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Logarithmic Convexity and Inequalities for the Gamma Function
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
203, 369]380 Ž1996.
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 Submitted by A. M. Fink Received June 27, 1995
We propose a method, based on logarithmic convexity, for producing sharp bounds for the ratio G Ž x q b .rG Ž x .. As an application, we present an inequality that sharpens and generalizes inequalities due to Gautschi, Chu, Boyd, Lazarevic´ Lupas¸, 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 G Ž x . can be expressed in terms of the series Žsee w2x or w10x. d2 dx
2
log G Ž x . s
1 x
2
q
1
Ž x q 1.
2
q
1
Ž x q 2.
2
q ???
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 2N
L Ž x . s L2 N Ž x . s y
Ý ks1
B2 k 2 k Ž 2 k y 1 . x 2 ky1
Ž N s 1, 2, . . . . , L0 Ž x . s 0, 2 Nq1
R Ž x . s R 2 Nq1 Ž x . s y
Ý
B2 k 2 k Ž 2 k y 1 . x 2 ky1
ks1
Ž N s 0, 1, 2, . . . . .
The functions
Ži.
F1 Ž x . s log G Ž x . F2 Ž x . s log G Ž x . y x log x F3 Ž x . s log G Ž x . y x y
ž ž
F Ž x . s log G Ž x . y x y
1 2 1 2
/ /
log x
/
log x q R Ž x .
log x q L Ž x .
are con¨ ex on x ) 0. Žii. The functions G Ž x . s log G Ž x . y x y
ž
1 2
are conca¨ e on x ) 0. Proof. We will apply the Euler]MacLaurin summation formula to the series in Ž1.. For a finite number of terms, m say, we have m
Ý f Ž x q k. s H ks0
xqm
x
f Ž t . dt q
N
q q
1 2
Ž f Ž x q m. q f Ž x . .
B2 k
Ý Ž 2 k . ! Ž f Ž2 ky1. Ž x q m . y f Ž2 ky1. Ž x . . ks1 my1
B2 Nq2
Ž 2 N q 2. !
Ý
f Ž2 Nq2. Ž x q k q u . ,
ks0
for some u g Ž0, 1., where Bj are Bernoulli numbers defined by t et y 1
s
`
Ý js0
Bj
tj j!
.
Ž 2.
371
INEQUALITIES FOR THE GAMMA FUNCTION
First five Bernoulli numbers with even indices are B2 s
1 6
B4 s y
,
1 30
B6 s
,
1 42
B8 s y
,
1 30
B10 s
,
5 66
.
Letting f Ž x . s xy2 in Ž2., we obtain Žsee also w7, 5.8x for this particular case. m
1
Ý ks0
Ž x q k.
2
s S Ž N . q T Ž m, N . q E Ž m, N .
Ž 3.
where SŽ N . s T Ž m, N . s
1
q
x
1 2
2Ž x q m.
N
1 2x
y
2
q
ks1
Ý ks0
B2 k x 2 kq1 N
1 xqm
my1
E Ž m, N . s Ž 2 N q 3 . B2 Nq2
Ý y
Ý ks1
B2 k
Ž x q m.
1
Ž xqkqu.
2 kq1
Ž 0 - u - 1.
2 Nq4
and the sign of EŽ m, N . is clearly equal to the sign of B2 Nq2 , which is Žy1. N Žsee w5, Sect. 449x.. This implies that for every m G 1, N G 1, and x ) 0 we have m
S Ž 2 N . q T Ž m, 2 N . F
1
Ý ks0
Ž x q k.
2
F S Ž 2 N q 1 . q T Ž m, 2 N q 1 . .
Ž 4. Now for x ) 0 and N G 1 being fixed, let m ª ` in Ž4.. Then T Ž m, 2 N . and T Ž m, 2 N q 1. converge to 0 and we obtain 1 x
q
2N
1 2x
2
q
Ý ks1
B2 k x
2 kq1
-
`
Ý ks0
1
Ž x q k.
2
-
1 x
q
2 Nq1
1 2x
2
q
Ý ks1
B2 k x 2 kq1
.
Ž 5. From Ž5. we see that, for positive x, F0 Ž x . ) 0 and G0 Ž x . - 0. As we already noticed, the expansion Ž1. implies that F1 is a convex function, so it remains to show convexity of F2 and F3 . Convexity of F3
372
MILAN MERKLE
follows from F3Y Ž x . s
`
1
Ý ks0
s
`
Ý ks0
Ž x q k.
ž
y
1
Ž x q k. y
s
2
1 x
y
2
y
`
2 x2
1 xqk
1 2Ž x q k .
1
2
q
q
ks0
xqkq1 1
2 Ž x q k q 1.
1
Ý
1
2
2 Ž x q k . Ž x q k q 1.
2
2
/
) 0.
Ž 6.
Since F2Y Ž x . ) F3Y Ž x . for x ) 0, the function F2 is also convex, which ends the proof. The functions F and G introduced in Theorem 1 are closely related to the well known asymptotic expansion log G Ž x . ; x y
ž
1 2
/
log x y x q
1 2
log 2p q
q`
Ý ks1
B2 k 2 k Ž 2 k y 1 . x 2 ky1
Ž x ª q`. . In fact, it follows from Theorem 1 that the function x ¬ log G Ž x . y x y
ž
1 2
/
log x q x y
1 2
n
log 2p y
Ý ks1
B2 k 2 k Ž 2 k y 1 . x 2 ky1
is convex on Ž0, q`. if n is even and it is concave for odd n. 2. BOUNDS FOR G Ž x q b .rG Ž x . In this section we will use the equalities x s b Ž x y 1 q b . q Ž1 y b . Ž x q b .
Ž 7.
x q b s Ž 1 y b . x q b Ž x q 1. .
Ž 8.
Let us also introduce the following notation: For a function S Žs L or . R let U Ž x, b , S . s S Ž x . y b S Ž x y 1 q b . y Ž 1 y b . S Ž x q b .
Ž 9.
V Ž x, b , S . s yU Ž x q b , 1 y b , S . s Ž 1 y b . S Ž x . q b S Ž x q 1. y S Ž x q b . .
Ž 10 .
373
INEQUALITIES FOR THE GAMMA FUNCTION
Further, let AŽ x, b . and B Ž x, b . Ždo not confuse the latter notation with the beta function. be defined by A Ž x, b . s
Ž Ž x y 1 q b . b Ž x q b . 1y b .
B Ž x, b . s s
xy1r2q b
Ž x ) 1 y b . , Ž 11 .
x xy1r2 x AŽ x q b , 1 y b .
Ž xqb.
xq b y1r2
x Ž1y b .Ž xy1r2. Ž x q 1 .
xb
Ž x ) 0. .
b Ž xq1r2 .
Ž 12 .
The ratio G Ž x q b .rG Ž x . will be denoted by QŽ x, b .. In this section we will derive general inequalities for QŽ x, b ., using LC with functions F and G as defined in Theorem 1. THEOREM 2.
For b g w0, 1x and for x ) 1 y b we ha¨ e
A Ž x, b . exp Ž U Ž x, b , L . . F Q Ž x, b . F B Ž x, b . exp Ž V Ž x, b , L . . Ž 13 . B Ž x, b . exp Ž V Ž x, b , R . . F Q Ž x, b . F A Ž x, b . exp Ž U Ž x, 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 w s, t x, where s - t, then by Jensen’s inequality we have
w Ž l s q Ž 1 y l . t . F lw Ž s . q Ž 1 y l . w Ž t . .
Ž 15 .
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 by Ž7. we get log G Ž x . y x y
ž
1 2
/
log x q L Ž x . 3
F b log G Ž x y 1 q b . y x y
ž
ž
2
qb
/
=log Ž x y 1 q b . q L Ž x y 1 q b .
/
q Ž 1 y b . log G Ž x q b . y x q b y
ž
ž
1 2
/
=log Ž x q b . q L Ž x q b . ,
/
Ž 16 .
374
MILAN MERKLE
or, after some manipulations, G b Ž x y 1 q b . G 1y b Ž x q b . GŽ x. G
Ž xy1qb.
Ž xy3r2q b . b
Ž x q b . 1y b Ž
.Ž xq b y1r2 .
exp Ž U Ž x, b , L . . .
x xy1r2
Ž 17 . Since G Ž x y 1 q b . s G Ž x q b .rŽ x y 1 q b ., we obtain GŽ x q b .
G
GŽ x.
Ž Ž x y 1 q b . b Ž x q b . 1y b .
xy1r2q b
exp Ž U Ž x, b , L . . ,
x xy1r2
Ž 18 . which is the left inequality in Ž13.. Similarly, the right inequality in Ž13. is obtained with F and Ž8.. Applying Jensen’s inequality to Žconcave function. G and using Ž7., one gets the right hand side of Ž14., and the left hand side of Ž14. is obtained with G and Ž8.. 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 0 or b s 1. 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 r s r Ž x, y, l, u. by
l u Ž x . q Ž 1 y l . u Ž y . s u Ž l x q Ž 1 y l . y . q r Ž x, y, l , u . . From the Taylor formula with the integral form of the remainder, it follows u Ž x . s u Ž a . q u9 Ž a . Ž x y a . q u Ž y . s u Ž a . q u9 Ž a . Ž y y a . q
x
Ha Ž x y t . u0 Ž t . dt
Ž 19 .
y
Ha Ž y y t . u0 Ž t . dt.
Ž 20 .
Letting a s l x q Ž1 y l. y Ž l g w0, 1x., multiplying Ž19. by l and Ž20. by 1 y l and adding, we obtain 0 F r Ž x, y, l , u . y
Hxl xq 1yl y Ž t y x . u0 Ž t . dt q Ž 1 y l. Hl xqŽ1yl. y Ž y y t . u0 Ž t . dt.
sl
Ž
.
Ž 21 .
375
INEQUALITIES FOR THE GAMMA FUNCTION
If we let x ª q`, y s x q 1, and if we assume that u0 Ž t . ª 0 as t ª q`, we get that r ª 0. Now replace u by F or yG to get the desired conclusion. Since QŽ x, b . ª q` as x ª q` and 0 - b - 1, relative errors also converge to zero. As one can see from the above proof, the restriction x ) 1 y b applies only to the left inequality in Ž13. and to the right inequality in Ž14., as AŽ x, b . is defined for x ) 1 y b . 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 Ž14., with R s R1 s y1r12 x reads
Ž xqb. x
Ž1y b .Ž xy1r2.
xq b y1r2
Ž x q 1.
xb
b Ž xq1r2 .
ž
exp y
b Ž1 y b . 12 x Ž x q 1 . Ž x q b .
/
- Q Ž x, b . ,
Ž 22 . for x ) 0, b g Ž0, 1.. Using the same technique with functions F1 and F2 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 inequality w6x Q Ž x, b . G Ž x q b y 1 .
b
Ž 23 .
follows easily upon applying Jensen’s inequality with F1 s log G and using Ž7.. 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 in w1x, 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 inequalities.
3. A NEW INEQUALITY Gautschi’s inequality Ž23., 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 article w4x gives the result G Ž nr2. G Ž nr2 y 1r2 .
ny1
2n y 3
2
2n y 2
( (
)
Ž 24 .
Ž n s 2, 3, . . . ., which, after letting x s Ž n y 1.r2, becomes G Ž x q 1r2 .
)
GŽ x.
(
xy
1 4
.
Ž 25 .
Chu indicates that, for x s 1, 2, . . . , there is a sharper lower bound: G Ž x q 1r2 . GŽ x.
)
(
xy
1 4
q
1
Ž 4 x q 2.
.
2
Ž 26 .
In 1967, Boyd w3x gives an inequality in the same spirit. The lower bound in our notation reads G Ž x q 1r2 . GŽ x.
)
(
xy
1 4
q
1 32 x q 16
,
Ž 27 .
for x s m q 1r2, m s 1, 2, . . . . Finally, Lazarevic´ and Lupas¸’ result w9x from 1979 reads GŽ x q b . GŽ x.
G xy
ž
1yb 2
b
/
,
Ž 28 .
for x ) Ž1 y b .r2 and b g w0, 1x. This inequality was rediscovered by Kershaw w8x in 1983. As we can see, Ž28. coincides with Ž25. where the latter holds, but it is much more general than Ž25.. On the other hand, Ž27. is sharper than Ž28. for a particular choice of x and b . 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 Ž1 y b .r2 and b g w0, 1x, Q Ž x, b . G x y
ž
1yb 2
q
with equality if and only if b s 0 or b s 1.
1yb2 24 x q 12
b
/
,
Ž 29 .
377
INEQUALITIES FOR THE GAMMA FUNCTION
Proof. If b s 0 or b s 1 it is easy to see that equality occurs in Ž29.. Therefore, we have to show that Ž29. holds as a strict inequality for b g Ž0, 1.. Let us denote the right hand side in Ž29. by C#Ž x, b . and let
r Ž x, b . s Q Ž x, b . rC# Ž x, b . . Borrowing an idea from Kershaw w8x, we will show that Ž29. holds by showing: Ži. lim x ªq` r Ž x, b . s 1, for every b g Ž0, 1.. Žii. r Ž x q 1, b . - r Ž x, b ., for every b g Ž0, 1. and every x G Ž1 y b .r2. Indeed, from Žii. it follows that for every natural number n, r Ž x q n, b . - r Ž x, b . and by letting n ª q` and using Ži. we get
r Ž x, b . ) lim r Ž x, b . s 1, xªq`
which gives the inequality Ž29.. Proof of Ži.. Letting t s 1rx and using Taylor’s expansion we find, after some elaboration, 1yb
log C# Ž x, b . s b ylog t y
ž
2
tq
y2 b 2 q 3 b y 1 12
t 2 q oŽ t 2 .
/
Ž t ª 0 . . Ž 30 . Let us denote the lower bound in inequality Ž22. by C Ž x, b . and let t s 1rx again. Then we have log C Ž x, b . s yb log t q yb
ž
1 t
q
ž 1 2
1 t
/
qby
1 2
/
log Ž 1 q b t .
log Ž 1 q t . y
b Ž1 y b . t3 12 Ž 1 q t . Ž 1 q b t .
. Ž 31 .
By the Taylor expansion in Ž31., one can see that the expansion Ž30. holds for log C too; therefore, log C# Ž x, b . y log C Ž x, b . s o Ž 1rx 2 .
Ž x ª q`. .
Ž 32 .
378
MILAN MERKLE
From the Theorem 2 we have that Q Ž x, b .
lim
C Ž x, b .
xªq`
s1
and by Ž32. we conclude that also Q Ž x, b .
lim
xªq`
C# Ž x, b .
s 1,
which ends the proof of Ži.. Proof of Žii.. Let DŽ x, b . s log r Ž x, b . y log r Ž x q 1, b .. We have to show that D Ž x, b . ) 0
for x G Ž 1 y b . r2.
Ž 33 .
It is clear from the previous considerations that lim x ªq` DŽ x, b . s 0. Therefore, to show Ž33. it suffices to show
x
D Ž x, b . - 0
for x G Ž 1 y b . r2, b g Ž 0, 1 . .
Ž 34 .
Writing DŽ x, b . in the form D Ž x, b . s b log 1 q
ž
48 x q 12 b q 24 24 x q 12 b x y Ž 1 y b . Ž 5 y b .
q b log 1 y
ž
2
2 2xq3
/
y log 1 q
ž
b x
/
/
,
we find the derivative
x
D Ž x, b . s
b Ž1 y b 2 . x Ž 2 x q 3. Ž 2 x q 1. Ž x q b .
?
U Ž x, b . V Ž x, b . W Ž x, b .
, Ž 35 .
where U Ž x, b . s 576 x 3 Ž b y 2 . q 8 x 2 Ž 11 b 2 q 120 b y 299 . y 4 x Ž b 3 y 22 b 2 y 115 b q 370 . y 3 Ž 19 y b . Ž 5 y b . , V Ž x, b . s 24 x 2 q 12 x Ž b q 4 . y b 2 q 18 b q 19, W Ž x, b . s 24 x 2 q 12 b x y Ž 1 y b . Ž 5 y b . .
INEQUALITIES FOR THE GAMMA FUNCTION
379
Let x 0 s Ž1 y b .r2. The sign of the expression in Ž35. depends only on the sign of U, V, W. We will show that for every b g Ž0, 1. and x G x 0 , the terms V and W are positive and the term U is negative. Term V. It is not difficult to see that V Ž x 0 , b . ) 0 for every b g Ž0, 1.. Further, for x ) x 0 we have
x
V Ž x, b . s 48 x q 12 Ž b q 4 . ) 12 Ž 6 y b . ) 0,
thus V Ž x, b . ) 0 for every b g Ž0, 1. and x G x 0 . Term W. By W Ž x 0 , b . s 1 y b 2 ) 0 and
x
W Ž x, b . s 48 x q 12 b ) 0,
we see that W Ž x, b . ) 0 for b g Ž0, 1. and x G x 0 . Term U. Here we evaluate UŽ x, b . at x 0 as U Ž x 0 , b . s y3 Ž 16 b 4 y 170 b 3 q 631 b 2 y 994b q 589 . .
Ž 36 .
By localizing zeros of the polynomial in Ž36., we conclude that UŽ x 0 , b . - 0 for b g Ž0, 1.. Further,
x
U Ž x, b . s y1728 x 2 Ž 2 y b . q 16 x Ž 11 b 2 q 120 b y 299 . y 4 Ž b 3 y 22 b 2 y 115 b q 370 . .
Ž 37 .
Both zeros of the quadratic trinomial in Ž37. are negative for b g Ž0, 1.; therefore the derivative of U with respect to x ) 0 is negative, and from UŽ x 0 , b . - 0 it follows that UŽ x, b . - 0 for b g Ž0, 1. and x G x 0 . This ends the proof of the theorem. Let us remark that the bound in Ž29. agrees with the bound Ž22. to the order of magnitude of 1rx 2 as x ª q`. Therefore, according to Theorem 2, the error converges to zero as x ª q`. In the following table we provide some sample numerical values of relative errors in Ž29., for x s 1 and x s 3.
380
MILAN MERKLE
Relative Errors in Ž29., in %
b
0.1
0.3
0.5
0.7
0.9
xs1 xs3
0.50 0.016
0.96 0.036
0.93 0.041
0.64 0.031
0.22 0.012
REFERENCES 1. H. Alzer, Some Gamma function inequalities, Math. Comp. 60 Ž1993., 337]346. 2. E. Artin, ‘‘The Gamma Function,’’ Holt, Rinehart & Winston, New York, 1964; translation from the German original of 1931. 3. A. V. Boyd, A note on a paper by Uppuluri, Pacific J. Math. 22 Ž1967., 9]10. 4. J. T. Chu, A modified Wallis product and some applications, Amer. Math. Monthly 69, No. 5 Ž1969., 402]404. 5. G. M. Fihtengolc, ‘‘Kurs differencialnogo i integralnogo iscisleniya,’’ Vol. II, Moskva, ˇ 1970. ŽIn Russian.. 6. W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 Ž1959., 77]81. 7. F. B. Hildebrand, ‘‘Introduction to Numerical Analysis,’’ 2nd ed., McGraw]Hill, New York, 1974. 8. D. Kershaw, Some extensions of W. Gautchi’s inequalities for the gamma function, Math. Comp. 41 Ž1983., 607]611. 9. I. Lazarevic´ and A. Lupas¸, Functional equations for Wallis and Gamma functions, Uni¨ . Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 461]497 Ž1979., 245]251. 10. D. S. Mitrinovic, ´ ‘‘Analytic Inequalities,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1970. 11. V. R. R. Uppuluri, A stronger version of Gautschi’s inequality satisfied by the Gamma function, Scand. Actuar. J. 1–2 Ž1965., 51]52.
203, 369]380 Ž1996.
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 Submitted by A. M. Fink Received June 27, 1995
We propose a method, based on logarithmic convexity, for producing sharp bounds for the ratio G Ž x q b .rG Ž x .. As an application, we present an inequality that sharpens and generalizes inequalities due to Gautschi, Chu, Boyd, Lazarevic´ Lupas¸, 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 G Ž x . can be expressed in terms of the series Žsee w2x or w10x. d2 dx
2
log G Ž x . s
1 x
2
q
1
Ž x q 1.
2
q
1
Ž x q 2.
2
q ???
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 2N
L Ž x . s L2 N Ž x . s y
Ý ks1
B2 k 2 k Ž 2 k y 1 . x 2 ky1
Ž N s 1, 2, . . . . , L0 Ž x . s 0, 2 Nq1
R Ž x . s R 2 Nq1 Ž x . s y
Ý
B2 k 2 k Ž 2 k y 1 . x 2 ky1
ks1
Ž N s 0, 1, 2, . . . . .
The functions
Ži.
F1 Ž x . s log G Ž x . F2 Ž x . s log G Ž x . y x log x F3 Ž x . s log G Ž x . y x y
ž ž
F Ž x . s log G Ž x . y x y
1 2 1 2
/ /
log x
/
log x q R Ž x .
log x q L Ž x .
are con¨ ex on x ) 0. Žii. The functions G Ž x . s log G Ž x . y x y
ž
1 2
are conca¨ e on x ) 0. Proof. We will apply the Euler]MacLaurin summation formula to the series in Ž1.. For a finite number of terms, m say, we have m
Ý f Ž x q k. s H ks0
xqm
x
f Ž t . dt q
N
q q
1 2
Ž f Ž x q m. q f Ž x . .
B2 k
Ý Ž 2 k . ! Ž f Ž2 ky1. Ž x q m . y f Ž2 ky1. Ž x . . ks1 my1
B2 Nq2
Ž 2 N q 2. !
Ý
f Ž2 Nq2. Ž x q k q u . ,
ks0
for some u g Ž0, 1., where Bj are Bernoulli numbers defined by t et y 1
s
`
Ý js0
Bj
tj j!
.
Ž 2.
371
INEQUALITIES FOR THE GAMMA FUNCTION
First five Bernoulli numbers with even indices are B2 s
1 6
B4 s y
,
1 30
B6 s
,
1 42
B8 s y
,
1 30
B10 s
,
5 66
.
Letting f Ž x . s xy2 in Ž2., we obtain Žsee also w7, 5.8x for this particular case. m
1
Ý ks0
Ž x q k.
2
s S Ž N . q T Ž m, N . q E Ž m, N .
Ž 3.
where SŽ N . s T Ž m, N . s
1
q
x
1 2
2Ž x q m.
N
1 2x
y
2
q
ks1
Ý ks0
B2 k x 2 kq1 N
1 xqm
my1
E Ž m, N . s Ž 2 N q 3 . B2 Nq2
Ý y
Ý ks1
B2 k
Ž x q m.
1
Ž xqkqu.
2 kq1
Ž 0 - u - 1.
2 Nq4
and the sign of EŽ m, N . is clearly equal to the sign of B2 Nq2 , which is Žy1. N Žsee w5, Sect. 449x.. This implies that for every m G 1, N G 1, and x ) 0 we have m
S Ž 2 N . q T Ž m, 2 N . F
1
Ý ks0
Ž x q k.
2
F S Ž 2 N q 1 . q T Ž m, 2 N q 1 . .
Ž 4. Now for x ) 0 and N G 1 being fixed, let m ª ` in Ž4.. Then T Ž m, 2 N . and T Ž m, 2 N q 1. converge to 0 and we obtain 1 x
q
2N
1 2x
2
q
Ý ks1
B2 k x
2 kq1
-
`
Ý ks0
1
Ž x q k.
2
-
1 x
q
2 Nq1
1 2x
2
q
Ý ks1
B2 k x 2 kq1
.
Ž 5. From Ž5. we see that, for positive x, F0 Ž x . ) 0 and G0 Ž x . - 0. As we already noticed, the expansion Ž1. implies that F1 is a convex function, so it remains to show convexity of F2 and F3 . Convexity of F3
372
MILAN MERKLE
follows from F3Y Ž x . s
`
1
Ý ks0
s
`
Ý ks0
Ž x q k.
ž
y
1
Ž x q k. y
s
2
1 x
y
2
y
`
2 x2
1 xqk
1 2Ž x q k .
1
2
q
q
ks0
xqkq1 1
2 Ž x q k q 1.
1
Ý
1
2
2 Ž x q k . Ž x q k q 1.
2
2
/
) 0.
Ž 6.
Since F2Y Ž x . ) F3Y Ž x . for x ) 0, the function F2 is also convex, which ends the proof. The functions F and G introduced in Theorem 1 are closely related to the well known asymptotic expansion log G Ž x . ; x y
ž
1 2
/
log x y x q
1 2
log 2p q
q`
Ý ks1
B2 k 2 k Ž 2 k y 1 . x 2 ky1
Ž x ª q`. . In fact, it follows from Theorem 1 that the function x ¬ log G Ž x . y x y
ž
1 2
/
log x q x y
1 2
n
log 2p y
Ý ks1
B2 k 2 k Ž 2 k y 1 . x 2 ky1
is convex on Ž0, q`. if n is even and it is concave for odd n. 2. BOUNDS FOR G Ž x q b .rG Ž x . In this section we will use the equalities x s b Ž x y 1 q b . q Ž1 y b . Ž x q b .
Ž 7.
x q b s Ž 1 y b . x q b Ž x q 1. .
Ž 8.
Let us also introduce the following notation: For a function S Žs L or . R let U Ž x, b , S . s S Ž x . y b S Ž x y 1 q b . y Ž 1 y b . S Ž x q b .
Ž 9.
V Ž x, b , S . s yU Ž x q b , 1 y b , S . s Ž 1 y b . S Ž x . q b S Ž x q 1. y S Ž x q b . .
Ž 10 .
373
INEQUALITIES FOR THE GAMMA FUNCTION
Further, let AŽ x, b . and B Ž x, b . Ždo not confuse the latter notation with the beta function. be defined by A Ž x, b . s
Ž Ž x y 1 q b . b Ž x q b . 1y b .
B Ž x, b . s s
xy1r2q b
Ž x ) 1 y b . , Ž 11 .
x xy1r2 x AŽ x q b , 1 y b .
Ž xqb.
xq b y1r2
x Ž1y b .Ž xy1r2. Ž x q 1 .
xb
Ž x ) 0. .
b Ž xq1r2 .
Ž 12 .
The ratio G Ž x q b .rG Ž x . will be denoted by QŽ x, b .. In this section we will derive general inequalities for QŽ x, b ., using LC with functions F and G as defined in Theorem 1. THEOREM 2.
For b g w0, 1x and for x ) 1 y b we ha¨ e
A Ž x, b . exp Ž U Ž x, b , L . . F Q Ž x, b . F B Ž x, b . exp Ž V Ž x, b , L . . Ž 13 . B Ž x, b . exp Ž V Ž x, b , R . . F Q Ž x, b . F A Ž x, b . exp Ž U Ž x, 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 w s, t x, where s - t, then by Jensen’s inequality we have
w Ž l s q Ž 1 y l . t . F lw Ž s . q Ž 1 y l . w Ž t . .
Ž 15 .
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 by Ž7. we get log G Ž x . y x y
ž
1 2
/
log x q L Ž x . 3
F b log G Ž x y 1 q b . y x y
ž
ž
2
qb
/
=log Ž x y 1 q b . q L Ž x y 1 q b .
/
q Ž 1 y b . log G Ž x q b . y x q b y
ž
ž
1 2
/
=log Ž x q b . q L Ž x q b . ,
/
Ž 16 .
374
MILAN MERKLE
or, after some manipulations, G b Ž x y 1 q b . G 1y b Ž x q b . GŽ x. G
Ž xy1qb.
Ž xy3r2q b . b
Ž x q b . 1y b Ž
.Ž xq b y1r2 .
exp Ž U Ž x, b , L . . .
x xy1r2
Ž 17 . Since G Ž x y 1 q b . s G Ž x q b .rŽ x y 1 q b ., we obtain GŽ x q b .
G
GŽ x.
Ž Ž x y 1 q b . b Ž x q b . 1y b .
xy1r2q b
exp Ž U Ž x, b , L . . ,
x xy1r2
Ž 18 . which is the left inequality in Ž13.. Similarly, the right inequality in Ž13. is obtained with F and Ž8.. Applying Jensen’s inequality to Žconcave function. G and using Ž7., one gets the right hand side of Ž14., and the left hand side of Ž14. is obtained with G and Ž8.. 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 0 or b s 1. 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 r s r Ž x, y, l, u. by
l u Ž x . q Ž 1 y l . u Ž y . s u Ž l x q Ž 1 y l . y . q r Ž x, y, l , u . . From the Taylor formula with the integral form of the remainder, it follows u Ž x . s u Ž a . q u9 Ž a . Ž x y a . q u Ž y . s u Ž a . q u9 Ž a . Ž y y a . q
x
Ha Ž x y t . u0 Ž t . dt
Ž 19 .
y
Ha Ž y y t . u0 Ž t . dt.
Ž 20 .
Letting a s l x q Ž1 y l. y Ž l g w0, 1x., multiplying Ž19. by l and Ž20. by 1 y l and adding, we obtain 0 F r Ž x, y, l , u . y
Hxl xq 1yl y Ž t y x . u0 Ž t . dt q Ž 1 y l. Hl xqŽ1yl. y Ž y y t . u0 Ž t . dt.
sl
Ž
.
Ž 21 .
375
INEQUALITIES FOR THE GAMMA FUNCTION
If we let x ª q`, y s x q 1, and if we assume that u0 Ž t . ª 0 as t ª q`, we get that r ª 0. Now replace u by F or yG to get the desired conclusion. Since QŽ x, b . ª q` as x ª q` and 0 - b - 1, relative errors also converge to zero. As one can see from the above proof, the restriction x ) 1 y b applies only to the left inequality in Ž13. and to the right inequality in Ž14., as AŽ x, b . is defined for x ) 1 y b . 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 Ž14., with R s R1 s y1r12 x reads
Ž xqb. x
Ž1y b .Ž xy1r2.
xq b y1r2
Ž x q 1.
xb
b Ž xq1r2 .
ž
exp y
b Ž1 y b . 12 x Ž x q 1 . Ž x q b .
/
- Q Ž x, b . ,
Ž 22 . for x ) 0, b g Ž0, 1.. Using the same technique with functions F1 and F2 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 inequality w6x Q Ž x, b . G Ž x q b y 1 .
b
Ž 23 .
follows easily upon applying Jensen’s inequality with F1 s log G and using Ž7.. 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 in w1x, 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 inequalities.
3. A NEW INEQUALITY Gautschi’s inequality Ž23., 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 article w4x gives the result G Ž nr2. G Ž nr2 y 1r2 .
ny1
2n y 3
2
2n y 2
( (
)
Ž 24 .
Ž n s 2, 3, . . . ., which, after letting x s Ž n y 1.r2, becomes G Ž x q 1r2 .
)
GŽ x.
(
xy
1 4
.
Ž 25 .
Chu indicates that, for x s 1, 2, . . . , there is a sharper lower bound: G Ž x q 1r2 . GŽ x.
)
(
xy
1 4
q
1
Ž 4 x q 2.
.
2
Ž 26 .
In 1967, Boyd w3x gives an inequality in the same spirit. The lower bound in our notation reads G Ž x q 1r2 . GŽ x.
)
(
xy
1 4
q
1 32 x q 16
,
Ž 27 .
for x s m q 1r2, m s 1, 2, . . . . Finally, Lazarevic´ and Lupas¸’ result w9x from 1979 reads GŽ x q b . GŽ x.
G xy
ž
1yb 2
b
/
,
Ž 28 .
for x ) Ž1 y b .r2 and b g w0, 1x. This inequality was rediscovered by Kershaw w8x in 1983. As we can see, Ž28. coincides with Ž25. where the latter holds, but it is much more general than Ž25.. On the other hand, Ž27. is sharper than Ž28. for a particular choice of x and b . 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 Ž1 y b .r2 and b g w0, 1x, Q Ž x, b . G x y
ž
1yb 2
q
with equality if and only if b s 0 or b s 1.
1yb2 24 x q 12
b
/
,
Ž 29 .
377
INEQUALITIES FOR THE GAMMA FUNCTION
Proof. If b s 0 or b s 1 it is easy to see that equality occurs in Ž29.. Therefore, we have to show that Ž29. holds as a strict inequality for b g Ž0, 1.. Let us denote the right hand side in Ž29. by C#Ž x, b . and let
r Ž x, b . s Q Ž x, b . rC# Ž x, b . . Borrowing an idea from Kershaw w8x, we will show that Ž29. holds by showing: Ži. lim x ªq` r Ž x, b . s 1, for every b g Ž0, 1.. Žii. r Ž x q 1, b . - r Ž x, b ., for every b g Ž0, 1. and every x G Ž1 y b .r2. Indeed, from Žii. it follows that for every natural number n, r Ž x q n, b . - r Ž x, b . and by letting n ª q` and using Ži. we get
r Ž x, b . ) lim r Ž x, b . s 1, xªq`
which gives the inequality Ž29.. Proof of Ži.. Letting t s 1rx and using Taylor’s expansion we find, after some elaboration, 1yb
log C# Ž x, b . s b ylog t y
ž
2
tq
y2 b 2 q 3 b y 1 12
t 2 q oŽ t 2 .
/
Ž t ª 0 . . Ž 30 . Let us denote the lower bound in inequality Ž22. by C Ž x, b . and let t s 1rx again. Then we have log C Ž x, b . s yb log t q yb
ž
1 t
q
ž 1 2
1 t
/
qby
1 2
/
log Ž 1 q b t .
log Ž 1 q t . y
b Ž1 y b . t3 12 Ž 1 q t . Ž 1 q b t .
. Ž 31 .
By the Taylor expansion in Ž31., one can see that the expansion Ž30. holds for log C too; therefore, log C# Ž x, b . y log C Ž x, b . s o Ž 1rx 2 .
Ž x ª q`. .
Ž 32 .
378
MILAN MERKLE
From the Theorem 2 we have that Q Ž x, b .
lim
C Ž x, b .
xªq`
s1
and by Ž32. we conclude that also Q Ž x, b .
lim
xªq`
C# Ž x, b .
s 1,
which ends the proof of Ži.. Proof of Žii.. Let DŽ x, b . s log r Ž x, b . y log r Ž x q 1, b .. We have to show that D Ž x, b . ) 0
for x G Ž 1 y b . r2.
Ž 33 .
It is clear from the previous considerations that lim x ªq` DŽ x, b . s 0. Therefore, to show Ž33. it suffices to show
x
D Ž x, b . - 0
for x G Ž 1 y b . r2, b g Ž 0, 1 . .
Ž 34 .
Writing DŽ x, b . in the form D Ž x, b . s b log 1 q
ž
48 x q 12 b q 24 24 x q 12 b x y Ž 1 y b . Ž 5 y b .
q b log 1 y
ž
2
2 2xq3
/
y log 1 q
ž
b x
/
/
,
we find the derivative
x
D Ž x, b . s
b Ž1 y b 2 . x Ž 2 x q 3. Ž 2 x q 1. Ž x q b .
?
U Ž x, b . V Ž x, b . W Ž x, b .
, Ž 35 .
where U Ž x, b . s 576 x 3 Ž b y 2 . q 8 x 2 Ž 11 b 2 q 120 b y 299 . y 4 x Ž b 3 y 22 b 2 y 115 b q 370 . y 3 Ž 19 y b . Ž 5 y b . , V Ž x, b . s 24 x 2 q 12 x Ž b q 4 . y b 2 q 18 b q 19, W Ž x, b . s 24 x 2 q 12 b x y Ž 1 y b . Ž 5 y b . .
INEQUALITIES FOR THE GAMMA FUNCTION
379
Let x 0 s Ž1 y b .r2. The sign of the expression in Ž35. depends only on the sign of U, V, W. We will show that for every b g Ž0, 1. and x G x 0 , the terms V and W are positive and the term U is negative. Term V. It is not difficult to see that V Ž x 0 , b . ) 0 for every b g Ž0, 1.. Further, for x ) x 0 we have
x
V Ž x, b . s 48 x q 12 Ž b q 4 . ) 12 Ž 6 y b . ) 0,
thus V Ž x, b . ) 0 for every b g Ž0, 1. and x G x 0 . Term W. By W Ž x 0 , b . s 1 y b 2 ) 0 and
x
W Ž x, b . s 48 x q 12 b ) 0,
we see that W Ž x, b . ) 0 for b g Ž0, 1. and x G x 0 . Term U. Here we evaluate UŽ x, b . at x 0 as U Ž x 0 , b . s y3 Ž 16 b 4 y 170 b 3 q 631 b 2 y 994b q 589 . .
Ž 36 .
By localizing zeros of the polynomial in Ž36., we conclude that UŽ x 0 , b . - 0 for b g Ž0, 1.. Further,
x
U Ž x, b . s y1728 x 2 Ž 2 y b . q 16 x Ž 11 b 2 q 120 b y 299 . y 4 Ž b 3 y 22 b 2 y 115 b q 370 . .
Ž 37 .
Both zeros of the quadratic trinomial in Ž37. are negative for b g Ž0, 1.; therefore the derivative of U with respect to x ) 0 is negative, and from UŽ x 0 , b . - 0 it follows that UŽ x, b . - 0 for b g Ž0, 1. and x G x 0 . This ends the proof of the theorem. Let us remark that the bound in Ž29. agrees with the bound Ž22. to the order of magnitude of 1rx 2 as x ª q`. Therefore, according to Theorem 2, the error converges to zero as x ª q`. In the following table we provide some sample numerical values of relative errors in Ž29., for x s 1 and x s 3.
380
MILAN MERKLE
Relative Errors in Ž29., in %
b
0.1
0.3
0.5
0.7
0.9
xs1 xs3
0.50 0.016
0.96 0.036
0.93 0.041
0.64 0.031
0.22 0.012
REFERENCES 1. H. Alzer, Some Gamma function inequalities, Math. Comp. 60 Ž1993., 337]346. 2. E. Artin, ‘‘The Gamma Function,’’ Holt, Rinehart & Winston, New York, 1964; translation from the German original of 1931. 3. A. V. Boyd, A note on a paper by Uppuluri, Pacific J. Math. 22 Ž1967., 9]10. 4. J. T. Chu, A modified Wallis product and some applications, Amer. Math. Monthly 69, No. 5 Ž1969., 402]404. 5. G. M. Fihtengolc, ‘‘Kurs differencialnogo i integralnogo iscisleniya,’’ Vol. II, Moskva, ˇ 1970. ŽIn Russian.. 6. W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 Ž1959., 77]81. 7. F. B. Hildebrand, ‘‘Introduction to Numerical Analysis,’’ 2nd ed., McGraw]Hill, New York, 1974. 8. D. Kershaw, Some extensions of W. Gautchi’s inequalities for the gamma function, Math. Comp. 41 Ž1983., 607]611. 9. I. Lazarevic´ and A. Lupas¸, Functional equations for Wallis and Gamma functions, Uni¨ . Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 461]497 Ž1979., 245]251. 10. D. S. Mitrinovic, ´ ‘‘Analytic Inequalities,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1970. 11. V. R. R. Uppuluri, A stronger version of Gautschi’s inequality satisfied by the Gamma function, Scand. Actuar. J. 1–2 Ž1965., 51]52.