ON A GAMMA FUNCTION INEQUALITY OF GAUTSCHI

Abstract

We prove the following.

(1) The inequalities

$$ \biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!a}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!a}\leq 2\leq\biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!b}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!b} $$

hold for all $x>0$ if and only if

$$ -1.204\,64\ldots=2+\frac{1}{\gamma}-\frac{1}{6}\biggl(\frac{\pi}{\gamma}\biggr)^{\!2}\leq a\leq0\leq b. $$

(2) For all real numbers $x\in(0,1]$ we have

$$ x^{\alpha}\leq\frac{1}{2}\biggl(\frac{1}{\varGamma(x)}+\frac{1}{\varGamma(1/x)}\biggr)\leq x^{\beta}, $$

with the best possible constants

$$ \alpha=1.321\,76\dots\link{and}\beta=0. $$

These theorems extend and complement a result of Gautschi (from 1974), who proved that for all $x>0$ the harmonic mean of $\varGamma(x)$ and $\varGamma(1/x)$ is greater than or equal to $1$.

AMS 2000 Mathematics subject classification: Primary 33B15; 26D15