An elementary inequality for psi function
by Yuming Chu   Xiaoming Zhang   Xiaomin Tang  

Vol. 3 No. 3 (2008) P.373~P.380

ABSTRACT

For $x>0$, let $\Gamma(x)$ be the Euler's gamma function, and $\Psi(x) = \frac {\Gamma'(x)}{\Gamma(x)}$ be the psi function. In this paper, we prove that $(b-L(a,b)) \Psi(b) + (L(a,b)-a) > (b-a) \Psi(\sqrt{ab})$ for $ b > a \ge 2$, where $L(a,b) = \frac {b-a}{\log b - \log a}$.

KEYWORDS
Gamma function, psi function, generalized logarithmic mean, logarithmic mean

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 33B15, 26D15

MILESTONES

Received: 2007-10-04
Revised : 2007-10-30
Accepted: 2007-10-31

Download Full Content