Directional Convexity and Characterizations of Beta and Gamma Functions

Martin Himmel; Janusz Matkowski

Abstract

The logarithmic convexity of restrictions of the Beta function to rays parallel to the main diagonal and the functional equation

\varphi (x+1) = \frac{x(x+k)}{(2x+k+1)(2x+k)}\, \phi(x),\ \ x>0,

for

k>0

allow to get a characterization of the Beta function. This fact and the notion of the beta-type function lead to a new characterization of the Gamma function.

Author information

Contact details are reproduced from the original publication and may be historical.

Martin Himmel

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora, Szafrana 4A, 65-516 Zielona Gora, Poland

himmel@mathematik.uni-mainz.de

Janusz Matkowski

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora, Szafrana 4A, 65-516 Zielona Gora, Poland

j.matkowski@wmie.uz.zgora.pl

Keywords

Gamma function
Beta function
beta-type function
logarithmical convexity
geometrical convexity
directional convexity
functional equation

Mathematics Subject Classification

33B15, 26B25, 39B22

Suggested citation

M. Himmel, J. Matkowski. “Directional Convexity and Characterizations of Beta and Gamma Functions.” Journal of Convex Analysis 25 (2018), No. 3, 927–938.