An inequality for ratios of gamma functions

Abstract

Let Γ (x) denote Euler's gamma function. The following inequality is proved: for y > 0 and x > 1 we havefrac([Γ (x + y + 1) / Γ (y + 1)]1 / x, [Γ (x + y + 2) / Γ (y + 1)]1 / (x + 1)) < sqrt(frac(x + y, x + y + 1)) . The inequality is reversed if 0 < x < 1. This resolves an open problem of Guo and Qi [B.-N. Guo, F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2003) 239-247]. © 2008 Elsevier Inc. All rights reserved.