Inequalities for the beta function of n variables

Abstract

We present various inequalities for Euler's beta function of n variables. One of our theorems states that the inequalities an ≤ 1/∏i=1n xi - B(x1,...,x n) ≤ bn hold for all xi ≥ 1 (i = 1,... , n; n ≥ 3) with the best possible constants an = 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. © Australian Mathematical Society 2003.