A Study on Becker's Univalence Criteria - univie.ac.at · A Study on Becker’s Univalence Criteria Maslina Darus and Imran Faisal School of Mathematical Sciences, Faculty of Science

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2011, Article ID 759175, 13 pagesdoi:10.1155/2011/759175

Research ArticleA Study on Becker’s Univalence Criteria

Maslina Darus and Imran Faisal

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,Bangi, Selangor D. Ehsan 43600, Malaysia

Correspondence should be addressed to Maslina Darus, [email protected]

Received 26 January 2011; Accepted 11 May 2011

Academic Editor: Allan C. Peterson

Copyright q 2011 M. Darus and I. Faisal. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study univalence properties for certain subclasses of univalent functions K, K2, K2,μ, and S(p),respectively. These subclasses are associated with a generalized integral operator. The extendedBecker-typed univalence criteria will be studied for these subclasses.

1. Introduction and Preliminaries

Let A denote the class of analytic functions f in the open unit disk U = {z : |z| < 1} normal-ized by f(0) = f ′(0) − 1 = 0. Thus, each f ∈ A has a Taylor series representation

f(z) = z +∞∑

k=2

akzk· (1.1)

Let A2 be the subclass of A consisting of functions of the form

f(z) = z +∞∑

k=3

akzk· (1.2)

Let K be the univalent subclass of A which satisfies

∣∣∣∣∣z2f ′(z)(f(z))2 − 1

∣∣∣∣∣ < 1, z ∈ U. (1.3)

2 Abstract and Applied Analysis

Let K2 be the subclass of K for which f ′′(0) = 0. Let K2,μ be the subclass of K2 consisting offunctions of the form (1.2) which satisfy

∣∣∣∣∣z2f ′(z)(f(z))2 − 1

∣∣∣∣∣ < μ, 0 < μ ≤ 1, z ∈ U. (1.4)

Next, we define a subclass S(p) of A consisting of all functions f(z) that satisfy

∣∣∣∣∣

(z

(f(z)))′′∣∣∣∣∣ ≤ p, 0 < p ≤ 2, p ∈ R, z ∈ U. (1.5)

For functions f(z) = z +∑∞

k=2 akzk and g(z) = z +

∑∞k=2 bkz

k, the Hadamard product (orconvolution) f ∗ g is defined as usual by

(f ∗ g)(z) = z +

∞∑

k=2

akbkzk· (1.6)

Define the function ϕ(a, c; z) by

ϕ(a, c; z) = z2F(1, a, c; z) =∞∑

k=0

(a)k(c)k

zk−1, c /= 0,−1−, 2, . . . , (1.7)

where (a)kk is the famous Pochhammer symbol defined in terms of Gamma function. It iseasily seen that ϕ(2 − α, 2; z) is a convex function, since zϕ′(z) = ϕ(2 − α, 1; z) ∈ SV�(α)·

Using the fractional derivative of order α, Dαz [1], Owa and Srivastava [2] introduced

the operator Ωα : A → A which is known as an extension of fractional derivative and frac-tional integral, as follows:

Ωαf(z) = Γ(2 − α)zαDαzf(z), α /= 2, 3, 4, . . .

= z +∞∑

k=2

Γ(k + 1)Γ(2 − α)Γ(k + 1 − α)

akzk

= ϕ(2, 2 − α; z) ∗ f(z)·

(1.8)

Note that Ω0f(z) = f(z)·For a function f inA, we defineDn,ν

λ (α, β, μ)f(z) : A → A, the linear fractional differ-ential operator, as follows:

I0,νλ

(α, β, μ

)f(z) = f(z),

I1,νλ

(α, β, μ

)f(z) =

(ν − μ + β − λ

ν + β

)(Ωαf(z)

)+(μ + λ

ν + β

)z(Ωαf(z)

),

Abstract and Applied Analysis 3

I2,νλ

(α, β, μ

)f(z) = Iαλ

(I1,νλ

(α, β, μ

)f(z))

...

In,νλ

(α, β, μ

)f(z) = Iαλ

(In−1,νλ

(α, β, μ

)f(z))·

(1.9)

If f is given by (1.1), then by (1.8) and (1), we see that

In,νλ

(α, β, μ

)f(z) = z +

∞∑

k=2

((Γ(k + 1)Γ(2 − α)Γ(k + 1 − α)

)(ν +(μ + λ

)(k − 1) + β

ν + β

))n

akzk· (1.10)

From (1.8) and (1), Dn,νλ (α, β, μ)f(z) can be written in terms of convolution as

In,νλ

(α, β, μ

)f(z) =

[ϕ(2, 2 − α; z) ∗ gμ,ν

β,λ (z) · · ·ϕ(2, 2 − α; z) ∗ gμ,ν

β,λ (z)]

︸︷︷︸∗ f(z), (1.11)

where

gμ, ν

β,λ (z) =z − ((ν − μ + β − λ

)/(ν + β

))z2

(1 − z)2

= z −(ν − μ + β − λ

ν + β

)z2(1 + 2z + 3z2 + · · ·

)

= z +(1 +

μ + λ

ν + β

)z2 +(1 + 2

μ + λ

ν + β

)z3 · · ·

...

gμ, ν

β,λ (z) = z +∞∑

k=2

((ν +(μ + λ

)(k − 1) + β

ν + β

))zk,

(1.12)

ϕ(2, 2 − α; z) ∗ gμ,ν

β,λ (z) · · ·ϕ(2, 2 − α; z) ∗ gμ,ν

β,λ (z) = n-times product, (1.13)

which generalizes many operators. Indeed, if we choose suitably values of α, β, μ, and ν in(1.12), we have the following.

(i) β = 1, μ = 0, and α = 0, we obtain Dmα,λ

f(z) given by Aouf et al. [3].

(ii) ν = 1, β = 0, μ = 0, and α = 0, we obtain Dmλ f(z) given by Al-Oboudi [4].

(iii) ν = 1, β = 0, μ = 0, λ = 1, andα = 0, we obtain Dmf(z) given by Salagean [5].

(iv) ν = 1, β = 1, λ = 1,μ = 0, andα = 0, we obtain Imf(z) given by Uralegaddi andSomanatha [6].

(v) β = 1,λ = 1,μ = 0, andα = 0, we obtain Im(�)f(z) given by Cho and Srivastava [7]and Cho and Kim [8].


(vi) ν = 1, β = 0,μ = 0,λ = 0, andn = 1, we obtain Owa and Srivastava differentialoperator [2].

(vii) ν = 1, β = 0, andμ = 0, we obtain Dn,αλ f(z) given by Al-Oboudi and Al-Amoudi

[9, 10].

(viii) β = l, μ = 0, andα = p, we obtain Inp (λ, l)f(z) given by Catas [11].

(ix) β = l, μ = 0, α = p, and λ = 1, we obtain Inp (λ, l)f(z) given by Kumar et al. andSrivastava et al., respectively [12, 13].

Next, we introduce a new family of integral operator by using generalized differential oper-ator already defined above.

Form ∈ N∪{0} and γ1, γ2, γ3, . . . , γn, ρ ∈ C\{0,−1,−2, . . .}, we define a family of integraloperators Υγi,η,λ(n, ρ, ν, α, β) : A

m → Am by

Υγi,η,λ

(n, ρ, ν, α, β : z

)=

⎧⎨

⎩ρ

∫z

0tρ−1

m∏

i=1

(In,νλ

(α, β, μ, η

)fi(t)

t

)1/γi

dt

⎫⎬

⎭

1/ρ

, fi ∈ A, (1.14)

which generalize many integral operators. In fact, if we choose suitable values of parametersin this type of operator, we get the following interesting operators.

(i) ν = 1, β = 0,μ = 0,α = 0, γi = 1/αi, and ρ = 1, we obtain I(f1, . . . , fm) given by Bulut[14].

(ii) n = 0, ν = 1, β = 0, μ = 0,α = 0, γi = 1/(α− 1), and ρ = n(α− 1) + 1, we obtain Fn,α(z)given by Breaz et al. [15].

(iii) n = 0, ν = 1, β = 0,μ = 0,α = 0, γi = 1/αi, and ρ = 1, we obtain Fα(z) given by D.Breaz and N. Breaz [16].

For our main result, we need the following lemmas.

Lemma 1.1 (see [17, 18]). Let c be a complex number, |c| ≤ 1, c /= − 1. If f(z) = z + a2z2 + · · · is a

regular function in U and

∣∣∣∣c|z|2 +(1 − |z|2

)zf ′′(z)f ′(z)

∣∣∣∣ ≤ 1, ∀z ∈ U, (1.15)

then the function f is regular and univalent in U.

Lemma 1.2 (Schwarz Lemma). Let the function f(z) be regular in the disk UR = {z ∈ C : |z| < R}with |f(z)| < M. If f(z) has one zero with multiply ≥ m for z = 0, then

∣∣f(z)∣∣ ≤ M

Rm|z|m, ∀z ∈ UR, (1.16)

and equality holds only if f(z) = eiθ(M/Rm)|z|m, where θ is constant.


Lemma 1.3 (see [19]). Let δ be a complex number with Re δ > 0 such that c ∈ C, |c| ≤ 1, c /= − 1.If f ∈ A satisfies the condition

∣∣∣∣c|z|2δ +(1 − |z|2δ

)zf ′′(z)δf ′(z)

∣∣∣∣ ≤ 1, ∀z ∈ U, (1.17)

then the function

Fδ(z) ={δ

∫z

0tδ−1f ′(t)dt

}1/δ

(1.18)

is analytic and univalent in U.

Lemma 1.4 (see [20]). If a function f ∈ S(p), then

∣∣∣∣∣z2f ′(z)(f(z))2 − 1

∣∣∣∣∣ ≤ p|z|2, ∀z ∈ U. (1.19)

2. Univalence Properties

In this section, we will discuss the univalence properties of the new family of integraloperators mentioned above.

Theorem 2.1. Let c be a complex number, |In,νλ

(α, β, μ, η)fi(z)| ≤ Mi,Mi ≥ 1 for all i = {1, 2, 3, . . .}and In,ν

λ(α, β, μ, η)fi(t) ∈ S(pi) for i = {1, 2, 3, . . .} such that

R(ρ) ≥

∞∑

i=1

((Mi − 1)pi + 2

)Mi − 1

∣∣γi∣∣(Mi1)

, (2.1)

where ρ, γi are complex numbers. If

|c| ≤ 1 − 1Re(ρ)

∞∑

i=1

((Mi − 1)pi + 2

)Mi − 1

∣∣γi∣∣(Mi − 1)

, Mi ≥ 1, (2.2)

then the family Υγi,η,λ(n, ρ, ν, α, β : z) is univalent.

Proof. Since In,νλ (α, β, μ, η)fi(t) ∈ S(pi), so by Lemma 1.4, we have

∣∣∣∣∣∣

z2(In,νλ

(α, β, μ, η

)fi(t))′

(In,νλ

(α, β, μ, η

)fi(t))2 − 1

∣∣∣∣∣∣≤ pi|z|2, ∀z ∈ U. (2.3)

Now, by using hypothesis, we have

∣∣In,νλ

(α, β, μ, η

)fi(z)∣∣ ≤ Mi, (2.4)


so by Lemma 1.3, we get

∣∣In,νλ

(α, β, μ, η

)fi(z)∣∣ ≤ Miz, ∵ R = 1. (2.5)

Let

In,νλ

(α, β, μ

)f(z)

z= 1 +

∞∑

k=2

((Γ(k + 1)Γ(2 − α)Γ(k + 1 − α)

)(ν +(μ + λ

)(k − 1) + β

ν + β

))n

akzk−1

/= 0

= 1 if z = 0,(2.6)

so

m∏

i=1

(In,νλ

(α, β, μ, η

)fi(z)

z

)1/γi

=

(In,νλ

(α, β, μ, η

)f1(z)

z

)1/γ1

· · ·(

In,νλ

(α, β, μ, η

)fm(z)

z

)1/γm

= 1.(2.7)

Let

F(z) =∫z

0

⎛

⎝(

In,νλ

(α, β, μ, η

)f1(t)

t

)1/γ1

· · ·(

In,νλ

(α, β, μ, η

)fm(t)

t

)1/γm⎞

⎠dt, (2.8)

which implies that

F ′(z) =

⎛

⎝(

In,νλ

(α, β, μ, η

)f1(t)

t

)1/γ1

· · ·(

In,νλ

(α, β, μ, η

)fm(t)

t

)1/γm⎞

⎠′

zF ′′(z)F ′(z)

=1γ1

(z(In,νλ

(α, β, μ, η

)f1(z)

)′

In,νλ

(α, β, μ, η

)f1(z)

− 1

)+ · · · + 1

γm

(z(In,νλ

(α, β, μ, η

)fm(z)

)′

In,νλ

(α, β, μ, η

)fm(z)

− 1

),

(2.9)

zF ′′(z)F ′(z)

=m∑

i=1

1γi

(z(In,νλ

(α, β, μ, η

)fi(z))′

In,νλ

(α, β, μ, η

)fi(z)

− 1

). (2.10)

This implies that

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

(∣∣∣∣∣z(In,νλ

(α, β, μ, η

)fi(z))′

In,νλ

(α, β, μ, η

)fi(z)

∣∣∣∣∣ + 1

), (2.11)


or

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

⎛

⎝

∣∣∣∣∣∣

z(In,νλ

(α, β, μ, η

)fi(z))′

(In,νλ

(α, β, μ, η

)fi(z))2

∣∣∣∣∣∣

∣∣∣∣∣

(In,νλ

(α, β, μ, η

)fi(z))

(z)

∣∣∣∣∣ + 1

⎞

⎠· (2.12)

Using (2.5), we get

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

⎛

⎝

∣∣∣∣∣∣

z(In,νλ

(α, β, μ, η

)fi(z))′

(In,νλ

(α, β, μ, η

)fi(z))2

∣∣∣∣∣∣Mi + 1

⎞

⎠· (2.13)

This implies that

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

⎛

⎝

∣∣∣∣∣∣

z(In,νλ

(α, β, μ, η

)fi(z))′

(In,νλ

(α, β, μ, η

)fi(z))2 − 1

∣∣∣∣∣∣Mi +Mi + 1

⎞

⎠· (2.14)

By using (2.3), we get

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣(pi|z|2Mi +Mi + 1

), (2.15)

which implies that

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣(piMi +

(Mi +M2

i +M3i + · · ·

)+ 1), (2.16)

because Mi,M2i ,M

3i , . . . ,≥ 1 implies that

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

(piMi +

(Mi

Mi − 1

)+ 1)

=∞∑

i=1

1∣∣γi∣∣

(piMi +

(2Mi − 1Mi − 1

)),

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

((piM

2i − piMi + 2Mi − 1

Mi − 1

))≤

∞∑

i=1

1∣∣γi∣∣

(((piMi − pi + 2

)Mi − 1

Mi − 1

)),

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

((((Mi − 1)pi + 2

)Mi − 1

Mi − 1

))·

(2.17)

Now, we calculate

∣∣∣∣c|z|2ρ +(1 − |z|2ρ

)zF ′′(z)ρF ′(z)

∣∣∣∣ ≤ |c| + 1∣∣ρ∣∣

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤ |c| + 1R(ρ)∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣. (2.18)


This implies that

∣∣∣∣c|z|2ρ +(1 − |z|2ρ

)zF ′′(z)ρF ′(z)

∣∣∣∣ < |c| + 1R(ρ)

∞∑

i=1

1∣∣γi∣∣

((((Mi − 1)pi + 2

)Mi − 1

Mi − 1

))· (2.19)

By using (2.46), we conclude that

∣∣∣∣c|z|2ρ +(1 − |z|2ρ

)zF ′(z)ρF(z)

∣∣∣∣ ≤ |c| + 1∣∣ρ∣∣

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤ 1· (2.20)

Hence, by Lemma 1.3, the family of integral operators Υγi,η,λ(n, ρ, ν, α, β : z) is univalent.

Corollary 2.2. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ M,M ≥ 1 for all i = {1, 2, 3, . . .}and In,νλ (α, β, μ, η)fi(t) ∈ S(p), Mi = M ≥ 1, for all i = {1, 2, 3, . . .} such that

R(ρ) ≥

∞∑

i=1

((M − 1)p + 2

)M − 1

∣∣γi∣∣(M − 1)

, (2.21)


|c| ≤ 1 − 1Re(ρ)

∞∑

i=1

((M − 1)pi + 2

)M − 1

∣∣γi∣∣(M − 1)

, M ≥ 1, (2.22)


Corollary 2.3. Let c be a complex number, |In,νλ

(α, β, μ, η)fi(z)| ≤ M,M ≥ 1, for all i = {1, 2, 3, . . .}and the family In,νλ (α, β, μ, η)fi(t) ∈ S(p),Mi = M ≥ 1, |γi| = |γ |, for all i = {1, 2, 3, . . .} such that

R(ρ) ≥

∞∑

i=1

((M − 1)p + 2

)M − 1

∣∣γ∣∣(M − 1)

, (2.23)


|c| ≤ 1 − 1Re(ρ)

∞∑

i=1

((M − 1)pi + 2

)M − 1

∣∣γ∣∣(M − 1)

, M ≥ 1, (2.24)


Using the method given in the proof of Theorem 2.1, one can prove the followingresults.


Theorem 2.4. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ Mi,Mi ≥ 1 for all i = {1, 2, 3, . . .}and the family In,ν

λ(α, β, μ, η)fi(t) ∈ S(pi) for i = {1, 2, 3, . . .} and c such that

R(ρ) ≥

∞∑

i=1

(piMi − 1

)Mi + 1

∣∣γi∣∣piMi

, (2.25)


|c| ≤ 1 − 1Re(ρ)

∞∑

i=1

(piMi − 1

)Mi + 1

∣∣γi∣∣(piMi

) , Mi ≥ 1, (2.26)


Theorem 2.5. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ Mi, Mi ≥ 1 for all i ={1, 2, 3, . . . , n} and In,ν

λ(α, β, μ, η)fi(t) ∈ S(pi) for i = {1, 2, 3, . . . , n} such that

R(ρ) ≥

n∑

i=1

(pi(Mi − 1) +Mn

i − 2)Mi + 1

∣∣γi∣∣(Mi − 1)

, (2.27)


|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(pi(Mi − 1) +Mn

i − 2)Mi + 1

∣∣γi∣∣(Mi − 1)

, Mi ≥ 1, (2.28)


Theorem 2.6. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ Mi, Mi ≥ 1 for all i = {1, 2,3, . . . , n} and In,ν

λ(α, β, μ, η)fi(t) ∈ S(pi) for i = {1, 2, 3, . . . , n} such that

R(ρ) ≥

n∑

i=1

(pi + (n(n + 1)/2)

)Mi − 1

∣∣γi∣∣ , (2.29)


|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(pi + (n(n + 1)/2)

)Mi − 1

∣∣γi∣∣ , Mi ≥ 1, (2.30)



(α, β, μ, η)fi(z)| ≤ Mi, Mi ≥ 1 for all i = {1, 2,3, . . . , n} and In,ν

λ(α, β, μ, η)fi(t) ∈ K2,μi , for i = {1, 2, 3, . . . , n} such that

R(ρ) ≥

n∑

i=1

(μi + n(n + 1)

)Mi∣∣γi

∣∣ , (2.31)



|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(μi + n(n + 1)

)Mi∣∣γi

∣∣ , Mi ≥ 1, (2.32)


Proof. Using the proof of Theorem 2.1, we have

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

⎛

⎝

∣∣∣∣∣∣

z(In,νλ

(α, β, μ, η

)fi(z))′

(In,νλ

(α, β, μ, η

)fi(z))2

∣∣∣∣∣∣Mi + 1

⎞

⎠. (2.33)

Since In,νλ

(α, β, μ, η)fi(t) ∈ K2,μi , so by using (1.4), we get

∣∣∣∣∣∣

z2(In,νλ

(α, β, μ, η

)fi(z))′

(In,νλ

(α, β, μ, η

)fi(z))2 − 1

∣∣∣∣∣∣< μi, 0 < μ ≤ 1, z ∈ U. (2.34)

So from (2.33), we get

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣

⎛

⎝

∣∣∣∣∣∣

z(In,νλ

(α, β, μ, η

)fi(z))′

(In,νλ

(α, β, μ, η

)fi(z))2 − 1

∣∣∣∣∣∣Mi +Mi + 1

⎞

⎠, (2.35)

or

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣(μiMi + 2Mi

), Mi > 1,

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣(μiMi + 2Mi + 4Mi + · · · + n-times

), Mi > 1,

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤∞∑

i=1

1∣∣γi∣∣(μiMi + n(n + 1)Mi

), Mi > 1.

(2.36)

Now, we evaluate the expression

∣∣∣∣c|z|2ρ +(1 − |z|2ρ

)zF ′′(z)ρF ′(z)

∣∣∣∣ ≤ |c| + 1∣∣ρ∣∣

∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣ ≤ |c| + 1R(ρ)∣∣∣∣zF ′′(z)F ′(z)

∣∣∣∣,

∣∣∣∣c|z|2ρ +(1 − |z|2ρ

)zF ′′(z)ρF ′(z)

∣∣∣∣ ≤ |c| + 1R(ρ)

∞∑

i=1

1∣∣γi∣∣(μiMi + n(n + 1)Mi

).

(2.37)


Using (2.45) and (2.46), we conclude that

∣∣∣∣c|z|2ρ +(1 − |z|2ρ

)zF ′(z)ρF(z)

∣∣∣∣ ≤ 1. (2.38)

Hence by using Lemma 1.3, the family Υγi,η,λ(n, ρ, ν, α, β : z) is univalent.

Corollary 2.8. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ M, M ≥ 1 for all i = {1, 2,3, . . . , n} and In,νλ (α, β, μ, η)fi(t) ∈ K2,μi , for i = {1, 2, 3, . . . , n} such that

R(ρ) ≥

n∑

i=1

(μi + n(n + 1)

)M

∣∣γi∣∣ , (2.39)


|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(μi + n(n + 1)

)M

∣∣γi∣∣ , M ≥ 1, (2.40)


Corollary 2.9. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ M, M ≥ 1, |γi| = |γ | for alli = {1, 2, 3, . . . , n} and In,ν


R(ρ) ≥

n∑

i=1

(μi + n(n + 1)

)M

∣∣γ∣∣ , (2.41)


|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(μi + n(n + 1)

)M

∣∣γ∣∣ , M ≥ 1, (2.42)


Using a similar method as in the proof of Theorem 2.7, one can prove the followingresults.

Theorem 2.10. Let c be a complex number, |In,νλ (α, β, μ, η)fi(z)| ≤ Mi, Mi ≥ 1 for all i ={1, 2, 3, . . . , n} and In,ν


R(ρ) ≥

n∑

i=1

(μiMi − 1

)Mi +Mn

i Mi∣∣γi∣∣(Mi − 1)

, (2.43)



|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(μiMi − 1

)Mi +Mn

i Mi∣∣γi∣∣(Mi − 1)

, Mi ≥ 1, (2.44)



(α, β, μ, η)fi(z)| ≤ Mi, Mi ≥ 1 for all i ={1, 2, 3, . . .} and In,ν

λ(α, β, μ, η)fi(t) ∈ K2,μi , for i = {1, 2, 3, . . .} such that

R(ρ) ≥

n∑

i=1

(μiMi − μi + 2

)Mi − 1

∣∣γi∣∣(Mi − 1)

, (2.45)


|c| ≤ 1 − 1Re(ρ)

n∑

i=1

(μiMi − μi + 2

)Mi − 1

∣∣γi∣∣(Mi − 1)

, Mi ≥ 1, (2.46)


Note that some other related work involving integral operators regarding univalencecriteria can also be found in [21–23].

Acknowledgment

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010.

References

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