Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions
© Tie-Hong Zhao et al. 2009
Received: 14 October 2008
Accepted: 27 February 2009
Published: 5 March 2009
Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N.Guo and F.Qi. Several inequalities for the geometric means of natural numbers are established.
For extension of these functions to complex variables and for basic properties see .
In recent years, many monotonicity results and inequalities involving the Gamma and incomplete Gamma functions have been established. This article is stimulated by an open problem posed by Guo and Qi in . The extensions and generalizations of this problem can be found in [3–5] and some references therein.
For convenience of the readers, we recall the definitions and basic knowledge of convex function and logarithmically convex function.
The following criterion for convexity of function was established by Fichtenholz in .
Our main results are Theorems 1.4 and 1.5.
The following two corollaries can be derived from Theorems 1.4 and 1.5 immediately.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Therefore, Lemma 2.3(2) follows from (2.11) and (2.13).
Therefore, Lemma 2.4 follows from (2.14)–(2.17).
Therefore, Lemma 2.6(2) follows from (2.23)–(2.25) and (2.33)–(2.36).
3. Proofs of Theorems 1.4 and 1.5
The following three cases will complete the proof of Theorem 1.4(1).
then, Theorem 1.4(2) follows from (3.12) and Lemma 2.3.
Proof of Theorem 1.5.
Therefore, (3.16) follows from (3.19) and (3.20), and (3.17) and (3.18) follow from Lemma 2.6. The proof of Theorem 1.5 is completed.
This research is partly supported by 973 Project of China under grant 2006CB708304, N S Foundation of China under Grant 10771195, and N S Foundation Zhejiang Province under Grant Y607128.
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