A characterization of Euler's constant

We prove the following theorem. Let α and β be real numbers. The inequality Γ(xα+yβ)≤Γ(Γ(x)+Γ(y)) holds for all positive real numbers x and y if and only if α=β=−γ. Here, Γ and γ=0.57721… denote Euler's gamma function and Euler's constant, respectively.