An inequality for ratios of gamma functions

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