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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Inequalities for the gamma function


Author: Horst Alzer
Journal: Proc. Amer. Math. Soc. 128 (2000), 141-147
MSC (1991): Primary 33B15; Secondary 26D07
Published electronically: June 30, 1999
MathSciNet review: 1622757
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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

\begin{displaymath}M_r(\Gamma(x),\Gamma(1/x))\ge 1 \end{displaymath}

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).


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Additional Information

Horst Alzer
Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04993-X
PII: S 0002-9939(99)04993-X
Keywords: Gamma function, psi function, power mean, inequalities
Received by editor(s): March 10, 1998
Published electronically: June 30, 1999
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society