A superadditive property of Hadamard’s gamma function

@article{Alzer2009ASP,
  title={A superadditive property of Hadamard’s gamma function},
  author={Horst Alzer},
  journal={Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
  year={2009},
  volume={79},
  pages={11-23}
}
  • H. Alzer
  • Published 8 January 2009
  • Mathematics
  • Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
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… 
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