1. Introduction
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
(1)
(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
(2)
are called polygamma functions.
H. Minc and L. Sathre [1] proved that the inequality
(3)
is valid for all natural numbers n. The Inequality (3) can be refined and generalized as (see [2-4])
(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
(5)
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.
2. Proof of the Theorems
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
(6)
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.
3. Use the Theorem
From the proof above the following corollaries are obvious.
Corollary 1. Let fixed and be a real number, then for all real numbers,
(7)
Both bounds in (7) are best possible.
Corollary 2. Let fixed, and be real numbers, be an positive integer, then for all real numbers,
(8)
In particular, taking in (8), , we obtain the result that Minc-Sathre and C. P. Chen-G. Wang got
(9)
The inequality is an improvement of above, and we can extend it as the below form.
Corollary 3. Let, we have
(10)
In most particular, weobtain Corollary 4. Let t be an positive integer, we get
(11)
and for,
(12)
Corollary 5. Let t be an positive integer, we get
(13)
The Inequality (13) is an improvement of (3).
4. Acknowledgements
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).