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Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions
Journal of Inequalities and Applications volume 2009, Article number: 728612 (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 real and positive values of the Euler gamma function and its logarithmic derivative , the so-called digamma function, are defined as
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.
Using Stirling formula, for all nonnegative integers , natural numbers and , an upper bound of the quotient of two geometrical means of natural numbers was established in  as follows:
Since as a generalization of inequality (1.3), the following monotonicity result was obtained by Guo and Qi in . The function
is decreasing with respect to on for fixed Hence, for positive real numbers and , we have
Recently, in , Qi and Sun proved that the function
is strictly increasing with respect to for all
Now, we generalize the function in (1.4) as follows: for positive real numbers and , , let
The aim of this paper is to discuss the monotonicity and logarithmical convexity of the function with respect to parameter .
For convenience of the readers, we recall the definitions and basic knowledge of convex function and logarithmically convex function.
Let be a convex set, is called a convex function on if
for all , and is called concave if is convex.
Let be a convex set, is called a logarithmically convex function on if is convex on , and is called logarithmically concave if is concave.
The following criterion for convexity of function was established by Fichtenholz in .
Let be a convex set, if have continuous second partial derivatives, then is a convex (or concave) function on if and only if is a positive (or negative) semidefinite matrix for all , where
and for , .
In Definitions 1.1, 1.2 and Proposition 1.3, we denote by the points (or vectors) of , and denote by the real variables in the later.
Our main results are Theorems 1.4 and 1.5.
For any fixed , is strictly increasing (or decreasing, resp.) with respect to on if and only if (or , resp.);
For any fixed , is strictly increasing with respect to on if and only if .
If , then is logarithmically concave with respect to ;
If is a convex set with nonempty interior and , then is neither logarithmically convex nor logarithmically concave with respect to on .
The following two corollaries can be derived from Theorems 1.4 and 1.5 immediately.
If , then
Inequality (1.3) can be derived from Corollary 1.6 if we take . Although we cannot get the inequality (1.2) exactly from Corollary 1.6, but we can get the following inequality which is close to inequality (1.2):
If , then
We conjecture that the inequality (1.2) can be improved if we can choose two pairs of integers and properly.
It is well known that the Bernoulli numbers is defined  in general by
In particular, we have
In , the following summation formula is given:
for nonnegative integer , where denotes the Euler number, which implies
Lemma 2.1 (see ).
For real number and natural number , one has
Lemma 2.2 (see ).
hold in for .
Let , then the following statements are true:
(1) if , then for ;
(2) if , then for .
Making use of (2.6) we get(2.11)
for any fixed .
Since and , we have
for all .
Therefore, Lemma 2.3(1) follows from (2.11) and (2.12).
If , then (2.12) leads to(2.13)
Therefore, Lemma 2.3(2) follows from (2.11) and (2.13).
If , then for .
It is easy to see that
for all .
Let , then
for . On the other hand, from (2.10) we know that is strictly decreasing on .
Therefore, Lemma 2.4 follows from (2.14)–(2.17).
Then simple computation shows that
Let then the following statements are true:
(1) if , then
(2) if then
Then it is not difficult to verify
If , then making use of Lemmas 2.2, 2.4 and (2.25) we get(2.26)
Let , and . Then simple computation leads to
for all .
It is well known that , where is the Euler's constant. From this we get
From Lemma 2.2, (2.27)–(2.29), (2.31) and the assumption , we conclude that
Therefore, Lemma 2.6(1) follows from (2.23)–(2.26), (2.30), and (2.32).
If , then making use of (2.8), Lemma 2.4 and (2.25) we obtain(2.33)
for by Lemma 2.2, and
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
Proof of Theorem 1.4.
Let and , then(3.1)
The following three cases will complete the proof of Theorem 1.4(1).
If , then (3.1) and Lemma 2.2 imply
From (3.2) and the fact that for all we know that is strictly increasing with respect to on for any fixed .
If , then (3.1) and (2.7) imply
for , where and .
From (3.3) and the fact that for all we know that is strictly decreasing with respect to on for any fixed .
If , let
It is obvious that (3.6) implies
The continuity of with respect to for any fixed and (3.7) imply that there exists such that
From (3.5), (3.8) and we know that is strictly decreasing with respect to on for .
On the other hand, making use of (2.5) and (2.6) we have
for and .
Equation (3.9) implies that there exists such that
Hence, from (3.11) we know that is strictly increasing with respect to on for .
then, Theorem 1.4(2) follows from (3.12) and Lemma 2.3.
Proof of Theorem 1.5.
Let and , then simple calculation yields
where , and are defined in Remark 2.5 and Lemma 2.6.
According to the Definition 1.2 and Proposition 1.3, to prove Theorem 1.5 we need only to show that
for and , and
for and .
Next, let then
for by Lemma 2.2 and .
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.
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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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Zhao, T., Chu, Y. & Jiang, Y. Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions. J Inequal Appl 2009, 728612 (2009). https://doi.org/10.1155/2009/728612
- Natural Number
- Convex Function
- Nonnegative Integer
- Gamma Function
- Positive Real Number