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Gamma function inequalities

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Abstract

We prove various new inequalities for Euler’s gamma function. One of our theorems states that the double-inequality

$$\alpha \cdot \Bigl(\frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr) {\kern-1pt}<{\kern-1pt} \frac{1}{\Gamma\,( \sqrt{x+y-xy})}+ \frac{1}{\Gamma\,( \sqrt{xy})} {\kern-1pt} <{\kern-1pt} \beta \cdot \Bigl( \frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr)$$

is valid for all real numbers x,y ∈ (0,1) with the best possible constant factors \(\alpha=1/\sqrt{2}=0.707...\) and β = 1.

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Authors and Affiliations

  1. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

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  1. Horst Alzer
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Correspondence to Horst Alzer.

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To the memory of Professor Luigi Gatteschi.

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Alzer, H. Gamma function inequalities. Numer Algor 49, 53–84 (2008). https://doi.org/10.1007/s11075-008-9160-4

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  • DOI: https://doi.org/10.1007/s11075-008-9160-4

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