The classical gamma function 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 [2-4])

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 is strictly decreasing and strictly increasing on, from which the Inequality (3) can be derived. In 2003, B.-N. Guo and F. Qi [2] proved that the function is decreasing in for fixed, 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.

In this paper, our Theorem 1 considers the monotonicity and logarithmic convexity of the new function g on. This extends and generalizes B.-N. Guo and F. Qi’s [2] as well as C. P. Chen and G. Wang’s [6] results.

Theorem 1. Let fixed and be real number, then the new function

is strictly decreasing and strictly logarithmically convex on, Moreover,

and

Theorem 2. Let be an positive integer, be real number, then the function

is strictly increasing on.

Proof of Theorem 1. First, we define for fixed and,

From the differentiation of, we should have

Hence, the function is strictly decreasing and, for, which yields the desired result that for.

Using the asymptotic expansion [7, p. 257]

and

we can conclude that.

By L’Hospital rule, we conclude from (6) that

Then from the Differentiation of yields

Hence, the function is strictly increasing and for, which yields the desired result that for.

Proof of Theorem 2. Define for be an positive integer and,

Differentiation of gives

Hence, the function is strictly increasing and for which yields the desired result that for.

From the proof above the following corollaries are obvious.

Corollary 1. Let fixed and be a real number, then for all real numbers,

Both bounds in (7) are best possible.

Corollary 2. Let fixed, and be real numbers, be an positive integer, then for all real numbers,

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

In most particular, weobtain Corollary 4. Let t be an positive integer, we get

and for,

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

The Inequality (13) is an improvement of (3).

Foundation item: Supported by SFC (11071194), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No 12JK0880) Shaanxi Provincial Natural Foundation (2012JM1021), Weinan Normal University Foundation (12YKS024), Key help subjects of Shaanxi Provincial Foundation. State Key Laboratory of Information Security (Institute of Software, Chinese Academy of Sciences100190) (2011NO: 01-01- 2).