Directional convexity and characterizations of Beta and Gamma functions

The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation $ϕ\left( x+1\right) =\frac{x\left( x+k\right) }{\left( 2x+k+1\right) \left( 2x+k\right) }ϕ\left( x\right) ,\ \ \ \ \ \ x>0,$ for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function.

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