Date: 08 Jan 2009

A superadditive property of Hadamard’s gamma function

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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,yα if and only if αα 0=1.5031…. Here, α 0 is the only solution of H(2t)=2H(t) in [1.5,∞).

Communicated by O. Riemenschneider.