Geodesic
Directional Convexity and Characterizations of Beta and Gamma Functions
Journal of convex analysis, Tome 25 (2018) no. 3, pp. 927-938
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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.
Classification : 33B15, 26B25, 39B22
Mots-clés : Gamma function, Beta function, beta-type function, logarithmical convexity, geometrical convexity, directional convexity, functional equation 
@article{JCA_2018_25_3_JCA_2018_25_3_a10,
     author = {M. Himmel and J. Matkowski},
     title = {Directional {Convexity} and {Characterizations} of {Beta} and {Gamma} {Functions}},
     journal = {Journal of convex analysis},
     pages = {927--938},
     year = {2018},
     volume = {25},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2018_25_3_JCA_2018_25_3_a10/}
}
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M. Himmel; J. Matkowski. Directional Convexity and Characterizations of Beta and Gamma Functions. Journal of convex analysis, Tome 25 (2018) no. 3, pp. 927-938. http://geodesic.mathdoc.fr/item/JCA_2018_25_3_JCA_2018_25_3_a10/