Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 896483, 13 pages

http://dx.doi.org/10.1155/2011/896483

## Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

^{1}Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China^{2}Department of Mathematics, Changsha University of Science and Technology, Changsha 410076, China

Received 26 March 2011; Accepted 18 May 2011

Academic Editor: Narcisa C.Β Apreutesei

Copyright Β© 2011 Tie-Hong Zhao et al. 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.

#### Abstract

We prove that the function is strictly logarithmically completely monotonic on if and is strictly logarithmically completely monotonic on if , where and

#### 1. Introduction

It is well known that the classical Eulerβs gamma function is defined for as The logarithmic derivative of defined by is called the psi or digamma function and for are known as the polygamma or multigamma functions. These functions play central roles in the theory of special functions and have lots of extensive applications in many branches, for example, statistics, physics, engineering, and other mathematical sciences.

For extension of these functions to complex variable and for basic properties, see [1]. Over the past half century, many authors have established inequalities and monotonicity for these functions (see [2β22]).

Recall that a real-valued function is said to be completely monotonic on if has derivatives of all orders on and for all and . Moreover, is said to be strictly completely monotonic if inequality (1.3) is strict.

Recall also that a positive real-valued function is said to be logarithmically completely monotonic on if has derivatives of all orders on and its logarithm satisfies for all and . Moreover, is said to be strictly logarithmically completely monotonic if inequality (1.4) is strict.

Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. There has been a lot of literature about the (logarithmically) completely monotonic functions related to the gamma function, psi function, and polygamma function, for example, [17, 18, 23β37] and the references therein. In 1997, Merkle [38] proved that is strictly log-concave on . Later, Chen [39] showed that is strictly logarithmically completely monotonic on . In [40], Li and Chen proved that is strictly logarithmically completely monotonic on for , and is strictly logarithmically completely monotonic on for . Qi et al. in their article [41] showed that is strictly logarithmically complete monotonic on for , and is strictly logarithmically complete monotonic on for .

The aim of this paper is to discuss the logarithmically complete monotonicity properties of the functions and on where and . The function is the deformation of the functions in [40, 41] with respect to the parameters and . We show that the properties of logarithmically complete monotonic are also true for suitable extensions of near by two lines and , which generalizes the results of [40, 41].

For , we define two binary functions as follows:

For convenience, we need to define five subsets of and refer to Figure 2,

We summarize the result as follows.

Theorem 1.1. *Let , , and be defined as (1.5); then the following statements are true: *(1)* is strictly logarithmically completely monotonic on if *(2)* is strictly logarithmically completely monotonic on if **Note that is the constant 1 for since .*

#### 2. Lemmas

In order to prove our Theorem 1.1, we need two lemmas which we present in this section.

We consider and defined as (1.6) and discuss the properties for these functions, see Figure 1 more clearly.

##### 2.1. The Properties of Function

The function can be interpreted as a quadric equation with respect to . Let where , , and its discriminant function

If , then it is easy to see that for .

Let , be two real roots of with ; then we claim that . Indeed, From (2.5)β(2.7), we know that has only one root , which is Moreover, for and for , which implies that is strictly decreasing on and strictly increasing on . An easy computation shows that , , and . Combining with (2.4), there exist two real roots such that . Furthermore, we conclude that for or and for .

If , then since and .

If , then , , which implies .

If or , then . We can solve two roots of the equation , which are For , we know that for and for or . For , we know that for and for . Moreover, we see that as and as .

##### 2.2. The Properties of Function

The function can also be interpreted as a quadric equation with respect to . Let where , , and its discriminant function

If , then we have for .

If , then a simple calculation leads to for . This implies that . Notice that , , and ; for , then we have .

If , then we can solve the roots of the equation but only one of the roots is positive, that is,

Therefore, we conclude that for and for . Moreover, it is easy to see that as and as .

Finally, we calculate an intersection point of and , that is, the point

Lemma 2.1. *The psi or digamma function, the logarithmic derivative of the gamma function, and the polygamma functions can be expressed as
**
for and , where is Eulerβs constant.*

Lemma 2.2. *Let and
**
Then the following statements are true: *(1)*if then for ; *(2)*if then for ; *(3)*if , or , , then there exist such that for and for ; *(4)*if , , then there exist such that for and for .*

*Proof. *Let , and . Then simple calculations lead to

(1) If then we divide the proof into two cases. Note that , see Figure 2.*Case 1. *If then , , and it follows from (2.21) that

Therefore, for follows from (2.17), (2.18) together with (2.27).*Case 2. *If , then , , and . It follows from that and then (2.20) and (2.22) together with (2.24) lead to
This could not happen together for all qualities of (2.28)β(2.31) since the qualities of (2.29) and (2.30) hold only for , while the qualities of (2.29) and (2.30) hold only for .

Therefore, for follows from (2.17) and (2.18) together with (2.28)β(2.31).

(2) If then we divide the proof into three cases.*Case 1. *If then and . From (2.26), we clearly see that

In terms of the properties of , we know that for lying on the left-side of the green curve, see Figure 1. From (2.24), we see that
Combining (2.32) with (2.33) we get that is strictly increasing on .

If , then and follow from (2.22), which implies that is strictly increasing in . Thus we can obtain

If , then it follows from or that there exists such that for and for . Hence, is strictly decreasing in and strictly increasing in . Then we can obtain
Finally, we conclude that for follows from (2.17), (2.18) together with (2.34), (2.35).*Case 2. *If then and . It follows from (2.21) that

Therefore, for follows from (2.17), (2.18) together with (2.36).*Case 3. *If then and . From (2.26), we know that
In terms of the location of we know that . From (2.24), we see that
It follows from (2.37) and (2.38) that is strictly increasing on .

If , then and follow that from (2.22), which implies that is strictly increasing on . From (2.20) and (2.21), we see that
Thus there exists such that for and for , which implies that is strictly decreasing on and strictly increasing on . It follows from (2.18) and that such that for and for , which implies that is strictly decreasing on and strictly increasing on . Therefore, it follows from (2.17) and that
for .

If , then there exists such that for and for follows from or . This leads to being strictly decreasing in and strictly increasing in . From (2.20), we clearly see that

For special case of , that is, and , it follows from (2.41) and (2.21) that
which implies that for follows from (2.17) and (2.18).

For , it follows from (2.38) and that there exists such that for and for . Making use of the same arguments as the case of , then for follows from (2.17).

(3) If , or , , then we have

From (2.20), we know that

It follows from (2.44) that there exists such that for , which implies that is strictly increasing on . Therefore, for follows from (2.17) and (2.18).

From (2.43), we know that there exists such that for .

(4) If , , then we have

From (2.15), we know that

Making use of (2.45) and (2.46) together with the same arguments as in Lemma 2.2(3), we know that there exist such that for and for .

#### 3. Proof of Theorem 1.1

*Proof of Theorem 1.1. *From (2.15), we have
where

(1) If then from (3.1) and (3.2) together with Lemma 2.2(1) we clearly see that
Therefore, is strictly logarithmically completely monotonic on following from (3.3).

(2) If then from (3.1) we can get
where is defined as (3.2).

Therefore, is strictly logarithmically completely monotonic on following from (3.4) and Lemma 2.2 (2).

*Remark 3.1. *Note that neither nor is strictly logarithmically completely monotonic on for following from Lemma 2.2 (3) and (4), it is known that the logarithmically completely monotonicity properties of and are not completely continuously depended on and .

*Remark 3.2. *Compared with Theorem 9 of [40], we can also extend onto one component of its boundaries, which is
Then is strictly logarithmically completely monotonic on for

#### Acknowledgment

The first author is supported by the China-funded Postgraduates Studying Aboard Program for Building Top University.

#### References

- E. T. Whittaker and G. N. Watson,
*A Course of Modern Analysis*, Cambridge University Press, New York, NY, USA, 1962. View at Zentralblatt MATH - Y.-M. Chu, X.-M. Zhang, and Zh.-H Zhang, βThe geometric convexity of a function involving gamma function with applications,β
*Korean Mathematical Society*, vol. 25, no. 3, pp. 373β383, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X.-M. Zhang and Y.-M. Chu, βA double inequality for gamma function,β
*Journal of Inequalities and Applications*, vol. 2009, Article ID 503782, 7 pages, 2009. View at Google Scholar Β· View at Zentralblatt MATH - T.-H. Zhao, Y.-M. Chu, and Y.-P. Jiang, βMonotonic and logarithmically convex properties of a function involving gamma functions,β
*Journal of Inequalities and Applications*, vol. 2009, Article ID 728612, 13 pages, 2009. View at Google Scholar Β· View at Zentralblatt MATH - X.-M. Zhang and Y.-M. Chu, βAn inequality involving the gamma function and the psi function,β
*International Journal of Modern Mathematics*, vol. 3, no. 1, pp. 67β73, 2008. View at Google Scholar Β· View at Zentralblatt MATH - Y.-M. Chu, X.-M. Zhang, and X.-M. Tang, βAn elementary inequality for psi function,β
*Bulletin of the Institute of Mathematics*, vol. 3, no. 3, pp. 373β380, 2008. View at Google Scholar Β· View at Zentralblatt MATH - Y.-Q. Song, Y.-M. Chu, and L.-L. Wu, βAn elementary double inequality for gamma function,β
*International Journal of Pure and Applied Mathematics*, vol. 38, no. 4, pp. 549β554, 2007. View at Google Scholar Β· View at Zentralblatt MATH - B.-N. Guo and F. Qi, βTwo new proofs of the complete monotonicity of a function involving the psi function,β
*Bulletin of the Korean Mathematical Society*, vol. 47, no. 1, pp. 103β111, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Ch.-P. Chen, F. Qi, and H. M. Srivastava, βSome properties of functions related to the gamma and psi functions,β
*Integral Transforms and Special Functions*, vol. 21, no. 1-2, pp. 153β164, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, βA completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums,β
*The Australian & New Zealand Industrial and Applied Mathematics Journal*, vol. 48, no. 4, pp. 523β532, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Qi and B.-N. Guo, βMonotonicity and convexity of ratio between gamma functions to different powers,β
*Journal of the Indonesian Mathematical Society*, vol. 11, no. 1, pp. 39β49, 2005. View at Google Scholar - C.-P. Chen and F. Qi, βInequalities relating to the gamma function,β
*The Australian Journal of Mathematical Analysis and Applications*, vol. 1, no. 1, Article ID 3, 7 pages, 2004. View at Google Scholar Β· View at Zentralblatt MATH - B.-N. Guo and F. Qi, βInequalities and monotonicity for the ratio of gamma functions,β
*Taiwanese Journal of Mathematics*, vol. 7, no. 2, pp. 239β247, 2003. View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, βMonotonicity results and inequalities for the gamma and incomplete gamma functions,β
*Mathematical Inequalities & Applications*, vol. 5, no. 1, pp. 61β67, 2002. View at Google Scholar Β· View at Zentralblatt MATH - F. Qi and J.-Q. Mei, βSome inequalities of the incomplete gamma and related functions,β
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 18, no. 3, pp. 793β799, 1999. View at Google Scholar Β· View at Zentralblatt MATH - F. Qi and S.-L. Guo, βInequalities for the incomplete gamma and related functions,β
*Mathematical Inequalities & Applications*, vol. 2, no. 1, pp. 47β53, 1999. View at Google Scholar Β· View at Zentralblatt MATH - H. Alzer, βSome gamma function inequalities,β
*Mathematics of Computation*, vol. 60, no. 201, pp. 337β346, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Alzer, βOn some inequalities for the gamma and psi functions,β
*Mathematics of Computation*, vol. 66, no. 217, pp. 373β389, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. D. Anderson and S.-L. Qiu, βA monotoneity property of the gamma function,β
*Proceedings of the American Mathematical Society*, vol. 125, no. 11, pp. 3355β3362, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Kershaw, βSome extensions of W. Gautschi's inequalities for the gamma function,β
*Mathematics of Computation*, vol. 41, no. 164, pp. 607β611, 1983. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Merkle, βLogarithmic convexity and inequalities for the gamma function,β
*Journal of Mathematical Analysis and Applications*, vol. 203, no. 2, pp. 369β380, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. Palumbo, βA generalization of some inequalities for the gamma function,β
*Journal of Computational and Applied Mathematics*, vol. 88, no. 2, pp. 255β268, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Alzer and Ch. Berg, βSome classes of completely monotonic functions II,β
*The Ramanujan Journal*, vol. 11, no. 2, pp. 225β248, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Alzer, βSharp inequalities for the digamma and polygamma functions,β
*Forum Mathematicum*, vol. 16, no. 2, pp. 181β221, 2004. View at Publisher Β· View at Google Scholar - H. Alzer and N. Batir, βMonotonicity properties of the gamma function,β
*Applied Mathematics Letters*, vol. 20, no. 7, pp. 778β781, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - W. E. Clark and M. E. H. Ismail, βInequalities involving gamma and psi functions,β
*Analysis and Applications*, vol. 1, no. 1, pp. 129β140, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Á. Elbert and A. Laforgia, βOn some properties of the gamma function,β
*Proceedings of the American Mathematical Society*, vol. 128, no. 9, pp. 2667β2673, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Bustoz and M. E. H. Ismail, βOn gamma function inequalities,β
*Mathematics of Computation*, vol. 47, no. 176, pp. 659β667, 1986. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. E. H. Ismail, L. Lorch, and M. E. Muldoon, βCompletely monotonic functions associated with the gamma function and its $q$-analogues,β
*Journal of Mathematical Analysis and Applications*, vol. 116, no. 1, pp. 1β9, 1986. View at Publisher Β· View at Google Scholar - V. F. Babenko and D. S. Skorokhodov, βOn Kolmogorov-type inequalities for functions defined on a semi-axis,β
*Ukrainian Mathematical Journal*, vol. 59, no. 10, pp. 1299β1312, 2007. View at Publisher Β· View at Google Scholar - M. E. Muldoon, βSome monotonicity properties and characterizations of the gamma function,β
*Aequationes Mathematicae*, vol. 18, no. 1-2, pp. 54β63, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, Q. Yang, and W. Li, βTwo logarithmically completely monotonic functions connected with gamma function,β
*Integral Transforms and Special Functions*, vol. 17, no. 7, pp. 539β542, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, D.-W. Niu, and J. Cao, βLogarithmically completely monotonic functions involving gamma and polygamma functions,β
*Journal of Mathematical Analysis and Approximation Theory*, vol. 1, no. 1, pp. 66β74, 2006. View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, Sh.-X. Chen, and W.-S. Cheung, βLogarithmically completely monotonic functions concerning gamma and digamma functions,β
*Integral Transforms and Special Functions*, vol. 18, no. 6, pp. 435β443, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, βA class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality,β
*Journal of Computational and Applied Mathematics*, vol. 206, no. 2, pp. 1007β1014, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C.-P. Chen and F. Qi, βLogarithmically complete monotonicity properties for the gamma functions,β
*The Australian Journal of Mathematical Analysis and Applications*, vol. 2, no. 2, Article ID 8, 9 pages, 2005. View at Google Scholar Β· View at Zentralblatt MATH - Ch.-P. Chen and F. Qi, βLogarithmically completely monotonic functions relating to the gamma function,β
*Journal of Mathematical Analysis and Applications*, vol. 321, no. 1, pp. 405β411, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Merkle, βOn log-convexity of a ratio of gamma functions,β
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika*, vol. 8, pp. 114β119, 1997. View at Google Scholar Β· View at Zentralblatt MATH - C.-P. Chen, βComplete monotonicity properties for a ratio of gamma functions,β
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika*, vol. 16, pp. 26β28, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A.-J. Li and Ch.-P. Chen, βSome completely monotonic functions involving the gamma and polygamma functions,β
*Journal of the Korean Mathematical Society*, vol. 45, no. 1, pp. 273β287, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Qi, D. Niu, J. Cao, and S. Chen, βFour logarithmically completely monotonic functions involving gamma function,β
*Journal of the Korean Mathematical Society*, vol. 45, no. 2, pp. 559β573, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH