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 ⌫Ž.Ž.x q  r⌫ x . As an application, we present an inequality that sharpens and generalizes inequalities due to Gautschi, Chu, Boyd, Lazarevic-´ Lupas¸, and Kershaw. ᮊ 1996 Academic Press, Inc.
1. SOME CONVEX AND CONCAVE FUNCTIONS
It is well known that the second derivative of the function x ¬ log ⌫Ž.x can be expressed in terms of the seriesŽ seewx 2 or w 10 x.
d2 11 1 2222log ⌫Ž.x sqqqиии 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 ⌫.
* This research was supported by the Science Fund of Serbia, Grant 0401A, through the Mathematical Institute, Belgrade.
369
0022-247Xr96 $18.00 Copyright ᮊ 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 ⌫ Ž.x
Fx2Ž.slog ⌫ Ž.x y x log x 1 Fx3Ž.slog ⌫ Ž.x y x y log x ž/2 1 FxŽ.slog ⌫ Ž.x y x y log x q LxŽ. ž/2 are con¨ex on x ) 0. Ž.ii The functions 1 GxŽ.slog ⌫ 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.qkq,2 Ž.2N2! qks0 Ž. for some g 0, 1 , where Bj are Bernoulli numbers defined by ttϱ 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--1 s q 2Nq2Ý 2Nq4 ks0Ž.xqkq
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 ϱ inŽ. 4 . Then Tm Ž,2N . and TmŽ.,2Nq1 converge to 0 and we obtain
2 N ϱ 2 Nq1 11 B2k 111 B2k qqÝÝ--qqÝ. xx2xx22kq12222xxkq1 ks1 ks0Ž.xqk ks1 Ž.5
FromŽ. 5 we see that, for positive x, FЉ Ž.x ) 0 and GЉ 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 ϱ Y 111 Fx3Ž.syÝ y 22x2x ks0Ž.xqk ϱ 11 1 syqÝ2 žŽ.xk xqkxqkq1 ks0 q 11 yq22 2Ž.Žxqk2xqkq1 ./ ϱ 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
qϱ 11 B2k log ⌫Ž.x ; x y log x y x q log 2 q Ý ž/222kŽ.2k1x2ky1 ks1 y Ž.xªqϱ. In fact, it follows from Theorem 1 that the function n 11 B2k xlog ⌫Ž.x y x y log x q x y log 2 y Ý ¬ž/222kŽ.2k1x2ky1 ks1 y is convex onŽ. 0, qϱ if n is even and it is concave for odd n.
2. BOUNDS FOR ⌫Ž.Ž.x q  r⌫ x
In this section we will use the equalities x s  Ž.Ž.Ž.Ž.x y 1 q  q 1 y  x q  7 xqsŽ.Ž.1yxqxq1.Ž. 8 Let us also introduce the following notation: For a function S Žs L or R. let UxŽ.Ž.Ž,,SsSxySxy1q .Ž.Ž.Ž.y1ySxq 9 VxŽ.Ž,,SsyUxq,1y,S . sŽ.Ž.Ž.Ž.Ž.1ySxqSxq1ySxq.10 INEQUALITIES FOR THE GAMMA FUNCTION 373
Further, let AxŽ., and Bx Ž.Ž, do not confuse the latter notation with the beta function. be defined by
x 1 2   1y y r q Ž.Ž.Ž.x y 1 q  x q  AxŽ., s Žx)1y .Ž.,11 xxy1r2 x BxŽ., s AxŽ.q,1y
xqy1r2 Ž.xqx Ž.x)0. Ž. 12 sŽ1.Ž x 1 2. Ž.x12 xxyy r Ž.q1qr The ratio ⌫Ž.Ž.x q  r⌫ x will be denoted by QxŽ.,. In this section we will derive general inequalities for QxŽ.,, using LC with functions F and G as defined in Theorem 1.
THEOREM 2. For  g wx0, 1 and for x ) 1 y  we ha¨e AxŽ.,expŽ.Ux Ž,,L .FQx Ž., FBx Ž.,expŽ.Vx Ž,,L .Ž.13 BxŽ.,expŽ.Vx Ž,,R .FQx Ž., FAx Ž.,expŽ.Ux Ž,,R .Ž.14 with equalities if and only if  s 0 and  s 1. As x ª qϱ, the absolute and relati¨e error in all four inequalities tends to zero. Proof. If is a convex function on wxs, t , where s - t, then by Jensen’s inequality we have
Ž.sqŽ.1ytF Ž.Ž.Ž.s q 1 y t.15 Ž. If, in this inequality, we put s F, s s x y 1 q , t s x q , s , then byŽ. 7 we get 1 log ⌫Ž.x y x y log x q LxŽ. ž/2 3 Flog ⌫Ž.x y 1 q  y x yq ž ž/2
=logŽ.Ž.x y 1 q  q Lxy1q/ 1 qŽ.Ž.1ylog ⌫ x q  y x q  y ž ž/2
=logŽ.Ž.Ž.x q  q Lxq/,16 374 MILAN MERKLE or, after some manipulations,  1  ⌫ Ž.Ž.x y 1 q  ⌫ y x q  ⌫Ž.x
Ž.x32 Ž.1 Ž.x  12 Ž.Ž.xy1q yrqxq y qyr G expŽ.UxŽ.,,L . xxy1r2 Ž.17
Since ⌫Ž.Ž.Ž.x y 1 q  s ⌫ x q  r x y 1 q  , we obtain x 1 2   1y y r q ⌫Ž.Žx q  Ž.x y 1 q  .Ž.x q  G expŽ.UxŽ.,,L , ⌫Ž.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  s 0ors1. Let us now prove that absolute errors in inequalities converge to zero as x ª qϱ. 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 qϱ. Indeed, for any convex function u, define rsrxŽ.,y,,uby uxŽ.q Ž1y .Ž.uysuŽ.xq Ž1y .yqrx Ž,y,,u .. From the Taylor formula with the integral form of the remainder, it follows x uxŽ.sua Ž.quЈ Ž.Žaxya .qH Žxytu .Љ Ž.tdt Ž19 . a y uyŽ.sua Ž.quЈ Ž.Žayya .qH Žyytu .Љ Ž.tdt.20 Ž . a Letting a s x q Ž.Ž1 y y g wx0, 1.Ž.Ž. , multiplying 19 by and 20 by 1 y and adding, we obtain 0 F rxŽ.,y,,u y xqŽ.1yy sHHŽtyxu .Љ Ž.tdtq Ž1y . Žyytu .Љ Ž.tdt. x xqŽ.1yy Ž.21 INEQUALITIES FOR THE GAMMA FUNCTION 375
If we let x ª qϱ, y s x q 1, and if we assume that uЉ Ž.t ª 0as tªqϱ, we get that r ª 0. Now replace u by F or yG to get the desired conclusion. Since QxŽ., ªqϱas x ª qϱ and 0 -  - 1, relative errors also converge to zero.
As one can see from the above proof, the restriction x ) 1 y  applies only to the left inequality inŽ. 13 and to the right inequality in Ž. 14 , as AxŽ., is defined for x ) 1 y . The remaining two inequalities hold for x ) 0. Theorem 2 gives a variety of inequalities that can be produced by taking various number of terms in L and in R. For instance, the left inequality in Ž. 14 , with R s R1 sy1r12 x reads
xqy1r2  Ž.x q  x  Ž.1 y  exp - QxŽ.,, Ž1.Ž x 1 2. Ž.x12 y xxyy r Ž.q1qr ž/12 xxŽ.Ž.q1 xq Ž.22 for x ) 0,  g Ž.0, 1 . Using the same technique with functions F12and F Žwhich cannot be considered as particular forms of F and G., one can also obtain bounds for Q. For example, the celebrated Walter Gautschi’s inequalitywx 6
 QxŽ.Ž, G xqy123 . Ž. follows easily upon applying Jensen’s inequality with F1 s log ⌫ and using Ž.7. It turns out that a great deal of known bounds for the ratio Q can be derived from Theorem 2, or from logarithmic convexity in general. Some inequalities for Q that involve the digamma function, as inwx 1 , can also be produced in this way. These topics will be considered in detail in our forthcoming papers. In the next section we give a new sharp bound for Q, based on Theorem 2, and we show that this generalizes and sharpens several known inequali- ties.
3. A NEW INEQUALITY
Gautschi’s inequalityŽ. 23 , published in 1959, has been an object of various improvements. We focus on one particular branch among many of them. 376 MILAN MERKLE
J. T. Chu in the articlewx 4 gives the result
⌫Ž.nr2 n y 12ny3 )(( Ž.24 ⌫Ž.nr2y1r222ny2
Ž.ns2, 3, . . . , which, after letting x s Ž.n y 1 r2, becomes
⌫Ž.x q 1r21 )xy.25Ž. ⌫Ž.x (4
Chu indicates that, for x s 1, 2, . . . , there is a sharper lower bound:
⌫Ž.x q 1r211 )xyq .26Ž. ⌫Ž.x (4 Ž.4x 22 q
In 1967, Boydwx 3 gives an inequality in the same spirit. The lower bound in our notation reads
⌫Ž.x q 1r211 )(xyq ,27Ž. ⌫Ž.x 432xq16 for x s m q 1r2, m s 1, 2, . . . . Finally, Lazarevic´ and Lupas¸’ resultwx 9 from 1979 reads
 ⌫Ž.x q  1 y  Gxy ,28Ž. ⌫Ž.x ž/2 for x ) Ž.1 y  r2 and  g wx0, 1 . This inequality was rediscovered by Kershawwx 8 in 1983. As we can see,Ž. 28 coincides with Ž. 25 where the latter holds, but it is much more general thanŽ. 25 . On the other hand,Ž. 27 is sharper thanŽ. 28 for a particular choice of x and . We now give an inequality which is a generalization of Boyd’s result and a sharpening of the inequalities of Gautschi, Chu, and Lazarevic´ and Lupas¸. THEOREM 3. For any x G Ž.1 y  r2 and  g wx0, 1 ,
2  1 y  1 y  QxŽ., G xyq ,29Ž. ž/224xq12 with equality if and only if  s 0 or  s 1. INEQUALITIES FOR THE GAMMA FUNCTION 377
Proof. If  s 0ors1 it is easy to see that equality occurs inŽ. 29 . Therefore, we have to show thatŽ. 29 holds as a strict inequality for gŽ.0, 1 . Let us denote the right hand side inŽ. 29 by C# Žx,  .and let
Ž.x,  s Qx Ž.Ž.,rC#x,.
Borrowing an idea from Kershawwx 8 , we will show thatŽ. 29 holds by showing: Ž.i lim Žx,  .1, for every  Ž0, 1 . . x ªqϱ s g Ž.ii Žx q 1,  .- Žx,  ., for every  g Ž.0, 1 and every x G Ž1 y  .r2. Indeed, fromŽ. ii it follows that for every natural number n, Žx q n,  . -Ž.x,and by letting n ª qϱ and usingŽ. i we get
Ž.x,  ) lim Ž.x,  s 1, xªqϱ which gives the inequalityŽ. 29 .
Proof of Ž.i . Letting t s 1rx and using Taylor’s expansion we find, after some elaboration,
1  2 2 3 1 y y q y 22 log C#Ž.x,  s  ylog t y t q t q otŽ. ž/212 Ž.Ž.tª0. 30
Let us denote the lower bound in inequalityŽ. 22 by Cx Ž, .and let t s 1rx again. Then we have
11 log CxŽ., sy log t qqylogŽ. 1 q  t ž/t 2
3 11 Ž.1yt yqlogŽ. 1 q t y .31Ž. ž/t 2121Ž.Ž.qt1qt
By the Taylor expansion inŽ. 31 , one can see that the expansionŽ. 30 holds for log C too; therefore,
2 log C#Ž.x,  y log Cx Ž., soŽ.1rx Žxªqϱ ..32 Ž. 378 MILAN MERKLE
From the Theorem 2 we have that
QxŽ., lim s 1 xªqϱCxŽ., and byŽ. 32 we conclude that also
QxŽ., lim s 1, xªqϱ C#Ž.x,  which ends the proof ofŽ. i .
Proof of Ž.ii . Let Dx Ž, .slog Žx,  .y log Žx q 1,  .. We have to show that
DxŽ., )0 for x G Ž.1 y  r2. Ž.33 It is clear from the previous considerations that lim DxŽ., 0. x ªqϱ s Therefore, to showŽ. 33 it suffices to show
Ѩ DxŽ., -0 for x G Ž1 y  .r2,  g Ž.0,1 . Ž. 34 Ѩx
Writing DxŽ., in the form
48 x q 12 q 24 DxŽ., slog 1 q 2 ž/24 x q 12 x y Ž.Ž.1 y  5 y  2  qlog 1 yylog 1 q , ž/ž/2xq3 x we find the derivative
2 ѨŽ.1yUxŽ., DxŽ., s и ,35Ž. ѨxxŽ.Ž.Ž.Ž.Ž.2xq32xq1 xq Vx,Wx, where
3 2 2 UxŽ., s576 x Ž y 2 .q 8 x Ž.11 q 120 y 299 32 y4xŽ.y22 y 115 q 370 y 319Ž.Ž.y 5y , 2 2 VxŽ.,s24 x q 12 x Ž q 4 .y  q 18 q 19, 2 WxŽ., s24 x q 12 x y Ž1 y  .Ž.5 y  . INEQUALITIES FOR THE GAMMA FUNCTION 379
Ž. Ž. Let x0 s 1 y  r2. The sign of the expression in 35 depends only on Ž. the sign of U, V, W. We will show that for every  g 0, 1 and x G x0 , the terms V and W are positive and the term U is negative. Ž. Ž. Term V. It is not difficult to see that Vx0 ,)0 for every  g 0, 1 . Further, for x ) x0 we have
Ѩ VxŽ.,s48 x q 12 Ž q 4 .) 12 Ž 6 y  .) 0, Ѩx
Ž. Ž. thus Vx,)0 for every  g 0, 1 and x G x0. Ž. 2 Term W.ByWx0 , s1y )0 and
Ѩ WxŽ., s48 x q 12 ) 0, Ѩx
Ž. Ž. we see that Wx, )0 for  g 0, 1 and x G x0.
Term U. Here we evaluate UxŽ., at x0 as
4 3 2 UxŽ.0, sy316Ž. y170 q 631 y 994 q 589 .Ž. 36
By localizing zeros of the polynomial inŽ. 36 , we conclude that UxŽ.0, -0 for  g Ž.0, 1 . Further,
Ѩ UxŽ., Ѩx 2 2 sy1728 x Ž.2 y  q 16 xŽ.11 q 120 y 299 32 y4Ž.y22 y 115 q 370 .Ž. 37
Both zeros of the quadratic trinomial inŽ. 37 are negative for  g Ž.0, 1 ; therefore the derivative of U with respect to x ) 0 is negative, and from Ž. Ž. Ž. Ux0 , -0 it follows that Ux, -0 for  g 0, 1 and x G x0. This ends the proof of the theorem. Let us remark that the bound inŽ. 29 agrees with the bound Ž. 22 to the 2 order of magnitude of 1rx as x ª qϱ. Therefore, according to Theorem 2, the error converges to zero as x ª qϱ. In the following table we provide some sample numerical values of relative errors inŽ. 29 , for x s 1 and x s 3. 380 MILAN MERKLE
Relative Errors inŽ. 29 , in %  0.1 0.3 0.5 0.7 0.9
x s 1 0.50 0.96 0.93 0.64 0.22 x s 3 0.016 0.036 0.041 0.031 0.012
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