SOME ELEMENTARY INEQUALITIES, II

SOME ELEMENTARY INEQUALITIES, II

By GRAHAME BENNETT

[Received 13th May 1987]

1. Introduction

THIS paper is concerned with a classical inequality due to Copson, andwith certain of its variants. Copson's result ([6], page 255) asserts that

S t e j ^ Z < (0<r<l) (1)rt_l \k-n * / * = 1

whenever wlt. . ., wN are non-negative.We prove here two main results, the first a generalization of (1), the

second a sort of converse. Our first result is

THEOREM 1. Let 0 < r < s « l and let u, v, w be N-tuples with non-negative entries. If

sJ (2for m = 1, 2, . . . , N, then

N , N

Theorem 1 is to be compared with a similar result that has been provedalready in [3].

THEOREM 2. Let r>s^l and let u, v, w be N-tuples with non-negativeentries. If

( W )2 « - ( 5 > * W z v*) (4)for m = 1,2, . . . , N, then

) ( l ) ( Z A (5)

The most important special case of Theorem 2, obtained by settings = l, vk = l, un= n~r, is Hardy's inequality

N /-i n \r , \ r N

Quart. J. Math. Oxfofd (2), 39 (1988), 385-400 © 1988 Orfofd University Press

386 GRAHAME BENNETT

In fact the proof of Theorem 2 given in [3] is based upon a reduction of(5) to (6). (An alternative approach, via Linear Programming, has beengiven by Professor Boris Mitiagin. His proof is based upon a similarreduction.)

The motivation for Theorem 1 is threefold. First, Theorem 2 has manyapplications (see [3]), and the same is to be expected of any analogousresult. Second, the two theorems are very different, despite their similarappearances, and Theorem 1 demands that a new proof be found. Inparticular, it does not seem to be possible to prove (3) by means of areduction to (1). Third, once the "correct" hypothesis, (2), has beenformulated, the proof of Theorem 1 is quite straightforward, and it maybe modified to give a new proof of Theorem 2, a proof much shorter thanthe one given in [3].

If we take w in (5) to have the form (1, . . . , 1, 0,. . . , 0), it is obviousthat hypothesis (4) is "correct" for Theorem 2. Indeed there is theappearance, in passing from (4) to (5), of "gaining something fornothing". The same is not the case for Theorem 1 since no simplesubstitution for w in (3) leads us back to (2). Moreover, it is notimmediately obvious that Theorem 1 contains Copson's inequality as aspecial case. We shall see, however, that this is so, and that (2) is neededfor (3). Theorem 1, then, may be regarded as a definitive version ofCopson's inequality.

According to Hardy, Littlewood and Polya ([6], page 255), the resultsof Copson and Hardy are part of a "systematic group of four". (Theother two are mentioned in Section 3.) Our second main theorem is afifth inequality, very similar in spirit to these classical results. It is quitesurprising that Theorem 3 seems not to have been noticed before.

THEOREM 3. / / 0 < r < 1, then°° / I °° \ r JIT °°

2 I" 2 **) <7T— 2 supwi, (7)n-i\nk.n 1 sin nrnalk»n

unless wk = 0. The constant is exact.

A point of some interest is the determination of the constant, which ispossible here even in the N-dimensional version of (7) (see Lemma 4).

Theorem 1 is proved in Section 2 and then restated (Section 4) in termsof matrix transformations of V spaces. Applications are given in Sections3 and 5. Theorem 3 is proved in Section 6. In Section 7 we give a quickproof of an inequality of Astala, Gehring and Hayman.

It should be mentioned that all our results, with the exception ofCorollary 3, are finite-dimensional in nature, and have, for the most part,been stated in that form. Analogues are possible for infinite series, andfor integrals, but the formulation of these results is left to the reader.

SOME ELEMENTARY INEQUALITIES, II 387

2. Proof of Theorem 1

It will be convenient to assume that the N-tuples u, v, w have positiveentries. The non-negative case is then proved by a continuity argument.We need the following well-known lemmas.

LEMMA 1. Let a, b, c be N-tuples with non-negative entries, andsuppose that the ck's are decreasing. If

m m

X » ^ S ' ) * (m = l,2,...,N),

thenN N

E akCk ^ E bkCk.Proof. Summation by parts.

LEMMA 2. Let r be fixed, 0 < r « 1, and let a be an N-tuple with positiveentries. Then

N , N

(8)

The lemma is valid also for r ? l provided that the sign of inequality in(8) is reversed.

N

Proof. Setting ak = E at, we have

(with reversal when r > 1). Summing on k gives (8).In order to prove Theorem 1 we find it necessary to consider first the

special case, s = 1. Accordingly, we must prove

L^fR, (9)

whereN

n - l

and

(ii)

Y, un) , bk = Mt, and c t = ( E U/Wy I ,

388 GRAHAME BENNETT

our hypothesis, (2), givesN / k vr* / N \ r

( J (ZA (12)

(here, as usual, r* denotes the conjugate exponent of r, so thatr* = r / ( r - l ) . )

On the other hand, applying Lemma 2 to (10) gives

un E ( )k-n

By Holder's inequality we obtain, N , k xr', N \r\yr'

Ls.rW2«*(2«.) SW • (13)\k-l \n-\ I \-k ' I

From (12), (13), and the fact that r* < 0 we deduce that

which is equivalent to (9). This completes the proof of the special case,s = \.

To handle the general case we use a reduction to the special caseproved above. Assume, then, that 0<r<s< 1, and that (2) is satisfied.Applying Holder's inequality with exponents

p = ( l - r ) / ( l - j ) and p* = (1 - r)l(s - r), (14)

we have, for any positive /V-tuple x,

r!(r—s)\ \lp* m

) S

We may now apply the special case of the theorem, with vk replaced byxkvX"'\\M\\p\ to get

>t, (15)

for any positive N-tuple y. By setting xk = (ykwkl')Up, yk = wk''

p', and byrecalling (14), we see that (15) is equivalent to (3). This completes theproof.

We remark that Theorem 1 fails to hold when r >s; and when s > 1.This is the case no matter what constant K (positive, of course) is placedin the right-hand side of (3). If 0 < s < r, we take v = w = (1,0,. . . , 0)


and u = ev. It is clear that (2) is satisfied provided only that 0 < e ^ 1. Onthe other hand, (3) requires that e s* K, and this may be violated byletting e-*0. If s > 1 and s>rwe take un = 2" and vk = Cuk'

l'~r), wherethe constant C is chosen so that (2) is satisfied. Taking wk = vk'

lr, andallowing N—>°°, we see that the left-side of (3) grows like N, theright-side like N*. Thus (3) fails to hold no matter how small K > 0.

The constant, r7, of Theorem 1 is known to be exact when s = 1, butsome improvement may be possible when s < 1.

The following trivial modification of Theorem 1 is useful forapplications.

THEOREM V. Let 0<r<s^l and let u,v,w be N-tuples with non-negative entries. If

N i N y/(r-s)

22k-m

for m = 1, 2, . . . , N, then

(17)

Proof. Apply Theorem 1 to the N-tuples n ' ,v ' ,w' , obtained respec-tively from u, v, w by reversing the order of the coordinates.

The proof of Theorem 1 may be modified very easily to give

THEOREM 2'. Let r>s^\ and let u, v, w be n-tuples with non-negativeentries. If

m i k sr/(,r-i) , m \ (l-r)/(l-.r)

5>*(Z) ( 2 0for m = 1, 2, . . . , N, then

2un(2vkwk) ^ S r f . (19)

To prove Theorem 2' we need the following substitute for Lemma 2.

LEMMA 3. Let r be fixed, r & 1, and let a be an N-tuple with positiveentries. Then

rZOkllaj 3* I £ «y). (20)*=1 \y=i / \y-i /

The lemma is valid also for 0 < r < 1 provided that the sign of inequalityin (20) is reversed.

We remark that Theorem 2' is equivalent to Theorem 2, but the

390 GRAHAME BENNETT

equivalence is a good deal more subtle than that between Theorems 1'and 1. The proof uses the "method of sinister transposes," described in[3]. These observations explain why applications of Theorem 1 tend toarise in pairs while those of Theorem 2 occur in groups of four.

3. Copson-EUiot inequalities

In this section we show how to derive some classical inequalities fromTheorem 1. The derivation is routine, and involves nothing more thanmaking the appropriate substitutions for u, v and w. We find itnecessary, however, to pay some attention to detail since there arediscrepancies between results of Copson and ours. (See especially theremarks following Corollary 3 below.)

Throughout this section we suppose that xn > 0, An > 0, and An =(n = l, 2, . . . ) .

COROLLARY 1. If c^0<p<l, then

2n - l

n \P N

i 2A*ATex£ (21)-CI t _i

The constant is best possible.

Proof. We apply Theorem 1 with un = AnA~c, vk = A*(A?~C(1 -c)~p)m~"\ wk = Aic-"m-p)xk, s = landr=p. Under these substitutionswe see that (21) is equivalent to (3). To check that hypothesis (2) holdswe apply Lemma 3 with r = 1 — c.

Copson ([4], page 51) points out that the constant in (21) is sharp, byconsidering AB = 1, and by letting N—*™. The problem springs to lifeagain, however, if we fix N and klt . .. , kN, and we seek the largestpossible constant, K(p, c, N, A), say, so that (21) holds for all x. It can be

shown that K(p, c, <*>, A) = (- ) provided that An—xx> and AnA^"'^>0\1 — cl

(use Lemma 2), but the determination of K{p, c, °°, A) for general Aeludes me. The corresponding problem for N < °° is hopless^—even if weassume that kr = • • • = kN = 1. It is possible, however for this A, toestimate the rate of convergence of K(p, c, N, A) to ( ) . The

\ X C/

relevant techniques, which are quite ingenious, are described in Section4.2 of [11].

We remark that Corollary 1 continues to hold, with constant pp, evenwhen c>0 . (Copson [4], Theorem 2.3, insists that c be less than 1, butthis restriction is not needed.) I do not know the best constant in (21)when c > 0.

The most important special cases of Corollary 1 are obtained by setting


An = 1. With c = 0 we recover Copson's original inequality, (1); withc =p we obtain the following result ([6], page 255)

N /I N

2 (-2\p N

** j^p"!*"* (O<P<I) (22)

This inequality is weaker than (1), but is still of some interest since thebest constant in (22) has yet to be determined.

Our next result, due to Elliott ([6], page 252), is one of the "group offour" mentioned in Section 1. (The others are (1), (6) and its transpose.)

The interest here lies in the (first) constant, I 1 , which is known to\1 — pi

be exact. While Theorem 1 does not guarantee to give always the bestconstant, it is remarkable how often it does so. (If the first constant in(23) is held fixed, I do not know whether the second is exact.)

COROLLARY 2. IfO<p<l,then

N l\ N \P I n \P N 1 / N \P

2 (l 2 xk) *(j*-) 2 *S-:r-(2 xk) • (23)

Proof. Take ux = 1 + (1 -p)~\ un = n'p (n = 2, . . . , N), and vk = 1.

The next result, due to Copson ([4], Theorem 1.3), concerns infinite,rather than finite, series. It corresponds to Theorem 1' in the same waythat Corollary 1 does to Theorem 1. A complication arises, however,when we attempt to check hypothesis (16), and it is here that thehypothesis "An -> oo" is required.

COROLLARY 3. / / c > 1 > p > 0, and if An —*• oo, then

2 *nKc( 2 ^xX 3* l^-X 2 KK-Cx"n, (24)

where

L = inf(An/An+1).n

Proof. Omitted.

We remark that, in place of "An—• oo," Copson insists that " Z A x̂* =

oo." These hypotheses are not comparable, and it is obvious from hisproof that he meant "An—»<».".

The inequalities of this section have all been derived from the special

392 GRAHAME BENNETT

case, s = l, of Theorem 1. Allowing more general values of s leads tonatural and non-trivial extensions of these results. We close this sectionby stating one such extension (of Corollary 1) and leave the others for thereader to formulate.

COROLLARY 4. / / 0< <?=£/>< 1 and if c < 1, then

S U ^ E E ^ , (25)

where K is some positive constant that depends only on c, p and q.

4. Lower bounds for matrices

Let A be a matrix with non-negative entries. We say that A is p-*qbounded below if there exists a positive constant K = K(p, q) such that

||i4*||,&K||x||, (26)

for every non-negative sequence x. The largest such constant is denotedby L(A).

Lower bounds have received much less attention than upper bounds(i.e. norms). Their analyses have much in common, however, andproblems concerning lower bounds can lead to interesting new techniquesfor norms. (See, for example, Section 7 of [2], where a very simple proofof the classical Hilbert inequality is given.)

Our next result gives necessary and sufficient conditions for a fac-torable matrix to be bounded below. We recall ([3], Section 4) that afactorable matrix, A, is of the form

= lanbk l^k^n

* > „ • (27)

THEOREM 4. Let 0 < q ^ p < 1 and let A be the factorable matrix givenby (27). Then the following conditions are equivalent.

(i) A is p—*q bounded below;(ii) there exists a constant Kx such that

k-m

for m = 1,2, ...;(iii) There exists a positive constant K2 such that

(2 *•for m = 1,2, . . . .


Remarks. Since p*<0 , condition (ii) is to be interpreted as implying

that bk is positive, as is £ aqn, for each k.

n=k

Proof. (ii)=>(i). Suppose that (ii) holds and that x is a sequence withnon-negative entries. Setting r = q, s = q/p, un = aq

n and vk = b£"', we seethat all the hypotheses of Theorem 1' are satisfied. Applying (17) withWk = bw-p)Xk giVes \\Ai\L^KYp'q \\x\L, so that (i) holds with

q(iii)=£>(ii). Our proof uses condition (iii) twice, and Lemma 2.

Assuming, then, that (iii) holds, we have

k-m

Therefore (ii) holds with /C, = K^\l - q)~l.(i)=>(iii)- If (i) holds we have

\i\v<i / " \VP

Fixing m, and setting

ro if i=s

kSM*

shows that (iii) holds with K2 = L{A).We remark that the implication (ii)=>(i) shows that the converse to

Theorem 1' is valid. A similar result, corresponding to Theorem 1, maybe formulated for factorable matrices of the type

Ja"k = I0 1 * : ' : - (28)

The details are left to the reader.The results of this section are valid only for the range p^q. If p <q, I

do not know how to describe lower-boundedness for factorable matrices.(The most important special case however, in which an = n" and bk = k&,is solved in the next section.)

394 GRAHAME BENNETT

5. An inequality of Hardy-Littlewood

The original motivation for Copson's inequality, (21), was a desire togeneralize the following inequality of Hardy and Littlewood ([5],Theorem 1).

I, n-e(f, xX&K f, n-'iwcY (0<p,c<l). (29)n - l \ * = n ' n=l

Here K is a positive constant depending only on c and p, and the xn's arenon-negative numbers.

In this section we consider a generalization in a different direction. (29)may be viewed as asserting that the factorable matrix of type (28) isp ->p bounded below, where an =n~clp and bk = kc/p~1. Our next resultcompletes this description with

an=n'a, bk = k-P (30)

THEOREM 5. Let 0 <p,q < 1 and let A be the factorable matrix given by(28), (30). Then A is p^*q bounded below if and only if the followingcondition

(0 P*

(ii) p.

(iii) p<

is satisfied.

,Oora + P

1< — or a +

P*1

< - o r * +

/? =£

P<

1

1

(0<<

1

1V-o,

p* q p*

Proof. We label the points of the {a, /3)-plane as good, if the matrix Ahas a lower bound; bad, if not. If a point is good for a certain pair,(p, q), it will continue to be good when p is increased and q is decreased.This explains why the conditions of the theorem become more stringentas we read from (i) to (iv). A second general observation is that (a1, P')is good when (a, /3) is, provided that

a' + P'^a + P and /S'«/3. (31)

Part (i) is easy and will not be treated separately here. We remark thatthe exact lower bound, in this case, is


(Note that L(A) > 0 precisely when condition (i) is satisfied.) Part (ii) is aconsequence of Theorem 1.

The remaining case, p <q, is not covered by our previous work. In factan unexpected dichotomy arises according as whether p^q/(l + q) ornot. No such complication arises in the corresponding inequalities forl^p,q^co (see [3], Section 5).

A crucial role is played by the region G, where

Indeed, the only discrepancies between the four parts of the theorem alllie on the boundary of G.

We show that every point of G is good. Our proof uses firstMinkowski's inequality (in the space /*) and then Holder's inequality (inlp). We have , - , . » „,

( 2 ( 2 ' ) )\lq

. (32)

Thus the point (a, /3) is good provided that the series on the right isconvergent. This is certainly the case if (a, /3) e G (treat aq < 1 andaq > 1 separately).

The point (a, f$) = (-, —-), on the boundary of G, is what distin-\q p*l

guishes case (iii) from (iv). If we set

* * l 0 k>N' ( 3 3 )

and let Af—•<», ||x||p grows like (log log N)Vp and 11/4x11,, in case (iv),

converges. Thus (-, —) is a bad point, and this, together with our\q p )

second observation, (31), completes the proof of case (iv).Case (iii) is "sandwiched" between (ii) and (iv), by our first general

observation. It remains therefore, only to prove that ( - ,—-I is good,1 1 1 Kq P '

and that the points a< —, a + fi < - H—- are bad. The first follows as in<i IP

(32), the second by taking x as in (33), without the logarithm. An easycalculation then shows that \\A\\\q grows like (log N)Uq, \\x\\p like(logN)v".

396 GRAHAME BENNETT

We remark that there is an analogue of Theorem 5 for factorablematrices of the type (27). The details are left to the reader.

6. Proof of Theorem 3

An N-dimensional version of the theorem is given below in Lemma 4.Inequality (7), and the statement about the constant, follow by lettingN—K& in the lemma.

To see this, note that

siipk(N)= U

which we recognize as a /J-function,

0(1 + r, 1 - r) = rr(r)r( l -r) = rx cosec (rn).

This approach is to be contrasted with the methods used in proving the"group of four" ([6], Chapter 9). For these classical results there is noknown analogue of Lemma 4 (see [11]).

LEMMA 4. Let r be fixed, 0 < r < 1. Then

i i, (34)

for each natural number N, where

(35)

There is equality in (34) only when x is a multiple of (1 , . . . , 1).

Proof. Setting yn = max**, we have

N

ZJ X

k=n

\ r

7 ^N ,

n - 1 V

N

N

1 ZJk-n

where we have set yN+i = 0. The last inequality follows by "summationby parts in /" ' (see [2], Proposition 1).


Switching the order of summation and summing by parts again (in theusual sense, this time), gives

N

k=n

«max w£yi- (37)

This gives a weak version of (34) with h(N) replaced by max A(A:). We

shall see, however, in Lemma 8, that X(k) is strictly increasing, so wehave, in fact, proved the full result.

There is equality in (37) only when yk is constant, k = 1, 2, . . . , N\ andin (36) only when yk = xk.

The inequality represented by Lemma 8 seems to be genuinely difficult,and a simpler proof than the one given here would be most welcome.Our approach uses the theory of majorization, which is concerned withresults of the type

• • • + <p(xN) *£ 4>(yi) + ••• + <t>(yN), (38)

valid for all continuous convex functions <f>. Lemma 8, of course, is to beproved by making appropriate choices for x, y and <f>.

Accordingly, we set

which is (strictly) convex on (0,1). (Other, more "obvious" choices of <peither fail to be convex, or demand majorizations that are invalid.) Ourchoice, (39), in fact, suffers this last defect, and we find it necessary tomake an additional observation:

<p(x) decreases on (0,1/2). (40)

This allows us to work with weak majorization, x<wy, and the defectmentioned above disappears.

Our next result is due to Tomic, [9], see also Weyl, [10].

LEMMA 5. Suppose that a<xx<- • • <xN<b and a < yj < • • -<yN<b.Then, in order that (39) should hold for every continuous decreasingconvex function <p on (a, b), it is necessary and sufficient that

x1 + ---+xm^y1 + ---+ym (41)for m = 1, 2, . . . , N.

We remark that if <p is strictly convex, and if at least one of theinequalities in (41) is strict, then so is (38).

398 GRAHAME BENNETT

An excellent account of all these ideas is given by Marshall and Olkinin [8]. We follow their notation here by saying that x is weaklysupermajorized by y, and by writing x < wy, in case (41) holds.

The proof of our next result is left as an exercise.

LEMMA 6. Suppose that a and b are m-tuples whose entries areinterlaced, in the sense that

0<bx<ai<b2<a2<- • -<am<l. (42)

Let x be the N = 2m(m — 1) + m-tuple formed by repeating 2m timeseach of the first (m — 1) a's, and just m times the last. Similarly, let y bethe N = (2m — l)m-tuple obtained by repeating (2m — 1) times each ofthe b's. Then

x<"y (43)

provided that2m* 2m*

22 (44)

The point of the lemma, of course, is that we need to check only m — 1partial sums instead of all m(2m — 1) of them. It will be convenient towrite (43) symbolically as

2m(au . . . . «„,_,) ®m(am)<w(2m - 1)(6,, . . . . bm). (45)

LEMMA 7. We have

Proo/. We set a t = kl2m, bk = fc/(2/n + 1), and apply Lemma 6. It isclear that (42) holds; and (44) too, after an easy calculation.

LEMMA 8. Let r be fixed, 0 < r < «. Then the sequence X(N), defined in(35), is strictly increasing.

Proof. We must show that

for every positive integer N.If N is odd, say N = 1m - 1, we may rewrite (46) in the form

^) + mtf>(^-) < (2m - 1) f2ml r \2m/ v ' &

and this is an immediate consequence of Lemmas 5 and 7.


If N is even, a similar argument may be applied, using, in place ofLemma 7, the majorization

7. An inequality of Astala, Gehring and HaymanOur final inequality is a known result, similar in spirit to those

discussed above, but not directly related to them. Theorem 6 was firstproved by Astala and Gehring [1], the statements about the constant byHayman, [7]. Our purpose here is to give a quick proof of Hayman'sversion.

THEOREM 6. Let a be a sequence of non-negative numbers and supposethat oo

* » = 2 «*«*«„(« = 1,2, . . .) (47)k—n

for some constant fl(3=l). IfO<p< 1, thenoo

2«*«Ca?. (48)

where

BP-{B- \y •

There is equality in (48) only when

an=ai(l-l/By-1 (/i = l ,2 , . . . ) - (50)

Proof. We first observe that the function <p(x) = is strictly

increasing on 0<x=sl. If an^0, we deduce from (47) that<p(ana~l), which gives

^-<+1^(B-') f lX-' (51)Using (47) and (51) we have

D 1

B\-P

which is equivalent to (48), by (47) and (49).

400 GRAHAME BENNETT

Since <p is strictly increasing there can be (and there is) equality aboveonly when ana~l = B~x and this is the same as (50).

REFERENCES

1. K. Astala and F. W. Gehring, 'Quasiconformal analogues of theorems of Koebe andHardy-LJttlewood1, Mich. Math. Jour. 32 (1985), 99-107.

2. G. Bennett, 'Lower bounds for matrices', Linear Algebra 82 (1986), 81-98.3. G. Bennett, 'Some elementary inequalities', Quart. Jour. Math. Oxford (2), 38 (1987),

401-425.4. E. T. Copson, 'Note on series of positive terms', Jour. London. Math. Soc. 2 (1927),

9-12 and 3 (1928), 49-51.5. G. H. Hardy and J. E. Littlewood, 'Elementary theorems concerning power series with

positive coefficients and moment constants of positive functions', Jour, fur Math. 157(1927), 141-158.

6. G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd edition, CambridgeUniv. Press, 1967.

7. W. K. Hayman, 'A lemma in the theory of series due to Astala and Gehring', Analysis6(1986), 111-114.

8. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications,Academic Press, New York, 1979.

9. M. Tomic, 'Theortme de Gauss rclatif au centre de gravit6 et son application', Bull.Soc. Math. Phys. Serbie 1 (1949), 31-40.

10. H. Weyl, 'Inequalities, between two kinds of eigenvalues of a linear transformation',PTOC. Nat. Acad. Sci. U.S.A. 35 (1949), 408-411.

11. H. S. Wilf, Finite sections of some classical inequalities, Springer Verlag, Heidelberg,1970.

Department of MathematicsIndiana UniversityBloomingtonIndiana 47405U.S.A.

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SOME ELEMENTARY INEQUALITIES, II