SOME ELEMENTARY INEQUALITIES

SOME ELEMENTARY INEQUALITIES

By GRAHAME BENNETT

[Received 12 September 1986]

1. Introduction

OUR main result is the following inequality.

THEOREM 1. Let r>s^\ and let u, v, w be N-tuples with non-negativeentries. If

for m = l,2t...,N, (1)rt = l ^Jfc — l ^ V'Jt = l ^

H»/iere

(3)

It is to be noted that K is independent of N, and that hypothesis (1)does not involve w. There is thus the appearance, in passing from (1) to(2), of "getting something for nothing."

Examples are given to show that the result fails, no matter what thevalue of K, when s < 1, and when r ̂ s.

The most important special case of Theorem 1, obtained by settings = l, vk = 1, un = n~r, is Hardy's inequality:

N /I n \r i \r N

_i \n *_i / \r — 1/ t_i

Here it is known ([15], §9.8) that the constant is best possible, but thesame seems doubtful of (3) when s > 1.

Theorem 1 is proved in section 3, and then restated, in terms of matrixtransformations of lp-spaces, in section 4. The remaining sections dealwith applications.

Our work originated with our last application, which solves a problemraised by Littlewood. His problem is to decide whether a constant Kexists for which «

2 l 2 2 2 l (5)

Here An = ax + • • • + an, and the inequality is to hold for all non-negativenumbers alt a2, •. . •

Quart. J. Math. Oxford (2), 38 (1987), 401-425 © 1987 Oxford University Press

402 GRAHAME BENNETT

We discovered an easy proof of (5), with K = 2, under the additionalhypothesis that the a's form a decreasing sequence. The argument(section 7), a simple application of Hardy's inequality in the space I2, ismade possible by means of the trivial estimate: nanÂn. In theunrestricted case this estimate fails, and Hardy's inequality is no longerapplicable. It was the need for a substitute result, a "generalized Hardyinequality," that led us to study weighted mean matrices as operators onV-spaces. A complete description is given, in section 6, of the mappingproperties of these matrices.

We next realized that our results on weighted means carry over,without change, to a large class of matrices. These matrices, calledfactorable, are of the form: a^ = anbk if 1 «£ k «s n, 0 otherwise. Switchingto the larger class led to great simplifications in notation, but there wereother advantages as well. The class of factorable matrices enjoys certainsymmetries that are not shared by the weighted means. Exploiting thesesymmetries by, among others, the device of "sinister transposes," greatlyincreased the scope of our theorems, even for weighted means. All this isexplained in detail in section 4.

Theorem 2 of that section provides a unified approach to inequalities ofHardy-Copson-Leindler, Hardy-Littlewood, Izumi-Izumi-Petersen,Borwein-Jakimowski, Rhaly-Liebowitz, and of Cartridge. These resultsmay all be viewed as asserting the boundedness, on some V-space, of anappropriate matrix transformation. Theorem 2 gives analogous resultswhen more than one space may be involved, and often under reducedhypotheses. (See sections 5 and 6.)

These results enabled us to prove a very general version ofLittlewood's inequality (Theorem 4). In place of the "averaging" in t1,indicated above, it turns out, surprisingly, to be more efficient to work in/ i This route shows that (5) holds, in the general case, with K = \.

Our work is elementary in nature ([21], page 151) that is, it involvesonly (finite sets of) variables that are discrete and non-negative. All ourresults, however, are "homogeneous in 2 " ([15], page 4), and thereforeadmit analogues for integral transformations of LP-spaces. We leave theformulation of these integral inequalities to the reader.

I have discovered a simple, direct proof of Littlewood's inequality, (5),and this will be published in [3]. The result (Theorem 4) presented here,however, is more general than the one in [3], and we have added asupplement (Theorem 5) concerning the converse to Littlewood'sinequality.

2. NotationWe shall be concerned with matrix transformations of F-spaces. Here,

as usual, V denotes the space of real-valued sequences x satisfying

SOME ELEMENTARY INEQUALITIES 403

where

and

For l ^ p ^ o o , we denote the conjugate exponent by p* so that p*pl(p — \) with the obvious convention when p = 1 or /? = °°.

The norm of a matrix ^4, mapping P into lq, is given by

where

3. Proof of Theorem 1We make three reductions: first, to the case in which w is "decreas-

ing;" next, to the case 5 = 1; and finally to (4). The proof of (4) is left asan exercise!

The reader may wish to assume that n and v have positive entries, forthen the proof flows much more smoothly. The general case (u, vnon-negative) demands that considerable attention be paid to detail—themore so since we wish to determine the cases of equality in (2).

Our first reduction is based on an idea of Wiener, [29], subsequentlyrefined by Hardy ([10], Theorem B). The idea is to show that (2) holds ingeneral if it holds whenever w is decreasing.

Fixing N, and non-negative iV-tuples u, v, we say that w is"decreasing" provided that wt 2* MJ- whenever

i <j and both vit vt and at

least one of ut,..., uN are positive.

(If u, v are positive, this definition reduces to the usual one.) The bestpossible constant K, appearing in (2), is attained by some non-negativeN-tuple w; we show that such w must be "decreasing."

If w is not "decreasing," there exist integers i, j , with \î<j^N,such that

w, < Wj (6)and

N

v,>0, Vj>0 and 2 "n > 0. (7)

404 GRAHAME BENNETT

We alter the ith and yth coordinates of w to obtain a new TV-tuple, w', asfollows.

Let

w'l=w = w'j, (8)

where w is chosen so that

(u, + Vj)wr" = ViW"/* + Vjwr/'. (9)

We note, from (8) and (9), that the right side of (2) is unchanged when wis replaced by w'. Moreover, since w is an L^-average (with respect to atwo-point probability measure) of H; and wjt we have, from (6),

Wi < w < W). (10)

We complete the reduction by showing that the left side of (2) increasesstrictly when w is replaced by w'. To see this, we note that, by (8) and(10),

n n

E vkwk =£ X vk">'k if n <j. (11)* - i *=i

On the other hand, from (9),

(i/,- + Vj)w > ViW, + VM, (12)

since an Z//j-average is bigger than an L'-average. Using (8) and (12), wesee that (11) holds, with strict inequality, when n 2=7. Finally, from (7), atleast one of the summands,

( n \r

X vkwk) n=j,. . . ,N,k"\ 'of the left side of (2), increases strictly when w is replaced by w'. Thus K

is not attained at w.Our next step is the reduction to the case s = 1. Assume, then, that the

theorem has been proved when 5 = 1, with K(r, 1) = (r*)r. If, now, s > 1,let x be a non-negative TV-tuple with

11*11,- = 1- (13)

Assuming (1) holds, Holder's inequality gives1/j


for m = 1, . . . , N. Applying the theorem, assumed true, with un replacedby xnu)i', r by r/s, and s by 1, gives

£ xnu?(i, vkWlX"^K(r/s, 1) 2 vkwi'.

Taking the supremum on the left, over all x satisfying (13), and applyingthe converse to Holder's inequality ([15], page 26), we obtain

( N , n

2 «»(2« K(r/s, 1)

k-1

which is equivalent to (2). This completes our second reduction.We must now prove

2 M , ( 2 VkWk)'*(rmy 2 vkwrk, (14)

n-l \*=1 / * - l

under the hypotheses: r > 1,

^ 2 ^ . m = l , . . . , M (15)

and w "decreasing." If v = 0, our problem is trivial so we assume thatvÔ. If vl = 0, we may delete the first components of u, v and w toobtain an equivalent problem with (Af — l)-ruples. Repeating this proc-ess, as necessary, we may assume that v1>0. A similar argument,deleting last components, allows us to assume that M̂T > 0.

Since w is "decreasing," it now follows that y is decreasing (in theusual sense), where

Summation by parts gives

2 *»>>:> o (i6)n - l

for every iV-tuple x satisfyingm

2 * n > 0 for m = l,...,N. (17)n - l

Setting

xn = vn-un^2j vkj ,

406 GRAHAME BENNETT

we see, from (15), that (17) holds. Applying (16) gives

n _ i \ Vx + • • • +Vn

Our problem is thus reduced to showing that

under the hypotheses: r > 1, w "decreasing," vx > 0. If any vn is zero,then deleting vn and the corresponding wn does not affect (18). We maythus assume that all vn > 0, and hence that w is decreasing. Furthermore,we may assume that each vn is rational, and, by homogeneity, that eachvn is a positive integer. Since w is decreasing, we have

- • • + vn.lwn.1 + kwn

for k = 1, . . . , vn. Thus the left side of (18) is dominated by

/ A VH- + • • • + V^Wn-x + kWHY)

which is the rth-power-sum of the arithmetic averages of the sequence

Wi, . . . ,Wi, . . . , WN, . . . , WN.

(18) is therefore a consequence of Hardy's inequality. •

We complete this section by showing that Theorem 1 fails when r as s,and also when ^ < 1 . If 0<r^s, we take v = (1 ,1 , . . . , 1), w =(1,0,. . . , 0) and un = Cn'~r~x, where the constant C is chosen so that (1)is satisfied. Since the series Zun diverges, (2) fails to hold for large N, nomatter what the value of K.

If 0 < 5 < l and r>s, we take vk = 2k, wk = vk'/r and uk = Cv'k~

r,where, again, the constant C is chosen so that (1) is satisfied. As N—»°othe left side of (2) grows like N, the right side like N*, so that (2) fails tohold no matter how large K.

4. Factorable matrices

In this section we study the mapping properties of matrices, A =(an*)n,*-i> of the type

0 k>n


Such matrices have been called factorable (see, for example, [30], p.649). The entries of A may be complex numbers, but the norm of A isunchanged when the entries are replaced by their absolute values. Thuswe shall assume throughout this section that the a's and b's arenon-negative.

We begin with a comment that serves to illuminate Theorem 2 below.The matrix A may be regarded as the lower-triangular "piece" of therank-one operator, a ® b, so that a sufficient condition for the bounded-ness of A from V to lq is

(20)

This condition is not necessary, even when p = q, as may be seen bytaking an = n~x, bk = 1, and by recalling Hardy's inequality. The correctcondition, then, must be somewhat weaker than (20); Theorem 2provides three such conditions.

THEOREM 2. Let l<p<.q<<*>, let a and b be sequences of non-negativenumbers, and let A be the matrix given by (19). Then the followingconditions are equivalent.

(i) A maps I" into lq;(ii) there exists Kx such that, for m = 1, 2, . . . ,

n \ tj / m \ tj/pV hJ>'\ -a V I V W"lZJ °k } ~~ A-ll ZJ °k I .

n-1 \ k-\ I \*-l /

(iii) there exists K2 such that, for m = 1,2, . . . ,

( y a*^ ^ y /JP*^ < A!"

(iv) tfiere erwte /C3 5uc/t f/urt, /or m = 1, 2, . . . ,

V /k Vk—m ^ n«»*

Proof. Our proof follows the logical cycles (ii) ̂ (i) >̂ (iii) => (ii) and(iv)=>(i)=>(iii)=>(iv). Indeed, if the K's satisfying (ii), (iii) and (iv) arechosen as small as possible, we obtain the following quantitativeestimates: \\A\\p,q*ip*K\«, K2^\\A\\p,q, K^qKl and l l ^ l l p . ^ ^ ' " ' ,K3^p*K^\ (The multiplicative constants appearing in all five of theseestimates are sharp, at least when q =p. When q >p some improvementmay be possible.)

Our result is finite-dimensional in nature, and it will be convenient toprove it in this form. Accordingly, we take a and b to be N-tuples, andwe work with the N X N matrix A given by (19). The theorem, as stated,then follows by letting N—»<».

408 GRAHAME BENNETT

We prove only the first logical cycle, since both are equivalent. To seewhy this is so, simply replace (alf . . . , aN) by {bN, . . . ,bx) and(bu.. . , bN) by (aN, ..., a,). The resulting factorable matrix, which wedenote by A', is the one obtained from A by reflection in the sinisterdiagonal (i.e., in the line n + k = N + l). It follows, therefore, that||/4*||,.,p. = ||i4||Pj,. If, further, we replace p by q* and q by p*, we seethat the hypothesis, Kpss^ssoo, and the conditions (i) and (iii) areunchanged, while (ii) and (iv) are interchanged. It suffices, therefore, toprove the equivalence of conditions (i), (ii) and (iii).

(»)=>(/). Let x be an arbitrary non-negative N-tuple. If (ii) holds, wemay apply Theorem 1 with un = a^/Klt vk = b%', r = q and s = q/p.Defining w by \xkb

yk^ tf6 > 0

I 0 otherwise,inequality (2) gives

\\Ax\\lso that (i) holds and

(i) => (iii). Let m be fixed, l^m^N, and suppose that (i) holds. If xand y are N-tuples, we have, by Holder's inequality,

\\A\\p,q ||x||p ||y||,.. (21)n.k-l

Setting xk = bk/(p~l) when l=£Jt=sm and 0 otherwise, yn=aq

n~l when

and 0 otherwise, (21) givesS m i m \Vp I N \llq'

so that (iii) holds, with K2^ \\A\\Piq.

(iu)4>(u). Let Bn = E bi' for n = 1, . . . , N, and let Bo = 0. If (iii)

holds, and l^m^N, we have,

k-n

N

n - l

so that (ii) holds, with


Theorem 2 applies also to matrices of the type

Onk = flmax(n,*)&min(/i,*) («» k = 1, 2, . . . ) .

We call such matrices L-shaped. The lower triangular piece of A isfactorable, as is the transpose of the upper triangular piece. Thus, if aand b are both non-negative, Theorem 2, applied separately to the twopieces, gives necessary and sufficient conditions for A to map V into /',Kp *£q <°°. We leave the details to the reader.

Theorem 2 contains many known inequalities as special cases and welist some examples in the next section. These examples are all of the typeq =p, and are, in fact, consequences of the following, relatively crude,"pointwise" versions of conditions (ii) and (iv). Condition (iii) ofTheorem 2, despite its simpler appearance than those of (ii) and (iv),plays a relatively minor role in the examples.

COROLLARY 1. Let p be fixed, K p < ° ° . / /

(22)*-i

for n = 1, 2, . . . , then A is bounded on lp, and

\\A\\p,p**Kp*.

The constant p* is best possible.

Proof. Apply Theorem 2, (ii)=>(i), with q=p. The hypothesis (22)guarantees that (ii) holds with Kx = Kp.

By using the implication (iv) =£> (i) of Theorem 2 a similar argumentleads to

COROLLARY 2. Let p be fixed, Kp <°°. / /

^ (23)n-k

for k = 1, 2, . . . , then A is bounded on lp, and

\\A\\PiP*Kp.

The constant p is best possible.

Corollary 2 has been given by Izumi, Izumi and Petersen ([16],Corollary 2.5), but with an inferior constant, and only for the case p = 2.When p # 2 , their proof requires an additional hypothesis, "£ anbn <°°"([16], Corollary 2.6).

The equivalence of conditions (ii) and (iv) is of considerable interest,for it means that applications of Theorem 2 tend to occur in groups of

410 GRAHAME BENNETT

four. To see why this is so, let us add a fifth condition, obviouslyequivalent to (i) of Theorem 2:

(v) A' maps /«* to /"*.If a result of the type "( i i )^( i )" is known, then so must the resultcorresponding to "(u)=>(v)." We refer to these as transposes. Theremust also be results of the types "(iv)=> (i)" and "(iv)=>(v)," which werefer to as the sinister transposes.

These remarks, while trivial in the abstract setting of the presentsection, are not quite so obvious in concrete situations, especially when achange of variables is involved. For example, in the next section, we shalldiscuss three inequalities of Hardy, Copson and Leindler. The ine-qualities of Hardy and Copson are transposes, while Leindler's is asinister transpose of Copson's. The fourth inequality, missing from theirworks, is (of course) the sinister transpose of Hardy's. Thus all fourinequalities, despite their different appearances and proofs, are merelymanifestations of a single result.

5. ExamplesWe begin with some inequalities of Hardy, Copson and Leindler. For

these results we suppose that xn 3= 0, kn > 0, and An = kt + • • • + An

(n = 1, 2, . . •)•COROLLARY 3. / / 1 < c *s p, then

2 KKe( £ tek)" * (-^X 2 KK-exZ (24)„ \k-\ I \C-\I „

The constant is best possible.Proof. Using the substitution y£ = knK

pn~

cxprt> the inequality (24) may

be viewed as asserting the boundedness on V, with norm ^p/(c — 1), ofthe factorable matrix A, where

an =

For this matrix we have

* i

£ I*"» A L

(Ao =

c - 1


Thus Corollary 1 may be applied with K = (p- l)/(c - 1 ) , to give\\A\\p,p^p/(.c-l).

COROLLARY 4. / / 0 =s c < 1, then

2 AnAn-c( £ A***)' ^ (-*-)" 2 KK~cxp

n. (26)

77ie constant is best possible.

Proof. Inequality (26) asserts the boundedness on lp of the transposeof the factorable matrix A', where

" K~c""so Corollary 4 is equivalent to the assertion

\\A'\\P:P'*P/Q-C). (27)

Replacing p by p*, and then c by 1 + (p — 1)(1 — c), we see that A'becomes A, the matrix given by (25), and hence that (27) follows fromCorollary 3.

To see that the constants in Corollaries 3 and 4 are best possible itsuffices to take kk = ka and xk = kp for suitably chosen a and /S. Thesefacts, and the method of proof, are well-known.

Corollary 3 was given by Hardy ([10], [11]) in the special case c =p.We could have used this result to shorten slightly the proof of Theorem1. Instead, we preferred to derive Theorem 1 directly from the morefamiliar version of Hardy's inequality, (4).

Corollary 4, under the additional hypothesis, E kkxk <°°, and thegeneral version of Corollary 3, are due to Copson, [9]. The hypothesis,E kkxk < oo, is not required in (26) for it is, via Holder's inequality, asimple consequence of E kk\

pk~

cxpk < °°.

Hardy, [11], showed that his result and the special case, c = 0, ofCorollary 4 are equivalent. The equivalence argument given above, usingtransposes, is similar to his. Viewing these inequalities in terms offactorable matrices, however, gives an additional bonus. For example, if,in Corollary 3, c is increased, it is obvious from (25), that the matrixentries must decrease. Thus Corollary 3 is valid even when c>p—provided that the constant in (24) is replaced by (p*y. I do not know,however, that this constant is still the best possible. In a similar fashion,Corollary 4 remains valid, with constant p9, when c < 0.

An interesting variant of the Hardy-Copson results has been given byLeindler ([20], inequality (1)). We now assume that EA t<oo and set

A£ = E At (n = l, 2, . . . ) . Leindler's result, after a change of

412 GRAHAME BENNETT

variables, is the special case, c = 0, of

COROLLARY 5. / / 0 =£ c < 1, then

2 K(Krc{ £ A**,)" * ( r M ' 2 An(A:r%• (28)77ie constant is best possible.

Proof. This is similar to the proof of Corollary 3, using Corollary 2 inplace of Corollary 1.

The fourth result, needed to complete the Hardy-Copson-Leindlerinequalities is

COROLLARY 6. IfKc^p, then

2 UKT'l 2 KxX * (-?-:)" 2 An(A;r^. (29)n v*=/f / vc — 1/ „

The constant is best possible.

Leindler's result is a sinister transpose of Corollary 4, while Corollary 6is similarly related to Corollary 3. This remark, which serves to unify allfour inequalities, shows also that the constants in (28) and (29) are bestpossible, and that extensions to c < 0 (with constant pp), respectivelyc>p (with constant (p*Y) are valid.

Leindler gives what appears to be & fifth inequality in (2') of [20]. Thisresult, however, after a suitable change of variables, reduces to the casec = 0 of Corollary 4.

The Hardy-Copson-Leindler results have all been derived from thespecial case, q =p, of Theorem 2. Allowing more general values of q(q =*/>), leads to natural and non-trivial extensions of their results. Wegive here only the extension of Hardy's inequality (Corollary 3), leavingthe three companion results for the reader to formulate.

COROLLARY 7. If\<p^q<<x> and c>\, then

( n \1

2 A***) *Proof. Use Theorem 2,The constant K depends on p, q and c, but I have been unable to

determine its best possible value.The original motivation for the work of both Copson and Leindler was

a desire to generalize the following inequality of Hardy and Littlewood([14], Theorem 1).

2"~c('O/' iP.Ol). (30)


Here K is a constant depending only on c and p and the xn's arenon-negative numbers with partial sums Xn = xx + • • •+*„. Thus, forexample, Corollary 3 reduces to (30) when An = 1.

We next consider a generalization in a different direction. (30) may beviewed as asserting that the factorable matrix A is bounded on V, whereaH=n~dp and \\ = k~ll~c/p). From this point of view, the special casep = 2, c = 3, has been obtained by Izumi, Izumi and Petersen ([16], page608). Our next result gives a complete description of the mappingproperties of factorable matrices of the form

an = n-°, bk = k-f>. (31)

The Hardy-Littlewood result, (30), is part (iii) below with q=p andp>\.

COROLLARY 8. Let 1 *£/?, q =£ °° and let A be the factorable matrix givenby (31). Then A maps I" into lq if and only if

(i) a 3=0 and a + fiÔ (p = l,q = oo)

(ii) a>0 and a + P^l/p*or * = 0 and fi>l/p* lP>l.1=»)

(iii) a>l/q and a + fi^l/q + l/p* (1 =£p «£<? <°°)

(iv) a>\lq and a + P>l/q + l/p* (p>q).

Proof. Parts (i) and (ii) are elementary and part (iii) is a consequenceof Theorem 2.

To see that condition (iv) is sufficient, we use part (iii), as follows. If(iv) holds, let y = 1 — a. Applying part (iii), with p = q, to the factorablematrix A', where a'n = n~a, b'k = k~r, we see that A' is bounded on lq.On the other hand, the matrix A may be factored,

A=A'D, (32)

where D is the diagonal matrix with diagonal entries dk = kY~p. Sincea + p>l/q + l/p*, we have d e P * - " . It follows, from Hdlder'sinequality, that D maps /" into /*, and thus, from (32), the same must betrue of A

To prove that condition (iv) is necessary, we use the inequality

Setting x = y = ( l , 0 , 0 , . . .) shows that a>l/q; setting x ={\-Up,..., N-yp, 0, . . . ) and y = ( I " 1 " * , . . . , N-y*', 0,. . . ) , and allow-ing N->oo, shows that a + fi>l/q + l/p*.

Factorable matrices of the more complicated type,

(k + l), (33)

_ 1q'1

q'1

q

ip*'

a + p -

cc + P >

1

ql

?

y '

l

P*

I

414 GRAHAME BENNETT

have been studied by Borwein and Jakimowski. They consider only thespecial case, q=p, a=l/p, P = l/p*. y = l, and show that A isbounded on V if 6=0 ([7], Example (b)) and unbounded if <5 = - l(Example (a)). Both their results are contained in part (i) below.

COROLLARY 9. Let A be the factorable matrix given by (33), and letKp^q<°o. Then A maps V into lq if and only if one of the followingconditions holds:

(i)

(ii)

(iii)

Proof. This follows from Theorem 2 after some tedious calculations.Factorable matrices for which bk = 1 have been considered by Liebo-

witz, [19]. He calls them Rhaly matrices, in honor of H. C. Rhaly, Jr.who studied first the special case: an = n", bk = l, p=q=2, [24].Liebowitz's result is part (i) of

COROLLARY 10. Let A be a Rhaly matrix (i.e. a matrix of the form (19)with bk = 1) and let K p < » . Then a is bounded on V if either of thefollowing conditions holds:

(i) an = O(l/n)

(ii) Eapn = O(ar1)-

Proof. Part (i) is a consequence of Corollary 1; part (ii), of Corollary2.

We remark that Theorem 2 solves Liebowitz's problem ([19], section 3)of describing the V-mapping properties of Rhaly matrices. Indeed, threesuch solutions are given, and the case q >p is covered as well. It wouldbe interesting to have analogues for the case p>q.

6. Weighted meansIn this section we obtain simple necessary and sufficient conditions for

a weighted mean matrix to map V into lq, 1 ^p, q =e ».We recall that a weighted mean matrix, A, is an infinite matrix of the

form

10 k>n v '


where the an's are non-negative numbers with partial sums An = ax +\-an. (We insist that ax > 0 so that each An is positive.) Such matrices

arise naturally in summability theory and have been studied extensivelyfrom this point of view (see, for example, [13] and [31]).

The cases q = °° and/or p = 1 are well-known and there is then anexplicit expression for the norm ||-4||p>,. We list these cases here merelyfor the sake of completeness.

» = 1 (1^«») (35)

||i4||i,, = s u p a ^ 2 ^ » * ) (1«9<«) (36)

To prove (35), we note that the row-sums of A are all 1, so that||-<4||°°,<» = 1; moreover, the first entry is 1, so that ||>l||p,oo> 1.

The remaining cases are covered by

THEOREM 3. Let A be the weighted mean matrix given by (34).(a) Ifl<p*^q<co, the following conditions are equivalent:(i) A maps I" into I";

(ii) v ^ 7 — N<7 ' — ^qlpn \q

for some constant K and all m = 1, 2, . . . ;

(iii)

for s o m e constant K and all m = 1 , 2 , ...;

0V) £ LÂ-A"'^K(fJA-')P'">'

for some constant K and all m = 1,2, ....(b) Ifl^q<p^°°, then A does not map I" into I".Proof. Part (a) is merely a specialization of Theorem 2 to weighted

mean matrices. Part (b) is a consequence of Proposition 1 below. Weprefer to state this result separately since it is of some interest in its ownright.

PROPOSITION 1. Let A be a lower triangular matrix withn

p = lim inf X ank > 0. (37)

IfO<q<p*z<n, then A does not map V into V.

Proof. If A maps V into I", a result of Pitt ([23], Theorem 1) asserts

416 GRAHAME BENNETT

that A must be compact; in particular, A must be compact when viewedas mapping lq into itself.

On the other hand, we may choose m so that

n - 1S 2ank

* = i•• p ' m / 2

whenever m > M. Let x(/l) e /* be given by

lO if Jfc>n '

Then x(n) -»• 0 weakly in /*, so that

||Ax<">||,-»0,

since /I is compact. But if m >Af, we have, by (38),

n-l k-l

m

*nk

(38)

(39)

Letting m —* », it follows from (34) that p = 0.We remark that Proposition 2 does not require that the matrix entries

be non-negative. The result fails, however, if "lower-triangular" isdropped; furthermore, "liminf" may not be replaced by "limsup".Hypothesis (37) applies to all the common methods of summabilitywherein p = 1), and, in particular, to generalized Hausdorff matrices ofthe type considered in [17]. Proposition 1, then, solves the problemraised on page 179 of [17].

Theorem 3 seems to be new even in the special case q=p- The bestpreviously known results in this case are given in an unpublished thesis ofJames Cartlidge, [8]. Cartlidge's main result, from which he derives manyinteresting corollaries, is the following

THEOREM C. Let p be fixed, \<p <°°. Let A be given by (34), andsuppose that an > 0 for n = 1,2, . . . . / /

(40)an+l an

then a maps lp into V and

\\A\\p,p^p/(p-L).

Cartlidge's result is less satisfactory than Theorem 3 since his hypothe-


sis, (40), is highly susceptible to changes in individual an's. On the otherhand, Theorem C covers many cases of practical interest, wherein thean's behave in a "regular" fashion. A particularly interesting consequenceof Theorem C, again due to Cartlidge, is

COROLLARY C. / / (an)"-.i is an increasing sequence, then A is boundedon I" (

Our next two results improve Corollary C; it is interesting to note thatneither is deducible from Cartlidge's theorem.

COROLLARY 1. Let p be fixed, K p < ° ° . / /

then a is bounded on F.

Proof. Use Theorem 3(a), (ii)=>(i), with q=p.

COROLLARY 2. Let p be fixed, \<p < °°. / /

then A is bounded on lp.

Proof. Use Theorem 3(a), (iv)=>(i), with q =p.Our final two results give necessary conditions for the boundedness of

a weighted mean matrix A.

COROLLARY 3. Let p be fixed, \<p < °°. If A is bounded on lp then

Proof. Use Theorem 3(a), (i)^>(iii), and the elementary estimate

PROPOSITION 2. If A is bounded on lp for some p , 1 s£/? < oo, then

Proof. We fix n and define x e V by

' for k = l,...,n*" 10 for k>n.

418 GRAHAME BENNETT

Using the elementary estimate,

v-iwe see that, if A is bounded on V,

n n , kV* / A i • \ X"1 / * _i V

> I A I IT I "*^ j I a >/1 \**-kl^) '— £j ! • * * * £l *

k-l k-1 \ j-1

= \\A\\pp,pAn.

7. Littlewood's problems

In this section we study a class of inequalities formulated by Littlewoodin connection with some work on the general theory of orthogonal series.The simplest (non-trivial) examples are inequality (5) and a companionresult: 2

2 l ( ^ l ) 2> (41)

We recall that the a's are arbitrary non-negative numbers with partialsums An = ax + • • • + an, and K is an absolute constant.

The problem of deciding whether any such K exists was offered byLittlewood as an example of an elementary inequality that is difficult todecide. (The reader should consult [21] for background information andfor several other examples.) Attempts to solve this problem, ultimatelysuccessful, [3], were what led us to the results of this paper.

The special case of (5), in which the a's are decreasing, is worthconsidering separately, for then a particularly simple and natural proof isavailable. We assume that ar > 0, and hence that each An > 0, else (5) istrivial. The left-hand side of (5) may be re-written as

Since the a's are decreasing, we have - p 5 5 " - Applying Cauchy'sinequality, and then Hardy's, we see that "

so that (5) holds with K = 2.


To handle the unrestricted version of (5), a substitute for Hardy'sinequality is called for, and this is what led us to study weightedmean/factorable matrices. To illustrate these ideas we now prove a verygeneral version of (5).

THEOREM 4. Let p, q,r^l. If (an)"-i is a sequence of non-negativenumbers with partial sums An = ax + • • • + an, then

Uf\ (42)Proof. By relabelling, if necessary, we may assume that ar > 0, and

hence that each An > 0.Setting K = anA

q/p and yn = d£qAn, we see that An (=k1 + • • • + kn)«A\+qlp, and therefore

i . .

(43)

LetA,,

(44)p(q+r)-q'

so that 0< 6 =s 1. Using (43), we estimate the left-hand side of (42) by

where we have set

s = ^ ^ . (45)qd + r v '

Using Corollary 4 to Theorem 2 with c = 0, p =s, and xm =ym/Am, we

see that

) ( ) • (46)

Eliminating s and 6, by means of (44) and (45), we see that (46) reducesto (42).

The special case, r = 1, of Theorem 4 is proved by a different methodin [3]. Inequalities (5) and (41) are also special cases; for example (5) isobtained by setting p = 2, q = l and r -1 in (42), and by interchangingthe order of summation on the left. It is interesting to note that we obtaina sharper constant, \, in the general case, than we did in the decreasingcase.

I have been unable to determine the best possible constant in (42). Inattempting to solve this problem I ran some machine computations of

420 GRAHAME BENNETT

inequality (5) to gain insight into the behaviour of the constant K. Theseefforts were not successful, but the following, rather surprising possibilityemerged: is Littlewood's inequality reversible? In other words, doesthere exist a constant K such that

(47)

Unfortunately, inequality (47) is false. This can be seen by taking an = 0for n > 2N and

_ f l i f n = 2r f o r r = l , . . . , Nle otherwise

A tedious calculation shows, by taking e = N2I2N and letting N-*°°, thatK must be infinite.

We close this section with the observation that (47), and indeed thereversal of (42), hold if the a's are decreasing. We need the followingvariant of an idea that goes back to Abel.

LEMMA. Let s, tÔ. If a is an N-tuple of non-negative numbers withpartial sums An = ax + • • • + an, then

n - l

Proof. We have, upon setting Ao = 0,

2 a - ) r > ! x'(AN-x)ldx

Therefore,

N , N y Af2 <>nA'J 2 am) s x?{AN-x)'*x- i \ m _ n / J

0

THEOREM 5. Let p,q,r^l. If (an)"-i is a decreasing sequence ofnon-negative numbers with partial sums An=ax + • • • + an, then


Proof. Applying the lemma with s = q, t = r — 1, we obtain, form = 1, 2, . . . ,

n - l

Since the a's are decreasing, we have

p(q + 1,

Summing on m, and interchanging the order on the right, leads to (48).

8. ProblemsIn this section we list some problems for further research and give a

brief survey of related work on non-factorable matrices.We have pointed out already that the constant appearing in Theorem

1, K(r, s) = (r/(r — s))r, is known to be sharp only when 5 = 1.Presumably some improvement is possible when s > 1, but this I havebeen unable to decide.

A harder problem arises if we are given u and v, and we seek the bestconstant, K(u, v, r, s), such that (2) holds for all w. This amounts tofinding the norm of an arbitrary factorable matrix, and hence is a difficulttask even when 5 = 1.

Similar problems arise in Corollaries 3 and 6 (when Op), inCorollaries 4 and 5 (when c < 0), and in Theorems 4 and 5.

A more important problem is that of finding a substitute for Theorem 1when 5 < 1. This corresponds to giving necessary and sufficient conditionsfor a factorable matrix to map V to /*, for fixed p and q, with\<q<p < oo. The problem is solved for weighted mean matrices(Theorem 3) and for matrices of the form (31) (Corollary 8), but ageneral criterion is difficult to come by. I do not know the answer evenfor Rhaly matrices (see Corollary 10). We remark that conditions (ii),(iii) and (iv) of Theorem 2 are equivalent even when 1< q <p < oo, andthat they are implied by condition (i). It is the converse implication whichfails, and therein lies the difficulty.

The problem of finding a substitute for Theorem 1 when r =e 5 is alsoopen, but this case is less interesting.

Much work has been done describing the mapping properties ofvarious "classical" matrices. A standard reference here, of course, is tothe last three chapters of [15]. The known results deal almost exclusivelywith mappings of one lp -space into itself. It would be interesting todevelop analogues when two different spaces are involved, as has beendone here, for example, for weighted mean matrices.

422 GRAHAME BENNETT

A result of considerable beauty and generality has been proved byHardy, [12]. He gives an explicit description of the norm of a Hausdorffmatrix: x

j). (49)o

Here n denotes a probability measure on [0,1], and the entries of {H, n)are given by

i

J (* - l )"*~ 1 ( 1 " aT'k M<X- . - • • - ( 5 ° )

0 k>n

Taking fi to be Lebesgue measure leads (49) back to Hardy's originalinequality, (4). Another case of special interest is obtained by fixing a,0 < Q ' < 1 , and taking \i to be the point mass at a. This leads to theso-called Eider matrices, E(a), and to

a result first obtained by Bochner and, independently, by Knopp (see[18], Satz II and the footnote to page 19). Other substitutions, toocomplicated to describe here, give rise to the Cesaro means, C(a), and tothe Holder means, H(a) ([13], [31]). The /p-mapping problem istherefore completely solved for these classes of matrices.

Hausdorff matrices more general than those given by (50) have beentreated in [5], [7], [17], [25] and [26]. In addition, there are severalpapers dealing with the spectral properties of Hausdorff (and other)matrices; we do not list these here. An alternative approach to Hardy'sresult, (4), which leads also to Hilbert's inequality (see below), is given insection 7 of [2].

The question of determining the /p-norm of a weighted mean matrixhas been raised by Jakimowski, Rhoades and Tzimbalario ([17], page181). Theorem 3 gives only a qualitive solution to their problem, butmore can be said, perhaps, by recalling the proof of Theorem 1. We sawthere, in attempting to maximize expressions of the form

\\Ax\\p/\\x\\p,

that is suffices to consider only those x elp satisfying

xk/al'°'-1) decreasing in k. (51)

Theorem 3 involves taking x's of the form

- { * " ' " (52,0 k>n v '


Not much information is lost in specializing (51) to (52)—this is thecontent of Theorem 3—but just how much is lost is difficult to evaluate.

The F-mapping problem for Norlund matrices remains wide-open.Since this is the last outstanding class from among the "common"methods of summability, the problem is not without its glamour. Werecall that a Norlund matrix has the form

n-k+i'An k = 1,.. ., nk>n

The a's, here, are positive numbers with partial sums An=al + • • • + an.The best known result, due to Borwein and Cass ([6], Theorem 2),

asserts that A is bounded on V, \<p < °°, provided that

aJAn = O(l/n).

Their theorem illustrates very nicely a basic difference between N6rlundmatrices and weighted means: in the first case it is desirable that a bedecreasing; in the second, increasing (see Corollary C in section 6).

For Toeplitz matrices the lp -mapping problem is a triviality—at least inthe context (non-negative entries) in which we have been working.Toeplitz matrices are of the type

U = an-k (n,k = l,2,...)

where (an)"__oo is a sequence of (non-negative) numbers. It is easy to seethat the boundness of T on V (for any, or for all, p, l«/?s£oo) isequivalent to E an < <».

The situation for Hankel matrices is much less transparent. These arematrices of the form

hnk = an+k {n,k = 0,1,2,...), (53)

where, again, (an)n-o is a sequence of non-negative numbers. Feffermanhas shown (see [1], p. 264) that the matrix given by (53) is bounded on I2

if and only if

sup 2 2 a,) <». (54)

Two special cases of Fefferman's result are of considerable interest. Thefirst, obtained by setting an = (n +1)"1, is Hilbert's inequality ([15],Chapter LX), from which all researches in this field began. The second,obtained by taking an = 0 except when n is a power of two—in which case(54) reduces to E a2,<co—is Paley's inequality, [22].

Proofs of Fefferman's result are given in [4], [27] and [28]. These allinvolve "//'-BMO-duality theory" or, what is equivalent, the "atomicdecomposition of H*". We thus have the very curious phenomenon of an

424 GRAHAME BENNETT

elementary inequality whose proof requires some of the deepest ideas ofAnalysis. Our last problem, then, is that of finding an elementary proofof Fefferman's result—one that is free from the taint of function theory.(There are several such proofs of Hilbert's inequality, and of Paley's, butsince there is no point of contact the difficulty here goes quite deep.)

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9-12 and 3 (1928), 49-51.10. G. H. Hardy, 'Notes on some points in the integral calculus LX: An inequality between

integrals', Messenger of Math 54 (1925), 150-156.11. G. H. Hardy, 'Remarks on three recent notes in the Journal', Jour. London Math. Soc.

3 (1928), 166-169.12. G. H. Hardy, 'An inequality for Hausdorff means', Jour. London Math Soc. 18 (1943),

46-50.13. G. H. Hardy, Divergent Series, Oxford University Press, 1949.14. 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.

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

16. M. Izumi, S. Izumi and G. M. Petersen, 'On Hardy's inequality and its generalization',Tohokhu Math. Jour. 21 (1969), 601-613.

17. A. Jakimowski, B. E. Rhoades and J. Tzimbalario, 'Hausdorff matrices as boundedoperators over /,,', Math. Z. 138 (1974), 173-181.

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19. G. Leibowitz, 'Rhaly matrices', to appear in Jour. Math. Analysis and Applications.20. L. Leindler, 'Generalization of inequalities of Hardy and Littlewood', Act Sci. Math.

(Szeged) 31 (1970), 279-285.21. J. E. Littlewood, 'Some new inequalities and unsolved problems', Inequalities (editor:

O. Shisha) 151-162, Academic Press, New York, 1967.22. R. E. A. C. Paley, 'On the lacunary coefficients of power series', Ann. of Math. 34

(1933), 615-616.23. H. R. Pitt, 'A note on bilinear forms', Jour. London Math. Soc. 11 (1936), 174-180.


24. H. C. Rhaly, Jr., 'p-Cesaro matrices', to appear in Houston Jour. Math.25. B. E. Rhoades, 'Hausdorff matrices as bounded operators on T, Proc. Amer. Math.

Soc. 78 (1980), 210-212.26. B. E. Rhoades, 'Generalized Hausdorff matrices bounded on F and c', Acta Set. Math.

(Szeged) 43 (1981), 333-345.27. W. T. Sledd and D. A. Stegenga, 'An H'-multiplier problem', Ark. Mai. 19 (1981),

265-270.28. S. J. Szarek and T. Wolniewicz, 'A proof of Fefferman's theorem on multipliers',

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Added in proof, June 4, 1987. Progress has been made on some of theproblems mentioned in Section 8.

In a paper to appear in the Rocky Mountain Journal of Mathematics,F. P. Cass and W. Kratz have shown that the Borwein-Cass condition(see page 23) is necessary as well as sufficient for the Norlund matrix, A,to be bounded on F, \<p <<*>. Their proof applies to matrices of theform an =f{n), where / is a logarithmico-exponential function. Thegeneral problem remains open.

The question of finding analogues of Theorem 1 when r =£ s and whens < 1 (see page 21) has been solved by the author. The details, toocomplicated to describe here, will be published elsewhere. The case r ^s,despite the comment on page 21, is quite interesting. In fact, theargument used gives a new proof of Theorem 1, a proof much shorterthan the one given here!

This paper was presented at the "Ohio Informal Analysis Seminar"held at Kent State University, November 8, 1986. Shortly thereafter,Professor Boris Mitiagin supplied another proof of Theorem 1 using thetechniques of Linear Programming. Thus there are now three quitedistinct approaches to our main result.

Department of MathematicsIndiana UniversityBloomingtonIndiana 47405U.S.A.

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SOME ELEMENTARY INEQUALITIES