JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 220, 99᎐109Ž. 1998 ARTICLE NO. AY975822

Stolarsky and Hadamard’s Inequality

C. E. M. Pearce

Department of Applied Mathematics, The Uni¨ersity of Adelaide, Adelaide, SA 5005, Australia

View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector

J. Pecaricˇ´ and V.ˇ Simic ´

Faculty of Textile Technology, Uni¨ersity of Zagreb, Pierottije¨a 6, 11000, Zagreb, Croatia

Submitted by A. M. Fink

Received March 25, 1997

A generalization is given of the extension of Hadamard’s inequality to r-convex functions. A corresponding generalization of the Fink᎐Mond᎐Pecaricˇ´ inequalities for r-convex functions is established. ᮊ 1998 Academic Press

1. INTRODUCTION

One of the most fundamental inequalities for convex functions is that associated with the name of Hadamard. This states that if f: wxa, b ª ޒ is convex, then

1 b faŽ.qfb Ž. HftdtŽ. F . bya a 2

Hadamard’s inequality has recently been extended in two quite different

ways. Recall that the integral power Mp of a positive function f on

99

0022-247Xr98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 100 PEARCE, PECARICˇ´, AND ˇ´SIMIC wxa,bis a functional given by

1 1rp ¡ b p HftŽ. dt , p/0, bya a MfpŽ.s~ Ž.1.1 1 b expH ln ftdtŽ. , ps0. ¢ bya a

Further, the extended Lp of two positive numbers a, b is given for a s b by LapŽ.,asaand for a / b by

p 1 p 1 1rp ¡ b q y a q , p / y1,0, Ž.Ž.p q 1 b y a bya LapŽ.,bs~ , psy1, ln b y ln a 1Ž.b1 1bbry , ps0. ¢eaž/a

Hadamard’s inequality may now be recast as a relationship

Mf11Ž.FLfaŽ. Ž.,fb Ž. between integral power means and extended logarithmic means. Inwx 8 the following extension is derived for this suggestive result.

THEOREM A. If f: wxa, b ª ޒ is positi¨e, continuous, and con¨ex, then

MfppŽ.FLfaŽ. Ž.,fb Ž.,1.2Ž. while if f is conca¨e,Ž. 1.2 is re¨ersed. Ž. Remark. We note that Lay1 ,bis the well-known logarithmic mean LaŽ.,b and La0 Ž.,bis the identric mean IaŽ.,b.

The second extension involves the power mean MxrŽ.,y;␭of order r of positive numbers x, y, which is defined by

Ž.␭xrrŽ.1␭y1rr ,ifr/0, MxŽ.,y;␭ qy r s ␭1␭ ½xyy,ifrs0.

In the special case ␭ s 1r2 this notation is contracted to MxrŽ.,y. It involves also the alternative extended logarithmic mean FxrŽ.,yof two positive numbers x, y, which is prescribed by FxrŽ.,xsxand for STOLARSKY MEANS 101 x / y by

r 1 r 1 ¡ rxq yyq иrr,r/0,y1, rq1xyy xyy FxrŽ.,ys~, rs0, ln x y ln y ln x y ln y xy , r sy1. ¢ x y y

This includes the usual logarithmic mean as the special case r s 0. An idea of r-convexity may be introduced ¨ia power means. DEFINITION. A positive function f is said to be r-convex on an interval wxa,bif, for all x, y g wxa, b and ␭ g wx0, 1 ,

f Ž.Ž.␭ x q Ž.1 y ␭ y F Mfxr Ž.Ž.,fy;␭.1 Ž..3

This definition of r-convexity complements naturally the concept of r-concavity in which the inequality is reversedŽ seewx 10. . This concept plays an important role in statistics. Our definition of r-convexity can be expanded as the condition that

␭ f r x 1 ␭ f r y ,ifr/0, r Ž.q Žy . Ž. fŽ.␭xqŽ.1y␭yF ␭1␭ ½fxfŽ.y Ž. y,ifrs0. For a positive function f, it is applicable for nonintegral values of r. Also, suppose as is usual that f is nonnegative and possesses a second deriva- tive. If r G 2, then

22rr2XY2 r1 drdx f s rrŽ.Ž.y1ffyqrfy f , Y which is nonnegative if f G 0. Hence under the restrictions noted, ordi- nary convexity implies r-convexity. The reverse implication is not the case, 12 as is shown by the function fxŽ.sxr for x ) 0. We note that the standard definition of r-convexityŽ seew 5, Chap. 1, Sect. 6x. is quite different. Recall that when the f Ž r. exists, f is Žr. r-convex if and only if f G 0Ž seewx 5, Chap. 1, Theorem 1. . Consider the function

3 2 fxŽ.[xx Žyxq1 . Ž2.Ž3. on I s Ž.1r2, 1 . For x g I, we have f - 0 but f ) 0, so f is 3-convex but not convex. The function g syf on the same domain is a function which is convex but not 3-convex. 102 PEARCE, PECARICˇ´, AND ˇ´SIMIC

After this lengthy aside, we are ready to state the second extension of Hadamard’s inequality, which was established recently inwx 4 . This relaxes the assumption of convexity to one of r-convexity.

THEOREM B. Suppose f is a positi¨e function onwx a, b . If f is r-con¨ex, then 1 b HftdtŽ.FFfarŽ. Ž .,fb Ž .,1.4Ž. byaa while if f is r-conca¨e, the inequality is re¨ersed. In Section 3 we prove a result which subsumes Theorems A and B as special cases. The relevant generalization of Lprand F turns out to be the well-known Stolarsky mean ExŽ.Ž,y;r,s see Stolarskywx 9. . This is given by ExŽ.,x;r,ssxif x s y ) 0 and for distinct positive numbers x, y by ss1Ž.sr ryyx ry ExŽ.,y;r,ss rr , r/sand rs / 0, syyx

rr1 r 1yyxr ExŽ.Ž.,y;r,0 sEx,y;0,r s , r/0, rln y y ln x

rr r1Ž xy. xxry y1rr ExŽ.,y;r,rse r , r/0, ž/yy ExŽ.,y;0,0 s'xy .

Clearly ExŽ,y;1, pq1 .sLxpr Ž.,yand Ex Ž,y;r,rq1 .sFx Ž.,y. The key to our proof is a new integral representation of Stolarsky’s mean. This is of some interest in its own right and is presented in Sec- tion 2. In Section 4 we establish a related generalization of the Fink᎐Mond᎐Pecaricˇ´ inequalitieswx 3, 6 .

2. INTEGRAL REPRESENTATIONS

Carlsonwx 1 has established the integral representation y1 1 dt LxŽ.,ysH ,2.1Ž. 0tx q Ž.1 y ty while Neumanwx 7 has given the alternative integral representation

1 t 1 t LxŽ.,ysHxyy dt.2.2Ž. 0 STOLARSKY MEANS 103

Let MfpŽ.denote the integral meanŽ. 1.1 for a s 0, b s 1, and put etx, yŽ.[tx q Ž1 y ty .. A simple evaluation of the right-hand side shows that

LxppŽ.,ysMe Ž.x,y, which provides a generalization of the integral representationŽ. 2.1 for the extended logarithmic means LxpŽ.,y. Similarly we can derive

1 FxrrŽ.,ysHMx Ž,y;tdt . 0 as a natural extension ofŽ. 2.2 .

In the above we regard MxrŽ.,y;tas a function of the parameter t. Set

mtr,x,yrŽ.[Mx Ž,y;t ..

Then we may evaluate the integrals on the right-hand side of

1rŽ.syr ¡1 syr HŽ.MxrŽ.,y;tdt ,s/r 0 MmsyrrŽ.,x,ys~ Ž.2.3 1 expH ln MxrŽ.,y;tdt, ssr ¢0 to give a simple derivation of the representation

ExŽ.Ž.,y;r,ssMmsyrr,x,y.2.4 Ž.

3. HADAMARD’S INEQUALITY FOR r-CONVEX FUNCTIONS

THEOREM 3.1. Suppose f is a positi¨e function onwx a, b . Then if f is r-con¨ex,

MfpŽ.FEfaŽ. Ž.,fb Ž.;r,pqr,3Ž..1 while if f is r-conca¨e, the inequality is re¨ersed. 104 PEARCE, PECARICˇ´, AND ˇ´SIMIC

Proof. First we suppose r-convexity. Let p / 0. Then

1 1rp b p MfpŽ.s HftdtŽ. byaa

1rp 1p sHfsbŽ.qŽ.1ysads 0

1rp 1p FHMfbrŽ.Ž.,fa Ž.;sds . 0

FromŽ. 2.3 and Ž. 2.4 we deduce that

MfpprŽ.FMm Ž,fŽb.,fŽa. .sEfaŽ. Ž.,fb Ž.;r,rqp.

Similarly, for p s 0, we have

1 b Mf0Ž.sexpH ln ftdtŽ. byaa

1 sexpH ln fsbŽ.qŽ.1ysads 0

1 FexpH ln MfbrŽ.Ž.,fa Ž.;sds 0 and again fromŽ. 2.3 and Ž. 2.4 we derive

Mf00Ž.FMm Žr,fŽb.,fŽa. .sEfaŽ. Ž.,fb Ž.;r,r.

The proof in the case of r-concavity is exactly similar.

Remark. For p s 1,Ž. 3.1 becomes

Mf1Ž.FEfaŽ. Ž.,fb Ž.;r,rq1, that is,Ž. 1.4 , while for r s 1 we have

MfpŽ.FEfaŽ. Ž.,fb Ž.;1, pq1, which isŽ. 1.2 . Thus our result subsumes Theorems A and B. We observe that an r-convex function f can be defined on a convex set Uin a real linear space X withŽ. 1.3 holding whenever x, y g U and ␭gwx0, 1 . This leads to the following result. STOLARSKY MEANS 105

THEOREM 3.2. Suppose f is a positi¨e function on UŽ.; X , where U is con¨ex and X a linear space. Then if f is r-con¨ex,

MfepaŽ.Ž.,bFEfaŽ. Ž.Ž.,fb;r,pqr, while if f is r-conca¨e, the inequality is re¨ersed.

4. A FURTHER GENERALIZATION OF THE FINK᎐MOND᎐PECARICˇ´ INEQUALITIES

The following generalization of Fink᎐Mond᎐Pecaricˇ´ inequalitieswx 3, 6 was obtained inwx 4 . THEOREM C. Let w be a nonnegati¨e, integrable, e¨en function on wxy1, 1 with positi¨e integral and let f be a positi¨e function. Ž.a If f is r-con¨ex, then for r F 1,

1 Hy1fxŽ.Ž.Ž.Ž.q¨twtdt f xq¨ qfxy¨ 1 F ,4.1Ž. Hy1wtdtŽ. 2 while if r G 1,

1 Hy1fxŽ.Ž.q¨twtdt 1 FMfxrŽ.Ž.Ž.Ž.q¨,fxy¨ .4.2 Hy1wtdtŽ.

Ž.b Suppose f is r-conca¨e. Then if r F 1, the inequality Ž.4.2 is re¨ersed, while if r G 1, the inequality Ž.4.1 is re¨ersed. We extend these results to allow a power mean of order p on the left-hand sides ofŽ. 4.1 and Ž. 4.2 in place of an arithmetic integral mean. The power mean of order p is defined by

1 p 1rp ¡ Hy1ftwtdtŽ. Ž. 1 , p/0 H1wtdtŽ. Mf˜Ž.,w y p s~ 1 Hy1wtŽ.ln ftdt Ž. exp 1 , p s 0. ½5H1wtdtŽ. ¢y

We shall need the following useful result due to Fejer´ wx 2 . 106 PEARCE, PECARICˇ´, AND ˇ´SIMIC

LEMMA 4.1. Suppose h: wxa, b ª ޒ is con¨ex and w: wxa, b ª ޒ a nonnegati¨e, integrable function with positi¨e integral and such that 1 waŽ.Ž.qt swbyt,0FtF Žbya ..4.3 Ž. 2 Then b a q b HawthtdtŽ. Ž. ha Ž .qhb Ž . h FF.4.4Ž. ž/22HbwtdtŽ. a The inequality is re¨ersed if h is conca¨e. We shall derive the following generalization of Theorem C.

THEOREM 4.1. Let w be a nonnegati¨e, integrable, e¨en function with positi¨e integral o¨er wxy1, 1 , and let f be a positi¨e function. Put f˜Ž. t [ fxŽ.q¨t for t gywx1, 1 . Ž.aIf f is r-con¨ex and m s maxÄ4r, p , then

Mf˜˜pmŽ.,wFMfxŽ.Ž.Ž.q¨,fxy¨ .4 Ž..5

Ž.bIf f is r-conca¨e and m s minÄ4r, p , then the inequality is re¨ersed. Proof. Ž.a Take f to be positive and r-convex. First suppose that pr / 0. Since f is r-convex, we have 1 q t 1 y t fxŽ.q¨q Ž.xy¨ ž/22 1t1t1rr qrry FfxŽ.q¨q fx Ž.y¨ . 22 Therefore

Mf˜˜pŽ.,w

1 p 1rp Hy1fxŽ.Ž.q¨twtdt s 1 Hy1wtdtŽ.

1 p 1 rrprr r Hy1Ž.Ž.1qtr2fx Žq¨t .Ž.q Ž.1ytr2fx Žy¨ .wtdt Ž. F 1 Hy1wtdtŽ.

1 1rp Hy1htwtdtŽ. Ž. s 1 ,4.6Ž. Hy1wtdtŽ. r r pr where htŽ.swŽŽ1qt .r2 . fxŽ.ŽŽ..q¨q1ytr2fxŽ.y¨xr. STOLARSKY MEANS 107

We have that htŽ.is convex on wxy1, 1 for r F p and concave for r G p. Since w is even, wŽ.Ž.Ž.y1 q t s w 1 y t , so that 4.3 holds for a sy1, b s 1. Hence by Lemma 4.1

rrprr 1 fxŽ.Ž.q¨qfxy¨ Hy1wthtdtŽ. Ž. F 1 ž/2 H1wtdtŽ. y p p fŽ.Ž.xq¨ qf xy¨ F 2 applies if r F p and the reverse inequality holds for r G p. If r F p with p ) 0, we may take pth roots to derive

1 1rp Hy1wthtdtŽ. Ž. MfxrŽ.Ž.Ž.q¨,fxy¨ F H1wtdt ž/y1 Ž.

FMfxpŽ.Ž.Ž.Ž.q¨,fxy¨ .4.7

The same conclusion holds if r G p with p - 0. The inequalities inŽ. 4.7 are reversed if r F p with p - 0orrGpwith p ) 0. Coupling these results withŽ. 4.6 gives part Ž. a of the enunciation for the case pr / 0. Now suppose r s 0 and p / 0. Then we have

1 p H 1 fxp Ž.Ž.twtdt r ˜˜ y1 q¨ MfpŽ.,ws 1 Hy1wtdtŽ.

1 p 1 pŽŽ1qt .r2 .p ŽŽ1yt .r2 . r Hy1fxŽ.q¨ fx Ž.y¨ wtdt Ž. F 1 .4.8Ž. Hy1wtdtŽ.

Put

pŽŽ..1qtr2 pŽŽ..1ytr2 htŽ.Ž.sfxq¨ fx Ž.y¨ .

Then h is convex, and byŽ. 4.4

1 pp p Hy1wthtdtŽ. Ž. f Ž xq¨ .qfx Žy¨ . ž/'fxŽ.Ž.q¨fxy¨ FF1 . Hy1wtdtŽ. 2

For p ) 0, we getŽ. 4.7 with r s 0. For p - 0 the inequalities are reversed. So byŽ. 4.8 , we again have Ž. 4.5 . 108 PEARCE, PECARICˇ´, AND ˇ´SIMIC

Suppose r / 0 and p s 0. Then we have

Mf˜˜0Ž.,w

1 Hy1wtŽ.ln fx Žq¨tdt . Fexp 1 Hy1wtdtŽ.

1 rr1rr Hy1wtŽ .lnwx ŽŽ 1 q t .r2 .fx Žq¨ .q ŽŽ1yt .r2 .fx Žy¨ . dt F exp1 . Ž.4.9 Hy1wtdtŽ. The function

1 t 1 t 1rr q rry htŽ.[ln fx Ž.q¨q fx Ž.y¨ 22 is convex for r - 0 and concave for r ) 0. So, for r - 0,Ž. 4.4 gives

1 Hy1wthtdtŽ. Ž. ln MfxrŽ.Ž.Ž.q¨,fxy¨ F 1 Hy1wtdtŽ.

Fln Mfx0Ž.Ž.Ž.Ž.q¨,fxy¨ . 4.10 The inequalities are reversed if r ) 0. ByŽ. 4.9 and Ž 4.10 . we again have Ž.4.5 . Finally, suppose r s 0, p s 0. We have H 1 wtŽ.ln fx Žtdt . ˜˜ y1 q¨ Mf0Ž.,wsexp 1 Hy1wtdtŽ.

1 ŽŽ..1qtr21ŽŽ..ytr2 Hy1wtŽ.Ž.ln fxq¨ fx Ž.y¨ dt Fexp 1 Hy1wtdtŽ.

sMfx0Ž.Ž.q¨,fxy¨ . ThusŽ. a is established in all cases. Ž.b The proof is similar. Remark. Suppose U is a convex set in a real linear space X. Then the conclusion of Theorem 4.1 holds when x, ¨ g X are such that x q ¨, x y ¨ ޒ Ž. gUand f: U ª q is r-convex r-concave on U.

ACKNOWLEDGMENT

The authors thank the referee for drawing their attention to several small points in the text. STOLARSKY MEANS 109

REFERENCES

1. B. C. Carlson, The logarithmic mean, Amer. Math. Monthly 79 Ž.1972 , 615᎐618. 2. L. Fejer,´ Uber¨ die Fourierreihen, II, Math. Naturwiss. Anz. Ungar. Akad. Wiss. 24 Ž.1906 , 369᎐390Ž. In Hungarian ; Gesammelte Arbeiten I Ž.1970 , 280᎐297. 3. A. M. Fink, Two inequalities, Uni¨. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 6 Ž.1995 , 48᎐49. 4. P. M. Gill, C. E. M. Pearce, and J. Pecaric,ˇ´ Hadamard’s inequality for r-convex functions, J. Math. Anal. Appl. 216 Ž.1997 , 1᎐22. 5. D. S. Mitrinovic,´ˇ´ J. E. Pecaric, and A. M. Fink, ‘‘Classical and New Inequalities in Analysis,’’ Kluwer Academic, Dordrecht, 1993. 6. B. Mond and J. Pecaric,ˇ´ A companion to Fink’s inequality, Octogon, in press. 7. E. Neuman, The weighted logarithmic mean, J. Math. Anal. Appl. 188 Ž.1994 , 885᎐900. 8. C. E. M. Pearce and J. Pecaric,ˇ´ A continuous analogue and an extension of Rado’s´ formulae for convex and concave functions, Bull. Austral. Math. Soc. 53 Ž.1996 , 229᎐233. 9. K. B. Stolarsky, Generalizations of the logarithmic mean, Math. Mag. 48 Ž.1975 , 87᎐92. 10. B. Uhrin, Some remarks about the convolution of unimodal functions, Ann. Probab. 12 Ž.1984 , 640᎐645.