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
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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
(1)
The logarithmic derivative of the Gamma function in [1,3] is defined as follows
(2)
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
(3)
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
(4)
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,
(5)
We have
(6)
Then we can obtain
(7)
where we denotes
(8)
We can count the derivative of as follows
(9)
Base on above, we have
(10)
Combining (6) with the following formula,
(11)
we have
(12)
Here we let
(13)
Since for then
(14)
especially, we have
(15)
If we need, then we should have
(16)
while . In this case, we can obtain
(17)
then should satisfies the following necessary and sufficient condition
(18)
and the function is strictly increasing, then for any , we have
(19)
on , and.
On the other hand, we find
(20)
and
(21)
for .
We can also count as follows
(22)
then it means one must have, while in this second condition, we get
(23)
In this case, we can find that should satisfies the necessary and sufficient condition as following
(24)
Consider first conditionwith the second one, we have
(25)
But we should have
(26)
We can find that should satisfies
(27)
In fact, we can describe the condition (26) into the following form
(28)
We have the following fact that
(29)
Base on the fact above, we can get a result that there exist a such that the (28) holdsfor , we denote as follows
(30)
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
(31)
and
(32)
where it exists such that .
Proof. Consider the complete monotonicity of , we have
(33)
for . It means
(34)
Let we get
(35)
then
(36)
Using the following fact
(37)
we get
(38)
Theorem 3.2. If, , then we have
(39)
Especially, for a enough large positive integer, we have
(40)
where it existssuch that .
Proof. Using the completely monotonic property of again, we get
(41)
for all . If is a positive integer, and, we have a classical useful inequality
(42)
then
(43)
for a enough large.
Acknowledgements
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).
References