Inequalities for the gamma function

Abstract:

We prove the following two theorems: (i) Let $M_r(a,b)$ be the $r$th power mean of $a$ and $b$. The inequality \[ M_r(\Gamma (x),\Gamma (1/x))\ge 1 \] holds for all $x\in (0,\infty )$ if and only if $r\ge 1/C-\pi ^2/(6C^2)$, where $C$ denotes Euler’s constant. This refines results established by W. Gautschi (1974) and the author (1997). (ii) The inequalities \begin{equation*} x^{\alpha (x-1)-C}<\Gamma (x)<x^{\beta (x-1)-C}\tag {$*$} \end{equation*} are valid for all $x\in (0,1)$ if and only if $\alpha \le 1-C$ and $\beta \ge (\pi ^2/6-C)/2$, while $(*)$ holds for all $x\in (1,\infty )$ if and only if $\alpha \le (\pi ^2/6-C)/2$ and $\beta \ge 1$. These bounds for $\Gamma (x)$ improve those given by G. D. Anderson an S.-L. Qiu (1997).