15, 33B15, 26A51, 39B62.

1. Introduction

Let I be an interval in R. Then f : I → R is said to be convex (see [17, P.1]) if

f (tx+ (1− t) y) ≤ tf (x) + (1− t) f (y)

holds for all x, y ∈ I and t ∈ [0, 1].In [27], Toader defined m-convexity as follows:

1.1. Definition. The function f : [0, b] → R, b > 0 is said to be m-convex, wherem ∈ [0, 1], if

f(tx+m(1− t)y) ≤ tf(x) +m(1− t)f(y)

holds for all x, y ∈ [0, b] and t ∈ [0, 1].We say that f is m−concave if −f is m−convex.

In [18], Mihesan defined (α,m)− convexity as follows:

1.2. Definition. The function f : [0, b]→ R, b > 0, is said to be (α,m)− convex, where(α,m) ∈ [0, 1]2, if

f(tx+m(1− t)y) ≤ tαf(x) +m(1− tα)f(y)

holds for all x, y ∈ [0, b] and t ∈ [0, 1].

∗College of Mathematics and Statistics, Nanjing University of Information Science and Tech-nology, Nanjing 210044, China E-mail:[email protected]

290 W. Liu

Denote by Kαm(b) the class of all (α,m)− convex functions on [0, b] for which f(0) ≤ 0.

It can be easily seen that for (α,m) = (1,m), (α,m)− convexity reduces to m− convexityand for (α,m) = (1, 1), (α,m)− convexity reduces to the concept of usual convexitydefined on [0, b], b > 0. For recent results and generalizations concerning m−convex and(α,m)−convex functions see [4, 6, 10, 19, 21, 26].

We recall that the notion of quasi-convex functions generalizes the notion of convexfunctions. More precisely, a function f : [a, b]→ R is said to be quasi-convex on [a, b] if

f(λx+ (1− λ)y) ≤ maxf(x), f(y)holds for any x, y ∈ [a, b] and λ ∈ [0, 1]. Clearly, any convex function is a quasi-convexfunction. Furthermore, there exist quasi-convex functions which are not convex (see [14]).

One of the most famous inequalities for convex functions is Hadamard’s inequality.This double inequality is stated as follows: Let f be a convex function on some nonemptyinterval [a, b] of real line R, where a 6= b. Then

(1.1) f

(a+ b

2

)≤ 1

b− a

∫ b

a

f(x)dx ≤ f (a) + f (b)

2.

Hadamard’s inequality for convex functions has received renewed attention in recentyears and a remarkable variety of refinements and generalizations have been found (see,for example, [1]-[19], [22]-[26], [28]). In [4], Bakula et al. establish several Hadamardtype inequalities for differentiable m−convex and (α,m)−convex functions.

Recently, Ion [14] established two estimates on the Hermite-Hadamard inequality forfunctions whose first derivatives in absolute value are quasi-convex. Namely, he obtainedthe following results:

1.3. Theorem. Let f : I ⊂ R→ R be a differentiable mapping on I, a, b ∈ I with a < b.If |f ′| is quasi-convex on [a, b], then the following inequality holds:∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(u)du

∣∣∣∣ ≤ b− a4

max

∣∣f ′(a)∣∣ , ∣∣f ′(b)∣∣ .

1.4. Theorem. Let f : I ⊂ R→ R be a differentiable mapping on I, a, b ∈ I with a 1. If |f ′|p

p−1 is quasi-convex on [a, b], then the following inequality holds:∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(u)du

∣∣∣∣ ≤ b− a2(p+ 1)

1p

(max

∣∣f ′(a)∣∣ pp−1 ,

∣∣f ′(b)∣∣ pp−1

) p−1p.

In [2], Alomari et al. obtained the following result.

1.5. Theorem. Let f : I ⊂ R→ R be a differentiable mapping on I, a, b ∈ I with a < band let q ≥ 1. If |f ′|q is quasi-convex on [a, b], then the following inequality holds:∣∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(u)du

∣∣∣∣ ≤ b− a4

(max

∣∣f ′(a)∣∣q , ∣∣f ′(b)∣∣q) 1

q .

In [20], Ozdemir et al. used the following lemma in order to establish several integralinequalities via some kinds of convexity.

1.6. Lemma. Let f : [a, b] ⊂ [0,∞)→ R be continuous on [a, b] such that f ∈ L([a, b]),a < b. Then the equality

(1.2)

∫ b

a

(x− a)p(b− x)qf(x)dx = (b− a)p+q+1

∫ 1

0

(1− t)ptqf(ta+ (1− t)b)dt

holds for some fixed p, q > 0.

Especially, Ozdemir et al. [20] discussed the following new results connecting withm−convex function and quasi-convex function, respectively:

New integral inequalities via (α,m)-convexity and quasi-convexity 291

1.7. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 0, then∫ b

a

(x− a)p(b− x)qf(x)dx

≤(b− a)p+q+1 min

β(q + 2, p+ 1)f(a) +mβ(q + 1, p+ 2)f

(b

m

),

β(q + 1, p+ 2)f(b) +mβ(q + 2, p+ 1)f( am

),(1.3)

where β(x, y) is the Euler Beta function.

1.8. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 0, we have∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1 maxf(a), f(b)β(p+ 1, q + 1).(1.4)

The aim of this paper is to establish some new integral inequalities like those given inTheorems 1.7 and 1.8 for (α,m)−convex functions (Section 2) and quasi-convex functions(Section 3), respectively. Our results in special cases recapture Theorems 1.7 and 1.8,respectively. That is, this study is a continuation and generalization of [20].

2. New integral inequalities for (α,m)− convex functions

2.1. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 0, then∫ b

a


≤(b− a)p+q+1 min

β(q + α+ 1, p+ 1)f(a) +m[β(q + 1, p+ 1)− β(q + α+ 1, p+ 1)]f

(b

m

),

β(q + 1, p+ α+ 1)f(b) +m[β(p+ 1, q + 1)− β(q + 1, p+ α+ 1)]f( am

),(2.1)


Proof. Since f is (α,m)−convex on [a, b], we know that for every t ∈ [0, 1]

(2.2) f(ta+ (1− t)b) = f

(ta+m(1− t) b

m

)≤ tαf(a) +m (1− tα) f

(b

m

).

Using Lemma 1.6, with x = ta+ (1− t)b, then we have∫ b

a


≤(b− a)p+q+1

∫ 1

0

(1− t)ptq(tαf(a) +m (1− tα) f

(b

m

))dt

=(b− a)p+q+1

[f(a)

∫ 1

0

(1− t)ptq+αdt+mf

(b

m

)∫ 1

0

(1− t)ptq (1− tα) dt

].

Now, we will make use of the Beta function which is defined for x, y > 0 as

β(x, y) =

∫ 1

0

tx−1(1− t)y−1dt.

It is known that ∫ 1

0

tq+α(1− t)pdt = β(q + α+ 1, p+ 1),

292 W. Liu∫ 1

0

(1− t)ptq (1− tα) dt =

∫ 1

0

tq(1− t)pdt−∫ 1

0

tq+α(1− t)pdt

=β(q + 1, p+ 1)− β(q + α+ 1, p+ 1)].

Combining all obtained equalities we get∫ b

a


≤(b− a)p+q+1

β(q + α+ 1, p+ 1)f(a) +m[β(q + 1, p+ 1)− β(q + α+ 1, p+ 1)]f

(b

m

).(2.3)

If we choose x = tb+ (1− t)a, analogously we obtain∫ b

a


≤(b− a)p+q+1β(q + 1, p+ α+ 1)f(b) +m[β(q + 1, p+ 1)− β(q + 1, p+ α+ 1)]f

( am

).(2.4)

Thus, by (2.3) and (2.4) we obtain (2.1), which completes the proof.

2.2. Remark. As a special case of Theorem 2.1 for α = 1, that is for f be m−convexon [a, b], we recapture Theorem 1.7 due to the fact that

β(q + 1, p+ 1)− β(q + 2, p+ 1) =β(q + 1, p+ 1)− q + 1

p+ q + 2β(q + 1, p+ 1)

=p+ 1

p+ q + 2β(q + 1, p+ 1) = β(q + 1, p+ 2)

and

β(q + 1, p+ 1)− β(q + 1, p+ α+ 1) = β(q + 2, p+ 1).

2.3. Corollary. In Theorem 2.1, if p = q, then (2.1) reduces to∫ b

a

(x− a)p(b− x)pf(x)dx

≤(b− a)2p+1 min

β(p+ α+ 1, p+ 1)f(a) +m[β(p+ 1, p+ 1)− β(p+ α+ 1, p+ 1)]f

(b

m

),

β(p+ 1, p+ α+ 1)f(b) +m[β(p+ 1, p+ 1)− β(p+ 1, p+ α+ 1)]f( am

).

2.4. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 1. If |f |

kk−1 is (α,m)−convex on [a, b], for some fixed (α,m) ∈ (0, 1]2

and p, q > 0, then∫ b

a


≤ (b− a)p+q+1

(α+ 1)k−1k

[β(kp+ 1, kq + 1)]1k min

[|f(a)|

kk−1 + αm

∣∣∣∣f ( b

m

)∣∣∣∣ kk−1

] k−1k

,

[|f(b)|

kk−1 + αm

∣∣∣f ( am

)∣∣∣ kk−1

] k−1k

.(2.5)


Proof. Since |f |k

k−1 is (α,m)−convex on [a, b] we know that for every t ∈ [0, 1]

|f(ta+ (1− t)b)|k

k−1 =

∣∣∣∣f (ta+m(1− t) bm

)∣∣∣∣ kk−1

≤tα|f(a)|k

k−1 +m (1− tα)

∣∣∣∣f ( b

m

)∣∣∣∣ kk−1

.

Using Lemma 1.6, with x = ta+ (1− t)b, then we have∫ b

a


≤(b− a)p+q+1

[∫ 1

0

(1− t)kptkqdt] 1

k[∫ 1

0

|f(ta+ (1− t)b)|k

k−1 dt

] k−1k

≤(b− a)p+q+1 [β(kq + 1, kp+ 1)]1k

[∫ 1

0

tα|f(a)|k

k−1 dt+m

∫ 1

0

(1− tα)

∣∣∣∣f ( b

m

)∣∣∣∣ kk−1

dt

] k−1k

=(b− a)p+q+1 [β(kq + 1, kp+ 1)]1k

[1

α+ 1|f(a)|

kk−1 +m

α

α+ 1

∣∣∣∣f ( b

m

)∣∣∣∣ kk−1

] k−1k

.


a


≤(b− a)p+q+1 [β(kp+ 1, kq + 1)]1k

[1

α+ 1|f(b)|

kk−1 +m

α

α+ 1

∣∣∣f ( am

)∣∣∣ kk−1

] k−1k

,

which completes the proof.


a


≤ (b− a)2p+1

(α+ 1)k−1k

[β(kp+ 1, kp+ 1)]1k min

[|f(a)|

kk−1 + αm

∣∣∣∣f ( b

m

)∣∣∣∣ kk−1

] k−1k

,

[|f(b)|

kk−1 + αm

∣∣∣f ( am

)∣∣∣ kk−1

] k−1k

.

2.6. Corollary. In Theorem 2.4, if α = 1, i.e., if |f |k

k−1 is m−convex on [a, b], then(2.5) reduces to∫ b

a


≤ (b− a)p+q+1

2k−1k

[β(kp+ 1, kq + 1)]1k min

[|f(a)|

kk−1 +m

∣∣∣∣f ( b

m

)∣∣∣∣ kk−1

] k−1k

,

[|f(b)|

kk−1 +m

∣∣∣f ( am

)∣∣∣ kk−1

] k−1k

.

294 W. Liu

2.7. Remark. As a special case of Corollary 2.6 for m = 1, that is for |f |k

k−1 be convexon [a, b], we get∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1

2k−1k

[β(kp+ 1, kq + 1)]1k

[|f(a)|

kk−1 + |f (b)|

kk−1

] k−1k.

2.8. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 0, then∫ b

a


≤(b− a)p+q+1 [β(p+ 1, q + 1)]l−1l

×min

[β(q + α+ 1, p+ 1)|f(a)|l +m[β(q + 1, p+ 1)− β(q + α+ 1, p+ 1)]

∣∣∣∣f ( b

m

)∣∣∣∣l] 1

l

,

[β(q + 1, p+ α+ 1)|f(b)|l +m[β(q + 1, p+ 1)− β(q + 1, p+ α+ 1)]

∣∣∣f ( am

)∣∣∣l] 1l

.(2.6)

Proof. Since |f |l is (α,m)−convex on [a, b], we know that for every t ∈ [0, 1]

|f(ta+ (1− t)b)|l =

∣∣∣∣f (ta+m(1− t) bm

)∣∣∣∣l ≤ tα|f(a)|l +m (1− tα)

∣∣∣∣f ( b

m

)∣∣∣∣l .Using Lemma 1.6, with x = ta+ (1− t)b, then we have∫ b

a


=(b− a)p+q+1

∫ 1

0

[(1− t)ptq]l−1l [(1− t)ptq]

1l f(ta+ (1− t)b)dt

≤(b− a)p+q+1

[∫ 1

0

(1− t)ptqdt] l−1

l[∫ 1

0

(1− t)ptq|f(ta+ (1− t)b)|ldt] 1

l

≤(b− a)p+q+1 [β(q + 1, p+ 1)]l−1l

×

[β(q + α+ 1, p+ 1)|f(a)|l +m[β(q + 1, p+ 1)− β(q + α+ 1, p+ 1)]

∣∣∣∣f ( b

m

)∣∣∣∣l] 1

l

.


a


≤(b− a)p+q+1 [β(p+ 1, q + 1)]l−1l

×[β(q + 1, p+ α+ 1)|f(b)|l +m[β(q + 1, p+ 1)− β(q + 1, p+ α+ 1)]

∣∣∣f ( am

)∣∣∣l] 1l

,




a


≤(b− a)2p+1 [β(p+ 1, p+ 1)]l−1l

×min

[β(p+ α+ 1, p+ 1)|f(a)|l +m[β(p+ 1, p+ 1)− β(p+ α+ 1, p+ 1)]

∣∣∣∣f ( b

m

)∣∣∣∣l] 1

l

,

[β(p+ 1, p+ α+ 1)|f(b)|l +m[β(p+ 1, p+ 1)− β(p+ 1, p+ α+ 1)]

∣∣∣f ( am

)∣∣∣l] 1l

.

2.10. Corollary. In Theorem 2.8, if α = 1, i.e., if |f |l is m−convex on [a, b], then (2.6)reduces to ∫ b

a


≤(b− a)p+q+1 [β(p+ 1, q + 1)]l−1l min

[β(q + 2, p+ 1)|f(a)|l +mβ(q + 1, p+ 2)

∣∣∣∣f ( b

m

)∣∣∣∣l] 1

l

,

[β(q + 1, p+ 2)|f(b)|l +mβ(q + 2, p+ 1)

∣∣∣f ( am

)∣∣∣l] 1l

.

2.11. Remark. As a special case of Corollary 2.10 for m = 1, that is for |f |l be convexon [a, b], we get∫ b

a


≤(b− a)p+q+1 [β(p+ 1, q + 1)]l−1l

[β(q + 2, p+ 1)|f(a)|l + β(q + 1, p+ 2) |f (b)|l

] 1l.

3. New integral inequalities for quasi-convex functions

3.1. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 1. If |f |

kk−1 is quasi-convex on [a, b], for some fixed p, q > 0, then∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1 [β(kp+ 1, kq + 1)]1k

(max

|f(a)|

kk−1 , |f(b)|

kk−1

) k−1k.(3.1)

Proof. By Lemma 1.6, Holder’s inequality, the definition of Beta function and the fact

that |f |k

k−1 is quasi-convex on [a, b], we have∫ b

a


≤(b− a)p+q+1

[∫ 1

0

(1− t)kptkqdt] 1

k[∫ 1

0

|f(ta+ (1− t)b)|k

k−1 dt

] k−1k

≤(b− a)p+q+1 [β(kq + 1, kp+ 1)]1k

[∫ 1

0

max|f(a)|

kk−1 , |f(b)|

kk−1

dt

] k−1k

=(b− a)p+q+1 [β(kq + 1, kp+ 1)]1k

[max

|f(a)|

kk−1 , |f(b)|

kk−1

] k−1k,


296 W. Liu

3.2. Corollary. Let f be as in Theorem 3.1. Additionally, if(1) f is increasing, then we have∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1 [β(kp+ 1, kq + 1)]1k f(b).

(2) f is decreasing, then we have∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1 [β(kp+ 1, kq + 1)]1k f(a).

3.3. Theorem. Let f : [a, b]→ R be continuous on [a, b] such that f ∈ L([a, b]), 0 ≤ a 0, then∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1β(p+ 1, q + 1)(

max|f(a)|l , |f(b)|l

) 1l,(3.2)


Proof. By Lemma 1.6, Holder’s inequality, the definition of Beta function and the factthat |f |l is quasi-convex on [a, b], we have∫ b

a


=(b− a)p+q+1

∫ 1

0

[(1− t)ptq]l−1l [(1− t)ptq]

1l f(ta+ (1− t)b)dt

≤(b− a)p+q+1

[∫ 1

0

(1− t)ptqdt] l−1

l[∫ 1

0

(1− t)ptq|f(ta+ (1− t)b)|ldt] 1

l

≤(b− a)p+q+1 [β(q + 1, p+ 1)]l−1l

[max

|f(a)|l , |f(b)|l

β(q + 1, p+ 1)

] 1l

=(b− a)p+q+1β(p+ 1, q + 1)(

max|f(a)|l , |f(b)|l

) 1l,


3.4. Corollary. Let f be as in Theorem 3.3. Additionally, if(1) f is increasing, then we have∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1β(p+ 1, q + 1)f(b).

(2) f is decreasing, then we have∫ b

a

(x− a)p(b− x)qf(x)dx ≤ (b− a)p+q+1β(p+ 1, q + 1)f(a).

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inequalities in Holder spaces, J. Appl. Math. Inform. 29 (2011), no. 3-4, 859–868.[14] D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex func-

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and Convex., Cluj-Napoca (Romania) (1993)

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STATISTICS


ON THE SEMI-MARKOVIAN

RANDOM WALK WITH DELAY AND

WEIBULL DISTRIBUTED INTERFERENCE

OF CHANCE

Tulay Kesemen ∗


Abstract

In this paper, a semi-Markovian random walk with delay and a discreteinterference of chance (X(t)) is considered. It is assumed that therandom variables ζn , n ≥ 1 which describe the discrete interferenceof chance have Weibull distribution with parameters (α, λ), α > 1, λ >0. Under this assumption, the ergodicity of this process is discussed andthe asymptotic expansions with three terms for the first four momentsof the ergodic distribution of the process X(t) are derived, when λ→ 0.Moreover, the asymptotic expansions for the skewness and kurtosis ofthe ergodic distribution of the process X(t) are established.

Keywords: Semi-Markovian random walk; a discrete interference of chance; Weibulldistribution; ergodic distribution; asymptotic expansion; ladder variables.

2000 AMS Classification: 60G50, 60K15, 60F99

1. Introduction

Many applied problems of the queueing, reliability, inventory control, insurance andother theories are formulated in the terms of random walks with various types of barrier.Some important studies on this topic exist in the literature (see, for example, [1–9]). Letus consider the following model before stating the problem mathematically.

The Model.Suppose that, the system is in state z = s+ x at the initial time t = 0.Here, s > 0 is a predefined control level, and x > 0. Demands and supplies are occurredat the random times Tn =

∑ni=1 ξi, n ≥ 1. System passes from a state to another one by

jumping at time Tn, according to quantities of demands and supplies ηn , n ≥ 1. Thischange of system continues until certain random time τ1, where τ1 is the first passage time

∗Karadeniz Technical University, Faculty of Sciences,Department of Mathematics, 61080 Tra-bzon, Turkey. E-mail:[email protected]

300 T. Kesemen

to the control level s. When this case happens, by interfering to the system from external,systems is stopped at the level s for a random time θ1. Usually the random variables θ1and K = Eθ1/Eξ1 are called as delaying time and delaying coefficient for the system,respectively. Then, as a consequence of external interference, system is brought from thecontrol level s to state ζ1. Thus, the first period has been completed. Afterwards, systemwill continue its function similar to the preceding period.

Note that, in the study [8], [2] and [1] the random variable ζ1, which describes thediscrete interference of chance, has an exponential, triangular and gamma distribution,respectively and the stationary moments of ergodic distribution were investigated whenthe delaying time is zero, i.e. θ1 = 0. But the delaying time is necessary for many realsystems. So, in this study, unlike [8], [2] and [1], the asymptotic expansions with threeterms for the first four moments of the ergodic distribution of the process X(t) will beinvestigated by taking into account the delaying time (θ1). Also in this study we assumethat the random variable ζ1 has Weibull distribution with parameters (α, λ), α > 1,λ > 0.

Our aim, in this paper, is to investigate the asymptotic behavior of the ergodic mo-ments of this process t ∈ [γn, τn+1), when λ→ 0.

2. Mathematical construction of the process X(t)

Let (ξn, ηn, θn, ζn) , n ≥ 1 be a sequence of independent and identically distributedvector of random variables defined on any probability space (Ω,F, P ), such that ξn andθn take only positive values, ηn takes negative values as well as positive ones; ζn hasWeibull distribution with parameters (α, λ), α > 1, λ > 0 . Suppose that ξ1, ηn, θ1, ζ1are mutually independent random variables and the distribution functions of them areknown, i.e.,

Φ (t) = P ξ1 ≤ t ; F (x) = P η1 ≤ x ; H(u) = P θ1 ≤ u ; t ≥ 0;

x ∈ (−∞,+∞)u ≥ 0 and π(z) = Pζ1 ≤ z = 1−e−(λz)α , z ≥ 0, α > 1, λ > 0.

Define renewal sequence Tn and random walk Sn as follows:

Tn =

n∑i=1

ξi, Sn =

n∑i=1

ηi, T0 = S0 = 0, n = 1, 2, . . . .

and a sequence of integer valued random variables Nn as:

N0 = 0,

N1 = inf n ≥ 1 : s+ x− Sn < s = inf n ≥ 1 : Sn > x = N(x), x ≥ 0;

Nn+1 = inf

k ≥ 1 : s+ ζn −

(N1+N2+...+Nn+k∑i=N1+N2+...+Nn+1

ηi

)< s

= infk ≥ 1 : SN1+N2+...+Nn+k − SN1+N2+...+Nn > ζn, n = 1, 2, . . .

Here s > 0 and inf∅ = +∞ is stipulated.

Let τ0 = γ0 = 0, τ1 = TN1 =∑N1i=1 ξi, γ1 = τ1 + θ1 = TN1 + θ1

τn = TN1+...+Nn +

n−1∑i=1

θi, γn = τn + θn = TN1+...+Nn +n∑i=1

θi, n ≥ 1.

Let’s construct the sequence of the counting processes:v0(t) ≡ v([0, t]) ≡ v(t) = maxn ≥ 0 : Tn ≤ t, t ∈ [0, τ1).vr(t) ≡ v([γr, t]) = maxn ≥ 0 : γr + (TN0+N1+...+Nn+n − TN0+N1+...+Nn) ≤ t,t ∈ [γr, τr+1), r = 0, 1, 2, . . .

On the semi-Markovian random walk with delay and . . . 301

By using these notations, the desired stochastic process X(t) is defined as follows:a)X(t) = s for all t ∈ [τn, γn)b)X(t) = s+ ζn − (SN0+N1+...Nn+vn(t) − SN0+N1+...Nn) for all t ∈ [γn, τn+1)

where n = 0, 1, 2, ...; γ0 = 0; ζ0 = x; N0 = 0.In this study, the process X(t) will be called ”a semi-Markovian random walk with

delay and Weibull distributed interference of chance”.The main purpose of this study is to investigate the asymptotic behavior of the sta-

tionary moments of the process X(t), as λ → 0. For this purpose, we first discuss theergodicity of the process X(t).

3. Preliminary discussions

Firstly, we can state the following lemma from [1].

3.1. Lemma. Assume that the initial sequence of the random vectors (ξn, ηn, θn, ζn),n ≥ 1 , satisfies the following supplementary conditions:1) Eξ1 <∞; 2) Eθ1 <∞; 3) 0 < Eη1 <∞; 4) E(η21) <∞5) η1 is non-arithmetic random variable;Then the process X(t) is ergodic.

3.2. Remark. Let’s now put ϕX(u) ≡ limt→∞

E exp(iuX(t)), u ∈ R. Using the basic

identity for the random walks (see, Feller W., [5], p.514) and 3.1 Lemma , we obtain thefollowing 3.3.

3.3. Lemma. Assume that assumptionts 3.1 are satisfied and the sequence of the randomvariables ζn , n ≥ 1, which describes the discrete interference of chance has Weibulldistribution with the parameters (α, λ), α > 1, λ > 0. Then for u ∈ R \ 0, the char-acteristic function ϕX(u) of the ergodic distribution of the process X(t) can be expressedby means of the characteristics of the pair (N(x), SN(x)) and the random variable η1 asfollows:

(3.1)

ϕX (u) =eius

EN(ζ1) +K

∫ ∞0

xα−1e−(λx)αeiuzϕSN(x)

(−u)− 1

ϕη (−u)− 1dx

+K eius

EN(ζ1) +K

∫ ∞0

xα−1e−(λx)αeiuzϕSN (x)(−u)dx,

where EN(ζ1) = αλα∫∞0xα−1e−(λx)αEN(x)dx; ϕSN(x)

(−u) = E exp(−iu SN(x));

ϕη(−u) = E exp(−iu η1); K = Eθ1/Eξ1.

4. Exact formulas for the first four moments of the ergodic dis-tribution of the process X(t)

The aim of this section is to express the first four moments of the ergodic distributionof the process X(t) by the characteristics of the boundary functional SN(x) and therandom variable η1. For this aim, introduce the following notations:

mk = E(ηk1 ), Mk(x) = E(SkN(x)), mk1 =mk

m1, Mk1(x) =

Mk(x)

M1(x), k = 1, 5, x ≥ 0;

E(ζn1Mk(ζ1)) = αλα∫ ∞0

xα+n−1e−(λx)αMk(x)dx, n = 0, 4, ek = E(ζk1 ), k = 1, 4;

and for the shortness of the expressions we put:

E(Xk) ≡ limt→∞

E((X(t))k), k = 1, 4 and X(t) ≡ X(t)− s.

We can now state the first main result of this section as follows.

302 T. Kesemen

4.1. Theorem. Let the conditions of 3.3 be satisfied and also E |η1|5 <∞. Then the first

four moments of the ergodic distribution of the process X(t) exist and can be expressed bymeans of the characteristics of the boundary functional SN(X) and the random variableη1 as follows:

(4.1)

E(X) =1

E(M1(ζ1)) +Km1

E(ζ1M1(ζ1)−

1

2E(M2(ζ1))

+1

2(m21 − 2Km1)E(M1(ζ1)) +Km1e1

;

(4.2)

E(X2) =

1

E(M1(ζ1)) +Km1

E(ζ

21M1(ζ1)− E(ζ1M2(ζ1)) +

1

3E(M3(ζ1))

+m21

(E(ζ1M1(ζ1))−

1

2E(M2(ζ1))

)+Km1 (E(M2(ζ1))

−2E(ζ1(M1(ζ1))) + A1E(M1(ζ1)) + Km1e2 ;

(4.3)

E(X3) =

1

E(M1(ζ1)) +Km1

E(ζ

31M1(ζ1)−

3

2E(ζ

21M2(ζ1)) + E(ζ1M3(ζ1))

−1

4E(M4(ζ1)) +

3

2(m21 − 2Km1)

(E(ζ

21M1(ζ1)− E(ζ1M2(ζ1))

)+

1

2(m21 − 2Km1)E(M3(ζ1)) + 3A1E(ζ1M1(ζ1))

−3

2A1E(M2(ζ1)) + 3A2E(M1(ζ1)) +Km1e3

;

E(X4) =

1

E(M1(ζ1)) +Km1

E(ζ

41M1(ζ1)− 2E(ζ

31M2(ζ1)) + 2E(ζ

21M3(ζ1)) − E(ζ1M4(ζ1))

+(m21 − 2Km1)

(2E(ζ

31)M1(ζ1))− 3E(ζ

21M2(ζ1)) + 2E(ζ1M3(ζ1)) +

1

2E(M4(ζ1))

)

+1

5E(M5(ζ1)) + 6A1

(E(ζ

21M1(ζ1))− E(ζ1M2(ζ1)) +

1

3E(M3(ζ1))

)(4.4) + 6A2 (2E(ζ1M1(ζ1))− E(M2(ζ1))) + 3A3E(M1(ζ1)) +Km1e4 ;

where A1 = m21

2; A2=

m221

2− m31

3; A3= m41

12− m31m21

3+

m321

4;

A4 =m4

21

4− m31m

221

2+m41m21

6+m2

31

9− m51

30,

ek = E(ζk1 ), k = 1, 4;K = Eθ1/Eξ1.

Proof. Note that the conditions of 4.1 provide the existence and finiteness of first fivemoments of SN(x) (see, Feller W., [5], p.514). And by using Taylor expansions for thecharacteristic functions of the variables η1 and SN(x), the exact expressions (4.1)-(4.4)for the first four ergodic moments of the process X(t) can be obtained.

5. Third-order asymptotic expansions for the first four momentsof the ergodic distribution

In this section, we will obtain asymptotic expansions for the first four moments of theergodic distribution of the process X(t). For this aim, we will use the ladder variables ofthe random walk Sn =

∑ni=1 ηi, n ≥ 1, with initial state S0 = 0.

Let ν+1 = minn ≥ 1 : Sn > 0, χ+1 = S

ν+1=∑ν+1i=1 ηi.

Note that, the random variables ν+1 and χ+1 are called the first strict ascending ladder

epoch and ladder height of the random walk Sn, n ≥ 0, respectively (see, Feller W., [5],p.391).

Let’s give the following lemma:


5.1. Lemma. Let g (x) (g : R+ → R) be a bounded function and limx→∞

g (x) = 0. Then

for any α > 1 the following asymptotic relation holds:

limλ→0

∫ ∞0

e−tg(t

1α

λ)dt = 0.

Proof. Under the conditions of 5.1, for any ε > 0, m(ε) > 0 exists such that for any

x ≥ m(ε), the inequality |g(x)| < ε holds. Choose b > 0, such that∫ b0e−tdt < ε. The

function g (x) is bounded. Therefore, for any λ < b1α

m(ε), we have:∣∣∣∣∣

∫ ∞0

e−tg(t

1α

λ)dt

∣∣∣∣∣ ≤∫ b

0

e−t

∣∣∣∣∣g(t

1α

λ)

∣∣∣∣∣ dt+

∫ ∞b

e−t

∣∣∣∣∣g(t

1α

λ)

∣∣∣∣∣ dt ≤≤ max

x≥0|g(x)|

∫ b

0

e−tdt+ ε

∫ ∞b

e−tdt ≤ εM + ε

∫ ∞0

e−tdt = ε(M + 1),

where M = maxx≥0|g(x)|.

Since M is finite and ε > 0 is arbitrary positive number, the proof of the 5.1 iscompleted.

Let’s give the following lemma, which proof is similar to proof of 5.1.

5.2. Lemma. Let g (x) be defined as in 5.1 and the function Rn(x) be defined as Rn(x) ≡xng(x), n = −1, 0, 1, 2, .... Then for each α > 1, the following asymptotic relation is true,when λ→ 0:∫ ∞

0

e−tRn(t

1α

λ)dt = o(

1

λn).

Now, we state the following auxiliary lemma, by using 5.1 in [7]:

5.3. Lemma. Let the condition E |η1|3 <∞ be satisfied. Then we can write the followingasymptotic expansions, as λ→ 0:

E (M1 (ζ1)) = E (ζ1) +1

2µ21 + o (λ) ,(5.1)

E (M2 (ζ1)) = E(ζ21)

+ µ21E (ζ1) +1

3µ31 + o (1) ,(5.2)

E (M3 (ζ1)) = E(ζ31)

+3

2µ21E

(ζ21)

+ µ31E (ζ1) + o

(1

λ

),(5.3)

E (M4 (ζ1)) = E(ζ41)

+ 2µ21E(ζ31)

+ 2µ31E(ζ21)

+ o

(1

λ2

),(5.4)

E (M5 (ζ1)) = E(ζ51)

+5

2µ21E

(ζ41)

+10

3µ31E

(ζ31)

+ o

(1

λ3

).(5.5)

Proof. Using 5.2 in this paper and 5.1 in [7], we can obtained the asymptotic expansions(5.1)-(5.5), as λ→ 0.

Now, we can state the first main result of this section as follows:

5.4. Theorem. Let the conditions of 4.1 be satisfied. Then the following asymptoticexpansion can be written for the first four moments of the ergodic distribution of the

304 T. Kesemen

process X(t), as λ→ 0:

E(X) = C21(α)1

λ+B11(α) +B12(α)λ+ o(λ),(5.6)

E(X2) = C31(α)

1

λ2+B21(α)

1

λ+B22(α) + o(1),(5.7)

E(X3) = C41(α)

1

λ3+B31(α)

1

λ2+B32(α)

1

λ+ o(

1

λ),(5.8)

E(X4) = C51(α)

1

λ4+B41(α)

1

λ3+B42(α)

1

λ2+ o(

1

λ2),(5.9)

where Ck(α) = Γ(1 + k/α); Ck1(α)= Ck(α)kC1(α)

, k = 1, 5, Γ(α) =∫∞0tα−1e−tdt,

B11(α) =1

2

[m21 −

C21(α)

C1(α)(µ21 + 2Km1)

],

B12(α) =C21(α)

4C21 (α)

(µ221 + 4µ21Km1 + 4K2m2

1)−

− 1

6C1(α)(µ31 + 3µ21m1K + 3m2K),

B21(α) =1

2

[2C21(α)m21 −

C31(α)

C1(α)(µ21 + 2Km1)

],

B22(α) =C31(α)

4C21 (α)

(µ221 + 4K2m2

1 + 4µ21Km1)− C21(α)

2C1(α)(m21µ21 − 2Km2)

+3m2

21 − 2m31

6,

B31(α) =3C31(α)

2m21 −

C41(α)

2C1(α)(µ21 + 2Km1),

B32(α) =C21(α)

2(3m2

21 − 2m31) +2C41(α)

3C21 (α)

(µ221 +

3

2µ21Km1 +

3

2K2m2

1)

− 3C31(α)

4C1(α)(m21µ21 + 2Km2),

B41(α) = 6C41(α)m21 −C51(α)

2C1(α)(µ21 + 2Km1),

B42(α) = C31(α) (3m221 − 2m31 + 6m21µ21 − 3µ31 − 12Km1µ21)

+C51(α)

4C21 (α)

(µ221 + 4µ21Km1 + 4K2m2

1)

− C41(α)

C1(α)(3m21µ21 − 3µ2

21 − 6Km2 − 10Km1µ21 − 8K2m21).

Proof. Firstly, we obtain the asymptotic expansion for the expectation of the ergodicdistribution of the process X(t), as λ → 0. For this aim, the exact formula (4.1) was

obtained for E(X) in 4.1. For the shortness, we put

(5.10) E(X) = R(λ) J(λ),

where R(λ)= 1E(M1(ζ1))+Km1

; J(λ) = J1(λ) + J2(λ);

J1(λ) = E (ζ1M1(ζ1))− 1

2E (M2(ζ1)) ;

J2(λ) =1

2(m21 − 2Km1)E(M1(ζ1)) +Km1e1.


By using Lemma 5.3 , we get the following expansion, as λ→ 0:

(5.11) J1(λ) =Γ(1 + 2/α)

2λ2− µ31

6+ o(1).

Using 5.3 , we obtain the following asymptotic expansion for J2(λ), as λ→ 0:

(5.12) J2(λ) =Γ(1 + 1/α)

2λm21 −

1

4(m21 − 2Km1)µ21 + o(1).

By using asymptotic expansions (5.11) and (5.12), we get:

(5.13) J(λ) =Γ(1 + 2/α)

λ2

[1

2+m21Γ(1 + 1/α)

Γ(1 + 2/α)λ+

+1

Γ(1 + 2/α)

(1

4(m21 − 2Km1)µ21 −

1

6µ31

)λ2 + o(λ2)

].

Analogically, we calculate:

(5.14)

R(λ) =λ

Γ((1 + 1/α)

[1− µ21 + 2Km1

2Γ(1 + 1/α)λ+

µ221 + 4µ21Km1 + 4K2m2

1

4Γ2(1 + 1/α)λ2 + o(λ2)

].

Taking into account the asymptotic expansions (5.13) and (5.14), we obtain the followingasymptotic expansion, as λ→ 0:

R(λ) J(λ) =Γ(1 + 2/α)

2Γ(1 + 1/α)

1

λ− 1

2

[m21 − (µ21 + 2Km1)

Γ(1 + 2/α)

Γ2(1 + 1/α)

]

+

[(µ2

21 + 4µ21Km1 + 4K2m21)

Γ(1 + 2/α)

8Γ3(1 + 1/α)

(5.15) − (µ31 + 3µ21Km1 + 3Km2)1

6Γ(1 + 1/α)

]λ+ o(λ).

Substituting (5.15) in (5.10), we finally get the asymptotic expansion (5.6) for E(X),as λ→ 0.

Now, we can analogically derive the asymptotic expansion for the second moment ofthe ergodic distribution of the process X(t). For this aim, the exact formula (4.2) was

obtained for E(X2) in 4.1. For the shortness, we put

(5.16) E(X2) = R(λ) J ′(λ),

where J ′(λ) = J3(λ) + J4(λ); J3(λ) = E(ζ21M1(ζ1)

)− E (ζ1M2(ζ1)) + 1

3E(M3(ζ1));

J4(λ) = (m21 − 2Km1)E(ζ1M1(ζ1))− 1

2(m21 − 2Km1)E(M2(ζ1))

+ A1E(M1(ζ1)) + Km1e2.

Using 5.3 , we obtain the following asymptotic expansion for J3(λ), as λ→ 0:

(5.17) J3(λ) =1

3

Γ(1 + 3/α)

λ3+ o(

1

λ).

Taking 5.3 into account, we write the following asymptotic expansion for J4(λ), as λ→ 0:

(5.18) J4(λ) =m21

2

Γ(1 + 2/α)

λ2+A1

Γ(1 + 1/α)

λ+A1µ21

2− (m21 −Km1)

6µ31 + o(1).

306 T. Kesemen

Using asymptotic expansions (5.17) and (5.18), we obtain the following asymptotic ex-pansion for J ′(λ), as λ→ 0:

(5.19)

J ′(λ) =Γ(1 + 3/α)

λ3

[1

3+m21

2

Γ(1 + 2/α)

Γ(1 + 3/α)λ+A1

Γ(1 + 1/α)

Γ(1 + 3/α)λ2

+

(A1µ21

2− (m21 −Km1)

6µ31

)1

Γ(1 + 3/α)α+ o(1)

].

Substituting asymptotic expansions (5.14) and (5.19) in the formula (5.16), and carry-ing out the corresponding calculation, we finally get the asymptotic expansion (5.7) for

E(X2), as λ→ 0.

Analogically, we can calculate the asymptotic expansions for the third and fourthmoments of the ergodic distribution of the process X(t).This completes the proof of 5.4.

5.5. Corollary. Let the conditions of 5.4 are satisfied. Then the following asymptoticexpansion can be written for the variance of the ergodic distribution of the process X(t),as λ→ 0:

V ar(X) =[C31(α)− C2

21(α)] 1

λ2+ [B21(α)− 2B11(α)C12(α)]

1

λ

+[B22(α)−B2

11(α)− 2C21(α)B12(α)]

+ o(1).

5.6. Remark. Thus, we obtained the asymptotic expansions for the first four ergodicmoments of the process X(t). Using these moments, it is possible to calculate skewness

(γ3) and kurtosis (γ4) of the ergodic distribution of X(t):

γ3 =E(X − a)3

σ3, γ4 =

E(X − a)4

σ4− 3 where a = E(X), σ2 = Var(X).

5.7. Corollary. Under the conditions of 5.4 , the following asymptotic expansions can bewritten for the skewness (γ3) and kurtosis (γ4) of the ergodic distribution of the process

X(t), as λ→ 0:

γ3 =C41(α)− 3C21(α)C31(α) + 2C3

21(α)

[C31(α)− C221(α)]

√C31(α)− C2

21(α)+O(λ),

γ4 =C51(α) + 6C31(α)C2

21(α)− 4C21(α)C41(α)− 3C421(α)

[C31(α)− C221(α)]2

− 3 +O(λ).

6. Conclusions

In this study, some stationary characteristics of the process X (t) are investigated byusing analytical and asymptotic methods, whenever the sequence of random variablesζn , n ≥ 1 which describes the discrete interference of chance has Weibull distributionwith parameters (α, λ), α > 1, λ > 0. The simple forms of the asymptotic expansionsallow us to observe how the initial random variables ξ1, η1, and ζ1 influence to the sta-tionary characteristics of the process X(t).

Note that it is important to obtain the similar results for other types of discreteinterference of chance by using the methods introduced in this paper.

Acknowledgments

I would like to express my regards to Prof. Dr. Tahir KHANIYEV (TOBB Univer-sity of Economics and Technology) and Asoc.Prof.Dr. Rovshan ALIYEV (Baku State


University, Department of Probability Theory and Mathematical Statistics) for their sup-port and valuable advices. Additionally, I would like to thank the referee, Editor andAssociate Editor for their careful reading, valuable comments, and patience.

References

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a semi-Markovian random walk with gamma distributed interference of chance, Communi-

cations in Statistics-Theory and Methods, 39 (1), 130–143, 2010.[2] Aliyev, R., Kucuk, Z. and Khaniyev, T. Three-term asymptotic expansions for the moments

of the random walk with triangular distributed interference of chance, Applied Mathematical

Modelling, 34 (11), 3599–3607, 2010.[3] Anisimov, V.V. and Artalejo, J.R. Analysis of Markov multiserver retrial queues with neg-

ative arrivals, Queueing Systems: Theory and Applic, 39 (2/3), 157–182, 2001.

[4] Borovkov, A.A. Stochastic Process in Queueing Theory, (Springer, New York, 1976) .[5] Feller, W. Introduction to Probability Theory and Its Appl. II,( New York, 1971 ).

[6] Gihman, I.I. and Skorohod, A.V. Theory of Stochastic Processes II, (Berlin, 1975 ).

[7] Khaniyev, T.A. and Mammadova, Z. On the stationary characteristics of the extended modelof type (s,S) with Gaussian distribution of summands, Journal of Statistical Computation

and Simulation,76 (10), 861–874 , 2006.[8] Khaniyev, T.A., Kesemen, T., Aliyev, R.T. and Kokangul, A. Asymptotic expansions for

the moments of a semi-Markovian random walk with exponential distributed interference of

chance, Statistics & Probability Letters, 78 (6), 785–793, 2008.[9] Lotov, V.I. On some boundary crossing problems for Gaussian random walks, The Annals

of Probability, 24 (4), 2154–2171, 1996.

[10] Rogozin, B.A. On the distribution of the first jump, Theory Probability and Its Applications,9 (3), 498–545, 1964.

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