Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 25 (2018), No. 3, 927--938Copyright 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 [ Fulltext-pdf  (102  KB)] for subscribers only.