An extension of an inequality for ratios of gamma functions

In this paper, we prove that for x+y>0x+y>0 and y+1>0y+1>0 the inequality [Γ(x+y+1)/Γ(y+1)]1/x[Γ(x+y+2)/Γ(y+1)]1/(x+1)<(x+yx+y+1)1/2 is valid if x>1x>1 and reversed if x<1x<1 and that the power 12 is the best possible, where Γ(x)Γ(x) is the Euler gamma function. This extends the result of [Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2) (2009) 967–970] and resolves an open problem posed in [B.-N. Guo, F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2) (2003) 239–247].