A superadditive property of Hadamard’s gamma function

Abstract

Hadamard’s gamma function is defined by

$$H(x)=\frac{1}{\Gamma(1-x)}\frac{d}{dx}\log \frac{\Gamma(1/2-x/2)}{\Gamma(1-x/2)},$$

where Γ denotes the classical gamma function of Euler. H is an entire function, which satisfies H(n)=(n−1)! for all positive integers n. We prove the following superadditive property.

Let α be a real number. The inequality

$$H(x)+H(y)\leq H(x+y)$$

holds for all real numbers x,y with x,y≥α if and only if α≥α ₀=1.5031…. Here, α ₀ is the only solution of H(2t)=2H(t) in [1.5,∞).