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Logarithmic Convexity and Inequalities for the Gamma Function* JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 203, 369]380Ž. 1996 ARTICLE NO. 0385 Logarithmic Convexity and Inequalities for the Gamma Function* Milan Merkle Department of Applied Mathematics, Faculty of Electrical Engineering, P.O. Box 816, 11001 Belgrade, Yugosla¨ia View metadata, citation and similar papers at core.ac.uk brought to you by CORE Submitted by A. M. Fink provided by Elsevier - Publisher Connector Received June 27, 1995 We propose a method, based on logarithmic convexity, for producing sharp bounds for the ratio GŽ.Ž.x q b rG x . As an application, we present an inequality that sharpens and generalizes inequalities due to Gautschi, Chu, Boyd, Lazarevic- LupasË, and Kershaw. Q 1996 Academic Press, Inc. 1. SOME CONVEX AND CONCAVE FUNCTIONS It is well known that the second derivative of the function x ¬ log GŽ.x can be expressed in terms of the seriesŽ seewx 2 or w 10 x. d2 11 1 2222log GŽ.x sqqq??? dx x Ž.Ž.x q 1 x q 2 Ž.x/0,y1,y2,... ,Ž 1. so the gamma function is a log-convex one. The following theorem gives a supply of convex and concave functions related to log G. * This research was supported by the Science Fund of Serbia, Grant 0401A, through the Mathematical Institute, Belgrade. 369 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 370 MILAN MERKLE THEOREM 1. Let B2 k be Bernoulli numbers and let L and R be generic notations for the sums 2 N B2 k LxŽ.sLx2N Ž.sy Ý 2kŽ.2k 1x2ky1 ks1 y Ž.Ž.Ns1,2,... , Lx0s0, 2Nq1 B2k RxŽ.sRx2N 1 Ž.sy Ý ŽNs0,1,2,.... q 2kŽ.2k 1x2ky1 ks1 y Ž.i The functions Fx1Ž.slog G Ž.x Fx2Ž.slog G Ž.x y x log x 1 Fx3Ž.slog G Ž.x y x y log x ž/2 1 FxŽ.slog G Ž.x y x y log x q LxŽ. ž/2 are con¨ex on x ) 0. Ž.ii The functions 1 GxŽ.slog G Ž.x y x y log x q RxŽ. ž/2 are conca¨eonx)0. Proof. We will apply the Euler]MacLaurin summation formula to the series inŽ. 1 . For a finite number of terms, m say, we have m xqm 1 ÝfxŽ.qksHft Ž.Ž dtq Ž.fxqm .Ž.qfx x 2 ks0 NB 2kŽ2k1.Ž2k1. qÝŽ.fxyŽ.qmyfxy Ž. Ž.2k! ks1 B my1 2Nq2 Ž2N2. q ÝfxqŽ.Ž.qkqu,2 Ž.2N2! qks0 Ž. for some u g 0, 1 , where Bj are Bernoulli numbers defined by tt` j sÝBj. et1j! yjs0 INEQUALITIES FOR THE GAMMA FUNCTION 371 First five Bernoulli numbers with even indices are 11115 B24s,Bsy , B 6s , B 8sy , B 10s . 630423066 2 Letting fxŽ.sxy inŽ. 2 , we obtain Ž see alsowx 7, 5.8 for this particular case. m 1 Ý 2s SNŽ.Ž.Ž.qTm,N qEm,N Ž.3 ks0Ž.xqk where N 11 B2k SNŽ.sq qÝ x2xx22kq1 ks1 N 11 B2k TmŽ.,N syyÝ 22xmkq1 2Ž.xqmxqks1Ž.qm my1 1 EmŽ.Ž.,N 2N 3B Ž.0-u-1 s q 2Nq2Ý 2Nq4 ks0Ž.xqkqu Ž. and the sign of Em,N is clearly equal to the sign of B2 Nq2 , which is N Ž.y1Ž seewx 5, Sect. 449. This implies that for every m G 1, N G 1, and x ) 0 we have m 1 SŽ.2 N q Tm Ž,2N .FFÝ 2 SŽ.Ž.2Nq1qTm,2Nq1. ks0Ž.xqk Ž.4 Now for x ) 0 and N G 1 being fixed, let m ª ` inŽ. 4 . Then Tm Ž,2N . and TmŽ.,2Nq1 converge to 0 and we obtain 2 N ` 2 Nq1 11 B2k 111 B2k qqÝÝ--qqÝ. xx2xx22kq12222xxkq1 ks1 ks0Ž.xqk ks1 Ž.5 FromŽ. 5 we see that, for positive x, F0 Ž.x ) 0 and G0 Ž.x - 0. As we already noticed, the expansionŽ. 1 implies that F1 is a convex function, so it remains to show convexity of F23and F . Convexity of F3 372 MILAN MERKLE follows from ` Y 111 Fx3Ž.syÝ y 22x2x ks0Ž.xqk ` 11 1 syqÝ2 žŽ.xk xqkxqkq1 ks0 q 11 yq22 2Ž.Žxqk2xqkq1 ./ ` 1 sÝ 22)0.Ž. 6 ks02Ž.Ž.xqkxqkq1 Y Y Since Fx23Ž.)FxŽ.for x ) 0, the function F2is also convex, which ends the proof. The functions F and G introduced in Theorem 1 are closely related to the well known asymptotic expansion q` 11 B2k log GŽ.x ; x y log x y x q log 2p q Ý ž/222kŽ.2k1x2ky1 ks1 y Ž.xªq`. In fact, it follows from Theorem 1 that the function n 11 B2k xlog GŽ.x y x y log x q x y log 2p y Ý ¬ž/222kŽ.2k1x2ky1 ks1 y is convex onŽ. 0, q` if n is even and it is concave for odd n. 2. BOUNDS FOR GŽ.Ž.x q b rG x In this section we will use the equalities x s b Ž.Ž.Ž.Ž.x y 1 q b q 1 y b x q b 7 xqbsŽ.Ž.1ybxqbxq1.Ž. 8 Let us also introduce the following notation: For a function S Žs L or R. let UxŽ.Ž.Ž,b,SsSxybSxy1qb .Ž.Ž.Ž.y1ybSxqb 9 VxŽ.Ž,b,SsyUxqb,1yb,S . sŽ.Ž.Ž.Ž.Ž.1ybSxqbSxq1ySxqb.10 INEQUALITIES FOR THE GAMMA FUNCTION 373 Further, let AxŽ.,b and Bx Ž.Ž,b do not confuse the latter notation with the beta function. be defined by x 1 2 b b 1yb y r q Ž.Ž.Ž.x y 1 q b x q b AxŽ.,b s Žx)1yb .Ž.,11 xxy1r2 x BxŽ.,b s AxŽ.qb,1yb xqby1r2b Ž.xqbx Ž.x)0. Ž. 12 sŽ1b.Ž x 1 2. bŽ.x12 xxyy r Ž.q1qr The ratio GŽ.Ž.x q b rG x will be denoted by QxŽ.,b. In this section we will derive general inequalities for QxŽ.,b, using LC with functions F and G as defined in Theorem 1. THEOREM 2. For b g wx0, 1 and for x ) 1 y b we ha¨e AxŽ.,bexpŽ.Ux Ž,b,L .FQx Ž.,b FBx Ž.,bexpŽ.Vx Ž,b,L .Ž.13 BxŽ.,bexpŽ.Vx Ž,b,R .FQx Ž.,b FAx Ž.,bexpŽ.Ux Ž,b,R .Ž.14 with equalities if and only if b s 0 and b s 1. As x ª q`, the absolute and relati¨e error in all four inequalities tends to zero. Proof. If w is a convex function on wxs, t , where s - t, then by Jensen's inequality we have wlŽ.sqŽ.1yltFlw Ž.Ž.Ž.s q 1 y lwt.15 Ž. If, in this inequality, we put w s F, s s x y 1 q b, t s x q b, l s b, then byŽ. 7 we get 1 log GŽ.x y x y log x q LxŽ. ž/2 3 Fblog GŽ.x y 1 q b y x yqb ž ž/2 =logŽ.Ž.x y 1 q b q Lxy1qb/ 1 qŽ.Ž.1yblog G x q b y x q b y ž ž/2 =logŽ.Ž.Ž.x q b q Lxqb/,16 374 MILAN MERKLE or, after some manipulations, b 1 b G Ž.Ž.x y 1 q b G y x q b GŽ.x Ž.x32bb Ž.1 bŽ.x b 12 Ž.Ž.xy1qb yrqxqb y qyr G expŽ.UxŽ.,b,L . xxy1r2 Ž.17 Since GŽ.Ž.Ž.x y 1 q b s G x q b r x y 1 q b , we obtain x 1 2 b b 1yb y r q GŽ.Žx q b Ž.x y 1 q b .Ž.x q b G expŽ.UxŽ.,b,L , GŽ.x xxy1r2 Ž.18 which is the left inequality inŽ. 13 . Similarly, the right inequality inŽ. 13 is obtained with F and Ž. 8 . Applying Jensen's inequality toŽ.Ž. concave function G and using 7 , one gets the right hand side ofŽ. 14 , and the left hand side of Ž. 14 is obtained with G andŽ. 8 . Since neither of the functions used for Jensen's inequality is a straight line, the equality in the obtained inequalities is possible if and only if b s 0orbs1. Let us now prove that absolute errors in inequalities converge to zero as x ª q`. This is a consequence of the fact that second derivatives of the convex function F and of the concave function G converge to zero as their arguments converge to q`. Indeed, for any convex function u, define rsrxŽ.,y,l,uby luxŽ.q Ž1yl .Ž.uysuŽ.lxq Ž1yl .yqrx Ž,y,l,u .. From the Taylor formula with the integral form of the remainder, it follows x uxŽ.sua Ž.qu9 Ž.Žaxya .qH Žxytu .0 Ž.tdt Ž19 . a y uyŽ.sua Ž.qu9 Ž.Žayya .qH Žyytu .0 Ž.tdt.20 Ž . a Letting a s l x q Ž.Ž1 y l y l g wx0, 1.Ž.Ž. , multiplying 19 by l and 20 by 1 y l and adding, we obtain 0 F rxŽ.,y,l,u y lxqŽ.1yly slHHŽtyxu .0 Ž.tdtq Ž1yl . Žyytu .0 Ž.tdt. x lxqŽ.1yly Ž.21 INEQUALITIES FOR THE GAMMA FUNCTION 375 If we let x ª q`, y s x q 1, and if we assume that u0 Ž.t ª 0as tªq`, we get that r ª 0.
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