Inequalities involving and

We prove that the double-inequality 1 x + 1 / x + α ( x + 1 / x ) 1 / 2 < 1 Γ ( x ) + 1 Γ ( 1 / x ) ⩽ 1 x + 1 / x + β ( x + 1 / x ) 1 / 2 holds for all x > 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 ) .