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

We prove that the double-inequality 1/x+1/x+x/(x+1/x)1/2 0 with the best possible constants α=0 and β=3/√2. The right-hand side refines 1/Γ(x)+1/Γ(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) and Γ(1/x).