Journal of Convex Analysis 25 (2018), No. 3, 927--938
Copyright Heldermann Verlag 2018
Directional Convexity and Characterizations of Beta and Gamma Functions
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
[Abstract-pdf]
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.
Keywords: Gamma function, Beta function, beta-type function, logarithmical convexity, geometrical convexity, directional convexity, functional equation.
MSC: 33B15, 26B25, 39B22
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