
Journal of Convex Analysis 25 (2018), No. 3, 927938 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, 65516 Zielona Gora, Poland himmel@mathematik.unimainz.de Janusz Matkowski Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora, Szafrana 4A, 65516 Zielona Gora, Poland j.matkowski@wmie.uz.zgora.pl [Abstractpdf] 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 betatype function lead to a new characterization of the Gamma function. Keywords: Gamma function, Beta function, betatype function, logarithmical convexity, geometrical convexity, directional convexity, functional equation. MSC: 33B15, 26B25, 39B22 [ Fulltextpdf (102 KB)] for subscribers only. 