Inequalities involving Γ(x)Γ(x) and Γ(1/x)Γ(1/x)

We prove that the double-inequality1x+1/x+α(x+1/x)1/2<1Γ(x)+1Γ(1/x)⩽1x+1/x+β(x+1/x)1/2holds for all x>0x>0 with the best possible constantsα=0andβ=3/2.The right-hand side refines1Γ(x)+1Γ(1/x)⩽2(x>0),which was proved in 1974 by W. Gautschi. Moreover, we present sharp inequalities for the product, the sum, the difference, and the ratio of Γ(x)Γ(x) and Γ(1/x)Γ(1/x).