Inequalities for the beta function

摘要

We prove the following inequalities involving Euler’s beta function. (i) Let α and β be real numbers. The inequalities (y~(z?x)/(x~(z?y) z~(y?x))~α≤ (B(x, x)~(z?y) B(z, z)~(y?x))/(B(y, y)~(z?x)) ≤(y~(z?x)/(x~(z?y) z~(y?x))~β hold for all x, y, z with 0 < x ≤ y ≤ z if and only if α ≤ 1/2 and β ≥ 1. (ii) Let a and b be non-negative real numbers. For all positive real numbers x and y we have δ(a, b) ≤ (x~aB(x + b, y) + y~aB(x, y + b))/((x + y)~aB(x, y))≤ Δ(a, b) with the best possible bounds δ(a, b) = min{2~(?a), 2~(1?a?b)} and Δ(a, b) = max{1, 2~(1?a?b)}.