Inequalities for the beta function of $n$ variables

ANZIAM J. 44 (2003), 609-623

Inequalities for the beta function of variables

Horst Alzer
Morsbacher Str. 10
51545 Waldbröl
Germany
alzerhorst@freenet.de

Abstract

We present various inequalities for Euler's beta function of

variables. One of our theorems states that the inequalities

$\begin{align} \label{ast} a_n\leq \frac{1}{\prod_{i=1}^{n}x_i}-B(x_1,\ldots,x_n)\leq{b_n} \tag{$*$}\end{align}$

(*)

hold for all $x_i\geq{1}$ ( $i=1,\ldots,n$ ; $n\geq{3}$ ) with the best possible constants

and

. This extends a recently published result of Dragomir et al., who investigated (*) for the special case