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
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
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
The functionis strictly completely monotonic onif and only if 1 ≤ α ≤ β, where it exists such that .
Proof. By using the Leibnitz’rule,
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,
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
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
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).