Some Properties on the Function Involving the Gamma Function

We studied the monotonicity and Convexity properties of the new functions involving the gamma function, and get the general conclusion that Minc-Sathre and C. P. Chen-G. Wang’s inequality are extended and refined.


Introduction
The classical gamma function 0 is one of the most important functions in analysis and its applications.The logarithmic derivative of the gamma function can be expressed in terms of the series (x > 0;  = 0.57721566490153286… is the Euler's constant), which is known in literature as psi or digamma function.We conclude from (1) by differentiation  are called polygamma functions.H. Minc and L. Sathre [1] proved that the inequality is valid for all natural numbers n.The Inequality (3) can be refined and generalized as (see [2][3][4]) where k is a nonnegative integer, n and m are natural numbers.For , the equality in (4) is valid.The Inequality (4) can be written as In 1985, D. Kershaw and A. Laforgia [5] showed the function 1 1 , from which the left-hand side inequality of (5) can be obtained.In the 2009, C. P. Chen-G.Wang had obtained the extended inequality of the function above.They gave the limits of it and other results.Theorem 1.Let fixed and be real number, then the new function is strictly decreasing and strictly logarithmically convex Theorem 2. Let be an positive integer, be real number, then the function is strictly increasing on .

Proof of the Theorems
Proof of Theorem 1. First, we define for fixed and , Hence, the function

 
A x is strictly decreasing and , for , which yields the desired result that we can conclude that By L'Hospital rule, we conclude from (6) that Then from the Differentiation of yields Proof of Theorem 2. Define for be an positive integer and , Hence, the function is strictly increasing and for which yields the desired result that for .

 
Corollary 5. Let t be an positive integer, we get

Use the Theorem
  (13) From the proof above the following corollaries are obvious.
Corollary 1.Let fixed and be a real number, then for all real numbers , The Inequality ( 13) is an improvement of (3).
x s s e e x s s

Acknowledgements
Both bounds in (7) are best possible.Corollary 2. Let fixed , 0 t  0   and be real numbers, be an positive integer, then for all real numbers ,

REFERENCES
In particular, taking in (8) , , we obtain the result that Minc-Sathre and C. P. Chen-G.Wang got The inequality is an improvement of above, and we can extend it as the below form.
Corollary 3. Let , we have can be derived.In 2003, B.-N. Guo and F. Qi [2] proved that the function In this paper, our Theorem 1 considers the monotonicity and logarithmic convexity of the new function g on   0,  .This extends and generalizes B.-N. Guo and F. Qi's [2] as well as C. P. Chen and G. Wang's[6] results.