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).