IJPAM: Volume 38, No. 4 (2007)

AN ELEMENTARY DOUBLE INEQUALITY
FOR GAMMA FUNCTION

Yingqing Song$^1$, Yuming Chu$^2$, Lingli Wu$^3$
$^1$Department of Mathematics
Hunan City University
Hunan, Yiyang, 413000, P.R. CHINA
$^{2}$Department of Mathematics
Huzhou Teachers College
Zhejiang, Huzhou, 313000, P.R. CHINA
e-mail: chuyuming@hutc.zj.cn
$^{3}$School of Educational Science and Technology
Huzhou Teachers College
Zhejiang, Huzhou, 313000, P.R. CHINA


Abstract.For $x>0$, let $\Gamma(x)$ denote the Euler's gamma function. In this paper, we shall prove $(b-L)\psi(b)+(L-a)\psi(a)+\frac{b-a}{2abL}(L^2-ab)\leq\log\frac{\Gamma(b)}{\Gam...
...\psi(a)+\frac{b-a}{2abL}(L^2-ab)+\frac{b-a}{4ab}\left(\frac{a+b}{2ab}L-1\right)$ for any $b>a>0$. Here $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ and $L=\frac{b-a}{\log b-\log a}$.

Received: June 17, 2007

AMS Subject Classification: 33B15, 26D15

Key Words and Phrases: gamma function, GA-convex function, GA-concave function, logarithmic mean

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 38
Issue: 4