# A Double Inequality for Gamma FunctionAcademic research paper on "Mathematics" 0 0
Share paper
J Inequal Appl
OECD Field of science
Keywords
{""}

## Academic research paper on topic "A Double Inequality for Gamma Function"

﻿Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 503782,7 pages doi:10.1155/2009/503782

Research Article

A Double Inequality for Gamma Function

Xiaoming Zhang1 and Yuming Chu2

1 Haining Radio and TV University, Haining 314400, Zhejiang, China

2 Department of Mathematics, Huzhou Teachers College, Huzhou 313000, Zhejiang, China

Correspondence should be addressed to Yuming Chu, chuyuming2005@yahoo.com.cn Received 12 June 2009; Revised 21 August 2009; Accepted 30 August 2009 Recommended by Ramm Mohapatra

Using the Alzer integral inequality and the elementary properties of the gamma function, a double inequality for gamma function is established, which is an improvement of Merkle's inequality.

Copyright © 2009 X. Zhang and Y. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

For real and positive values of x, the Euler gamma function r and its logarithmic derivative f, the so-called psi function, are defined by

respectively. For extensions of these functions to complex variables and for basic properties, see .

Recently, the gamma function has been the subject of intensive research, many remarkable inequalities for r can be found in literature [2-21]. In particular, the ratio (r(s)/r(r))(s > r > 0) have attracted the attention of many mathematicians and physicists. Gautschi  first proved that

i s r(n +1) .

n < r(„ + c) < exp [(1 - s)v(n +1)]

for 0 < s < 1 and n = 1,2,3.

Journal of Inequalities and Applications A strengthened upper bound was given by Erber :

r(n + 1) < 4(n + s)(n + 1)

r(n + s) 4n + (s + 1)2 In , KeCkiC and VasiC established the following double inequality for b > a > 0:

1 b r(b) bb-(1/2) b

-e < -b < b „,^ea-b.

aa-1 r(a) aa-(1/2)

In , Kershaw obtained

exp [(1 - s)f(^x + s1/2)j

r(x +1) < F(XXTs) <exp

(1 - s)y(x + 2 (s + 1)

x + - s 2

r(x + 1) < —-- <

r(x + s)

x -1+(s+4)

for x > 0 and 0 < s < 1.

The generalized logarithmic mean Lp(a, b) of order p of two positive numbers a and b with a = b is defined by

L„(a, b) =

bp+i - ap+i

(p + 1)(b - a) b - a log b - log a'

1/ (b-a)

, p= - 1, p / 0, p = -1,

p = 0.

It is well known that Lp(a, b) is strictly increasing with respect to p for fixed a and b.

If we denote A(a,b) = L1 (a,b) = (a + b)/2,1(a,b) = Lc(a, b) = (1/e)(bb/aa)1/(b-a} ,L(a,b) = L-1(a,b) = (b - a)/(log b - log a), and G(a,b) = L-2(a,b) = Vab the arithmetic mean, identric mean, logarithmic mean, and geometric mean of a and b with a =/ b, respectively, then

min{a,b} < G(a,b) < L(a,b) < I(a,b) < A(a,b) < max{a,b}. (1.7)

In 1996, Merkle  established

A(f (a),f(b)) < logmb - 0»r(a) (A(a, b))

for a,b > 0 with a = b.

It is the aim of this paper to present the new upper and lower bounds of inequality (1.8) in terms of I and L.

2. Lemmas

In order to establish our main result we need several lemmas, which we present in this section.

Lemma 2.1 (see [27, page 2670]). If x > 0, then

f(x) > x + 2X2 ■ (11)

Lemma 2.2 (see ). Let f e C[a,b] be a strictly increasing function. If 1/f 1 is strictly convex (or concave, resp.), then

b - a J a

[ f (t)dt > (or <,resp)f (L(a,b))■ (2.2)

Here, f 1 is the inverse of f. Lemma 2.3. If x > 0, then

0 < 2f'(x)+ xy"(x) <x■ (2.3)

Proof. It is well known that logr(x) = -jx + ^O=i [x/k - log(1 + (x/k))] - logx, where j = 0577 215 ■■■ is the Euler constant. Then, we have

V'(x) = X --3 (2.4)

k=0 (k + x)

From (2.4) and (2.5), we get

2w'(x) + xw"(x) = Y,-5- > 0,

^ W ^ X k=1 (k + x)3

y 2k 2f '(x) + xf"(x) = y

k=1 (k + x)3

< k=1 (k - 1 + x)(k + x)(k + 1 + x)

(k - 1 + x)(k + x) (k + x)(k + 1 + x) 1

k=1 (k - 1 + x)(k + x)

= y /J___L_

k=1\k - 1 + x k + x

Lemma 2.4. Suppose that b > a > 0 and f : [a, b] ^ R is a twice differentiable function. If f'(x) > 0 and 2f '(x) + xf"(x) > (or <, resp.) 0 for x e [a, b], then there exists the inverse function f-1 of f and 1/f-1 is strictly convex (or concave, resp.).

Proof. The existence of f-1 can be derived from f '(x) > 0 directly. Next, let y = f (x), then simple computation yields

f '(x)( f -1(y))' = 1, f "(x)[(f-1 (y))']2 + f '(x)(f -1(y))" = 0,

2[(f-1(y))']2 (f-1(y))''

J-1^)/ (/-1(y))3 (f-1(y)Y

From (2.7) and x = f ~l(y), we get

"_ 2/' (x) + xf "(x)

f-1( y)) x3J <(x))

Therefore, the strict convexity (or concavity, resp.) of 1/f 1 follows from (2.8) and the assumed condition 2f'(x) + xf"(x) > (or <, resp.) 0. □

3. Main Result

Theorem 3.1. For all a,b> 0 with a = b, one has

f^ < ^^ <fLiaMU .o8 ™ <3.1)

Proof. Without loss of generality, we assume that b > a > 0. From <2.4) and Lemma 2.3, together with Lemma 2.4, we clearly see that f is strictly increasing and 1/f-1 is strictly convex on [a,b]. Then, Lemma 2.2 leads to

(f <t)dt > f<L<a, b)). <3.2)

Therefore, the left-side inequality in <3.1) follows from <3.2).

Next, for x e [a,b], let g<x) = f<x) - logx. Then, Lemmas 2.1 and 2.3 lead to

g'<x) = f'<x) - x > 2x2 > 0, <3.3)

2g'<x) + xg"<x) = 2f'<x) + xf"<x) - x< 0. <3.4)

From <3.3) and <3.4), together with Lemma 2.4, we clearly see that g<x) is strictly increasing and 1/g-1 is strictly concave on [a,b]. Then, Lemma 2.2 implies

[ (f<t) - log t)dt < f<L<a, b)) - logL<a,b). <3.5)

Therefore, the right-side inequality in <3.1) follows from <3.5).

To compare the bounds in Theorem 3.1 with that in <1.8), we have the following two remarks. □

Remark 3.2. The lower bound in Theorem 3.1 is greater than that in <1.8), that is, f<L<a, b)) > A<f<a),f<b)) for a,b > 0 with a / b. In fact, for any b > a > 0 and x e [a,b], Lemmas 2.1 and 2.3 lead to

f'<x) + xf"<x) < -< 0. <3.6)

From <3.6) and , we know that f<x) is a strictly geometric-arithmetic concave function on [a,b], hence, we get

f <G<a,b)) > A(f<a),f<b)). <3.7)

Since f is strictly increasing and G<a, b) < L<a, b), so we have

f <L<a,b)) > f <G<a,b)).

Inequalities (3.7) and (3.8) show that f(L(a, b)) > A(f(a), f(b)) for a,b> 0 with a f b.

Remark 3.3. The upper bound in Theorem 3.1 is less than that in (1.8), that is, f (L(a,b)) + log I (a,b) - log L(a,b) < f (A(a,b)). In fact, for any b > a > 0 and x e [a,b], (3.3) and L(a,b) < I(a,b) imply

f(L(a,b)) - log L(a,b) <f (I(a,b)) - log I (a,b). (3.9)

On the other hand, the monotonicity of f and I (a, b) < A(a, b) leads to

f (I(a, b)) <f (A(a,b)). (3.10)

From (3.9) and (3.10), we get

f (L(a,b))+ log I (a, b) - log L(a,b) <f (A(a,b)). (3.11)

Acknowledgments

The authors wish to thank the anonymous referee for the very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China under Grants 60850005 and 10771195, and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.

References

 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1996.

 H. Alzer and G. Felder, "A Turan-type inequality for the gamma function," Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 276-282, 2009.

 T.-H. Zhao, Y.-M. Chu, and Y.-P. Jiang, "Monotonic and logarithmically convex properties of a function involving gamma functions," Journal of Inequalities and Applications, vol. 2009, Article ID 728612, 13 pages, 2009.

 H. Alzer, "Inequalities for Euler's gamma function," Forum Mathematicum, vol. 20, no. 6, pp. 955-1004, 2008.

 N. Batir, "Inequalities for the gamma function," Archiv der Mathematik, vol. 91, no. 6, pp. 554-563, 2008.

 A. Laforgia and P. Natalini, "Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions," Journal ofInequalities and Applications, vol. 2006, Article ID 48727, 8 pages, 2006.

 Y. Yu, "An inequality for ratios of gamma functions," Journal ofMathematical Analysis and Applications, vol. 352, no. 2, pp. 967-970, 2009.

 R. P. Agarwal, N. Elezovic, and J. Pecaric, "On some inequalities for beta and gamma functions via some classical inequalities," Journal of Inequalities and Applications, no. 5, pp. 593-613, 2005.

 M. Merkle, "Gurland's ratio for the gamma function," Computers & Mathematics with Applications, vol. 49, no. 2-3, pp. 389-406, 2005.

 H. Alzer, "On Ramanujan's double inequality for the gamma function," The Bulletin of the London Mathematical Society, vol. 35, no. 5, pp. 601-607, 2003.

 B.-N. Guo and F. Qi, "Inequalities and monotonicity for the ratio of gamma functions," Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 239-247, 2003.

 H. Alzer, "On a gamma function inequality of Gautschi," Proceedings of the Edinburgh Mathematical Society, vol. 45, no. 3, pp. 589-600, 2002.

 S. S. Dragomir, R. P. Agarwal, and N. S. Barnett, "Inequalities for beta and gamma functions via some classical and new integral inequalities," Journal of Inequalities and Applications, vol. 5, no. 2, pp. 103165, 2000.

 H. Alzer, "A mean-value inequality for the gamma function," Applied Mathematics Letters, vol. 13, no. 2, pp. 111-114, 2000.

 H. Alzer, "Inequalities for the gamma function," Proceedings of the American Mathematical Society, vol. 128, no. 1, pp. 141-147, 2000.

 M. Merkle, "Convexity, Schur-convexity and bounds for the gamma function involving the digamma function," The Rocky Mountain Journal of Mathematics, vol. 28, no. 3, pp. 1053-1066,1998.

 B. Palumbo, "A generalization of some inequalities for the gamma function," Journal of Computational and Applied Mathematics, vol. 88, no. 2, pp. 255-268,1998.

 J. Dutka, "On some gamma function inequalities," SIAM Journal on Mathematical Analysis, vol. 16, no.

1, pp. 180-185,1985.

 A. Laforgia, "Further inequalities for the gamma function," Mathematics of Computation, vol. 42, no. 166, pp. 597-600,1984.

 J. B. Selliah, "An inequality satisfied by the gamma function," Canadian Mathematical Bulletin, vol. 19, no. 1, pp. 85-87,1976.

 W. Gautschi, "A harmonic mean inequality for the gamma function," SIAM Journal on Mathematical Analysis, vol. 5, pp. 278-281,1974.

 W. Gautschi, "Some elementary inequalities relating to the gamma and incomplete gamma function," Journal ofMathematics and Physics, vol. 38, pp. 77-81, 1960.

 T. Erber, "The gamma function inequalities of Gurland and Gautschi," Scandinavian Aktuarietidskr, vol. 1960, pp. 27-28,1961.

 J. D. Keckic and P. M. Vasic, "Some inequalities for the gamma function," Institut Mathematique Publications, vol. 11(25), pp. 107-114,1971.

 D. Kershaw, "Some extensions of W. Gautschi's inequalities for the gamma function," Mathematics of Computation, vol. 41, no. 164, pp. 607-611,1983.

 M. Merkle, "Logarithmic convexity and inequalities for the gamma function," Journal of Mathematical Analysis and Applications, vol. 203, no. 2, pp. 369-380,1996.

 A. Elbert and A. Laforgia, "On some properties of the gamma function," Proceedings of the American Mathematical Society, vol. 128, no. 9, pp. 2667-2673, 2000.

 H. Alzer, "On an integral inequality," LAnalyse Numerique et la Theorie de lApproximation, vol. 18, no.

2, pp. 101-103,1989.

 R. A. Satnoianu, "Improved GA-convexity inequalities," Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 5, article 82, pp. 1-6, 2002.

Copyright of Journal of Inequalities & Applications is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.