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
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