JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 203, 369᎐380Ž. 1996 ARTICLE NO. 0385

Logarithmic Convexity and Inequalities for the Function*

Milan Merkle

Department of Applied , 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 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 , 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

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 xq␤sŽ.Ž.1y␤xq␤xq1.Ž. 8 Let us also introduce the following notation: For a function S Žs L or R. let UxŽ.Ž.Ž,␤,SsSxy␤Sxy1q␤ .Ž.Ž.Ž.y1y␤Sxq␤ 9 VxŽ.Ž,␤,SsyUxq␤,1y␤,S . sŽ.Ž.Ž.Ž.Ž.1y␤Sxq␤Sxq1ySxq␤.10 INEQUALITIES FOR THE GAMMA FUNCTION 373

Further, let AxŽ.,␤ and Bx Ž.Ž,␤ do not confuse the latter notation with the . 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␤

xq␤y1r2␤ Ž.xq␤x 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.1y␭tF␭␸ 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 F␤log ⌫Ž.x y 1 q ␤ y x yq␤ ž ž/2

=logŽ.Ž.x y 1 q ␤ q Lxy1q␤/ 1 qŽ.Ž.1y␤log ⌫ 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 0or␤s1. 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 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.1y␭y s␭HHŽtyxu .Љ Ž.tdtq Ž1y␭ . Žyytu .Љ Ž.tdt. x ␭xqŽ.1y␭y 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

xq␤y1r2 ␤ 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 xq␤y123 . 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 , 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 0or␤s1 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 # 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 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 qq␤ylogŽ. 1 q ␤ t ž/t 2

3 11 ␤Ž.1y␤t y␤qlogŽ. 1 q t y .31Ž. ž/t 2121Ž.Ž.qt1q␤t

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.,␤ s␤log 1 q 2 ž/24 x q 12␤ x y Ž.Ž.1 y ␤ 5 y ␤ 2 ␤ q␤log 1 yylog 1 q , ž/ž/2xq3 x we find the derivative

2 Ѩ␤Ž.1y␤UxŽ.,␤ 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

REFERENCES

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