ANZIAM  J.  44 (2003), 609-623
Inequalities for the beta function of $n$ variables

Horst Alzer
  Morsbacher Str. 10
  51545 Waldbröl

We present various inequalities for Euler's beta function of $n$ 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 $a_n=0$ and $b_n=1-1/(n-1)!$. This extends a recently published result of Dragomir et al., who investigated (*) for the special case $n=2$.
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