## Inequalities for the gamma function

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**128**(2000), 141-147 Request permission

## 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).## References

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## Additional Information

**Horst Alzer**- Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany
- MR Author ID: 238846
- Received by editor(s): March 10, 1998
- Published electronically: June 30, 1999
- Communicated by: Hal L. Smith
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**128**(2000), 141-147 - MSC (1991): Primary 33B15; Secondary 26D07
- DOI: https://doi.org/10.1090/S0002-9939-99-04993-X
- MathSciNet review: 1622757