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2015 Some characterizations of the Euler gamma function
Janusz Matkowski
Rocky Mountain J. Math. 45(4): 1225-1231 (2015). DOI: 10.1216/RMJ-2015-45-4-1225

Abstract

Assume that $f:(0,\infty)\rightarrow (0,\infty ) $ is bounded from above on a set of positive Lebesgue measures or on a set of the second category with the Baire property and satisfies the functional equation $f(x+1)=xf(x)$ for $x>0$ and $f(1)=1$. We prove that, if there is a positive sequence $(p_{n})$, $\lim_{n\rightarrow \infty }p_{n}=\infty$, such that for every $n\in \mathbb{N}$, the function $x\mapsto \log ( x^{p_{n}})$ is Jensen convex in the interval $(1,\infty ) $; or there are two positive sequences $(p_{n})$ and $(q_{n})$, $\lim_{n\rightarrow \infty }p_{n}=\infty$, $% \lim_{n\rightarrow \infty }q_{n}=0$ such that, for every $n\in \mathbb{N}$, the function $x\mapsto [f(x^{p_{n}})] ^{q_{n}}$ is Jensen convex in the interval $(1,\infty)$, then $f$ is the Euler gamma function.

Citation

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Janusz Matkowski. "Some characterizations of the Euler gamma function." Rocky Mountain J. Math. 45 (4) 1225 - 1231, 2015. https://doi.org/10.1216/RMJ-2015-45-4-1225

Information

Published: 2015
First available in Project Euclid: 2 November 2015

zbMATH: 1332.33002
MathSciNet: MR3418192
Digital Object Identifier: 10.1216/RMJ-2015-45-4-1225

Subjects:
Primary: 26A51‎, 26D07, 33B15, 39B22

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

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Vol.45 • No. 4 • 2015
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