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.
is strictly increasing with respect to for all
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.
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 .
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.
We conjecture that the inequality (1.2) can be improved if we can choose two pairs of integers and properly.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
hold in for .
Let , then the following statements are true:
(1) if , then for ;
(2) if , then for .
- (1)Making use of (2.6) we get(2.11)
for any fixed .
for all .
- (2)If , then (2.12) leads to(2.13)
Therefore, Lemma 2.3(2) follows from (2.11) and (2.13).
If , then for .
for all .
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).
Let then the following statements are true:
- (1)If , then making use of Lemmas 2.2, 2.4 and (2.25) we get(2.26)
for all .
- (2)If , then making use of (2.8), Lemma 2.4 and (2.25) we obtain(2.33)
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
- (1)Let and , then(3.1)
The following three cases will complete the proof of Theorem 1.4(1).
From (3.2) and the fact that for all we know that is strictly increasing with respect to on for any fixed .
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 .
From (3.5), (3.8) and we know that is strictly decreasing with respect to on for .
for and .
then, Theorem 1.4(2) follows from (3.12) and Lemma 2.3.
Proof of Theorem 1.5.
where , and are defined in Remark 2.5 and Lemma 2.6.
for and .
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.
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.
- Whittaker ET, Watson GN: A Course of Modern Analysis, Cambridge Mathematical Library. Cambridge University Press, Cambridge, UK; 1996:vi+608.View ArticleMATHGoogle Scholar
- Guo B-N, Qi F: Inequalities and monotonicity for the ratio of gamma functions. Taiwanese Journal of Mathematics 2003,7(2):239–247.MathSciNetMATHGoogle Scholar
- Qi F, Guo B-N: Monotonicity and convexity of ratio between gamma functions to different powers. Journal of the Indonesian Mathematical Society 2005,11(1):39–49.MathSciNetMATHGoogle Scholar
- Qi F: Inequalities and monotonicity of sequences involving . Soochow Journal of Mathematics 2003,29(4):353–361.MathSciNetMATHGoogle Scholar
- Qi F, Luo Q-M: Generalization of H. Minc and L. Sathre's inequality. Tamkang Journal of Mathematics 2000,31(2):145–148.MathSciNetMATHGoogle Scholar
- Qi F, Sun J-S: A mononotonicity result of a function involving the gamma function. Analysis Mathematica 2006,32(4):279–282. 10.1007/s10476-006-0012-yMathSciNetView ArticleMATHGoogle Scholar
- Fichtenholz GM: Differential- und Integralrechnung. II. VEB Deutscher Verlag der Wissenschaften, Berlin, Germany; 1966.MATHGoogle Scholar
- Abramowitz M, Stegun IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series. Volume 55. U.S.Government Printing Office, Washington, DC, USA; 1964:xiv+1046.Google Scholar
- Wang Zh-X, Guo D-R: Introduction to Special Function, The Series of Advanced Physics of Peking University. Peking University Press, Beijing, China; 2000.Google Scholar
- Guo B-N, Qi F: Generalization of Bernoulli polynomials. International Journal of Mathematical Education in Science and Technology 2002,33(3):428–431. 10.1080/002073902760047913MathSciNetView ArticleMATHGoogle Scholar
- Luo Q-M, Guo B-N, Qi F, Debnath L: Generalizations of Bernoulli numbers and polynomials. International Journal of Mathematics and Mathematical Sciences 2003,2003(59):3769–3776. 10.1155/S0161171203112070MathSciNetView ArticleMATHGoogle Scholar
- Luo Q-M, Qi F: Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics (Kyungshang) 2003,7(1):11–18.MathSciNetMATHGoogle Scholar
- Luo Q-M, Qi F, Debnath L: Generalizations of Euler numbers and polynomials. International Journal of Mathematics and Mathematical Sciences 2003,2003(61):3893–3901. 10.1155/S016117120321108XMathSciNetView ArticleMATHGoogle Scholar
- Qi F, Guo B-N: A new proof of complete monotonicity of a function involving psi function. RGMIA Research Report Collection 2008.,11(3, article 12):
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.