JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 220, 99]109Ž. 1998 ARTICLE NO. AY975822 Stolarsky Means 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. PecaricÏ and V.Ï 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]PecaricÏ inequalities for r-convex functions is established. Q 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 R 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 mean Mp of a positive function f on 99 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. 100 PEARCE, PECARICÏÂ, AND ÏÂSIMIC 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 logarithmic mean 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 R 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;lof order r of positive numbers x, y, which is defined by Ž.lxrrŽ.1ly1rr ,ifr/0, MxŽ.,y;l qy r s l1l ½xyy,ifrs0. In the special case l 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 ?rr,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 l g wx0, 1 , f Ž.Ž.l x q Ž.1 y l y F Mfxr Ž.Ž.,fy;l.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 l f r x 1 l f r y ,ifr/0, r Ž.q Žy . Ž. fŽ.lxqŽ.1ylyF l1l 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 derivative 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, PECARICÏÂ, AND ÏÂSIMIC 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]PecaricÏ 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, PECARICÏÂ, AND ÏÂSIMIC 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 lgwx0, 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]PECARICÏ INEQUALITIES The following generalization of Fink]Mond]PecaricÏ inequalitieswx 3, 6 was obtained inwx 4 . THEOREM C.