Pure and Applied Mathematics Journal
Volume 4, Issue 5-1, October 2015, Pages: 20-23

On The Parameter Function Involving Gamma Function and Its Applications

Bin Chen, Rong Miao

Department of mathematics and information science, Weinan Normal University, ShaanXi, China

Email address:

(Bin Chen)
(Rong Miao)

To cite this article:

Bin Chen, Rong Miao. On The Parameter Function Involving Gamma Function and Its Applications. Pure and Applied Mathematics Journal. Special Issue: Mathematical Aspects of Engineering Disciplines. Vol. 4, No. 5-1, 2015, pp. 20-23. doi: 10.11648/j.pamj.s.2015040501.14

Abstract: In this paper, the complete monotonic parameter function involving the Gamma function is considered. The necessary and sufficient condition of the parameter f is presented. As an application, two meaningful inequalities of Gamma function are obtained.

Keywords: Gamma function, Completely Monotonicity, Inequalities

1. Introduction

The classical Gamma function in [1,3] is defined by


The logarithmic derivative of the Gamma function in [1,3] is defined as follows


There are lots of literatures and applications about the completely monotonic functions and logarithmic completely monotonic functions, for example, [2,4,5,6-10] and the references therein.

The function  is said to be completely monotonic on an interval  if  has derivatives of all orders on and


If the inequality is strict for all and is said to be strictly completely monotonic function.

A positive function  is said to be logarithmic completely monotonic on an interval  if its logarithm satisfies


If the inequality is strict for all and is said to be strictly logarithmically completely monotonic function.

In this paper, we are about to consider the completely monotonic property of a parameter functions involving the Gamma function as follows.

2. Main Results

Theorem 2.1.

The functionis strictly completely monotonic onif and only if 1 ≤ α ≤ β, where it exists  such that .

Proof. By using the Leibnitz’rule,


We have


Then we can obtain


where we denotes


We can count the derivative of  as follows


Base on above, we have


Combining (6) with the following formula,


we have


Here we let


Since for then


especially, we have


If we need, then we should have


while . In this case, we can obtain


then  should satisfies the following necessary and sufficient condition


and the function  is strictly increasing, then for any , we have


on , and.

On the other hand, we find




for .

We can also count as follows


then it means one must have, while in this second condition, we get


In this case, we can find that  should satisfies the necessary and sufficient condition as following


Consider first conditionwith the second one, we have


But we should have


We can find that  should satisfies


In fact, we can describe the condition (26) into the following form


We have the following fact that


Base on the fact above, we can get a result that there exist a such that the (28) holdsfor , we denote as follows


Then the theorem follows directly. We can easily get the following corollary.

Corollary 2.2. The function is strictly logarithmically completely monotonic on if and only if, where it exists such that.

3. Two Applications

Following the notation above, by applying the complete monotonicity of, we can obtain the following inequalities.

Theorem 3.1. If, , then




where it exists  such that .

Proof. Consider the complete monotonicity of , we have


for . It means


Let we get




Using the following fact


we get


Theorem 3.2. If, , then we have


Especially, for a enough large positive integer, we have


where it existssuch that .

Proof. Using the completely monotonic property of again, we get


for all . If is a positive integer, and, we have a classical useful inequality




for a enough large.


The author thank the referees for their time and comments. This article is partly supported by NSFC (Grant No.61402335), NSFSXP(Grant No.2014JM1006) and WNU(Grant No.15YKF005).


  1. W. Magnus, F. Oberhettinger, R.P. Soni.Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag,Berlin, 1966.
  2. D. Kershaw, A. Laforgia.Monotonicity results for the gamma function .Atti Accad. Sci. Torino Cl. Sci. Fis.Mat. Natur.1985, 119:127-133.
  3. Zh.-X. Wang, D.-R. Guo.Special Functions .World Scientific, Singapore, 1989.translated into Chinese by D.-R. Guo, X.-J. Xia.
  4. Ch.-P.Chen.Some properties of functions related to the gamma, psi and tetragamma functions.Computers and Mathematics with Applications.2011,62:3389–3395.
  5. Ch.-P.Chen.Some properties of functions related to the gamma,psi and tetragamma functions. Computers and Mathematics with Applications.2011, 62:3389-3395.
  6. Bin Chen. Some properties on the function involving the Gamma function. Applied Mathematics. 2012, 3(6): 587-589.
  7. Bin Chen. Research on equation involving two newfunctionson information security. Journal of Convergence Information Technology. 2013,8(8) :284-291.
  8. Chao-Ping Chen,Long Lin.Remarks on asymptotic expansions for the gamma function. Applied Mathematics Letters. 2012,25: 2322–2326.
  9. Bin Chen, Haigang Zhou. Note on the problem of Ramanujanradial limits,Advances in difference equations. 2014,191:1-11.
  10. Bin Chen.A new equation involving the Eulerfunction.2010 IEEE InternationalConference on Information Theory and Information Security.2010,12:121-125.

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