A superadditive property of Hadamard’s gamma function

Article
First Online:
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DOI: 10.1007/s12188-008-0009-5

Cite this article as:
Alzer, H. Abh. Math. Semin. Univ. Hambg. (2009) 79: 11. doi:10.1007/s12188-008-0009-5

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≥α0=1.5031…. Here, α0 is the only solution of H(2t)=2H(t) in [1.5,∞).

Keywords

Hadamard’s and Euler’s gamma functions Psi function Superadditive Convex Inequalities 

Mathematics Subject Classification (2000)

33B15 39B62 

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2008

Authors and Affiliations

  1. 1.WaldbrölGermany

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