An inequality for ratios of gamma functions

Let Γ ( x ) denote Eulerʹs gamma function. The following inequality is proved: for y > 0 and x > 1 we have [ Γ ( x + y + 1 ) / Γ ( y + 1 ) ] 1 / x [ Γ ( x + y + 2 ) / Γ ( y + 1 ) ] 1 / ( x + 1 ) < 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].