Inequalities for the gamma function

We prove the following two theorems:

(i) Let $M_r(a,b)$ be the $r$th power mean of $a$ and $b$. The inequality

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

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).