Conditions for convexity of a derivative and some applications to the Gamma function

Abstract.

We consider necessary and sufficient conditions for the convexity of a function $x \to f '(x)$ in terms of some properties of the associated function of two variables F (x, y) = (f (y) - f (x)) / (y - x). In particular, we prove that f ' is convex if and only if F is convex and if and only if F is Schur-convex. These results are applied to the theory of the Gamma function. We complement a characterization of the Gamma function due to H. Kairies and present some inequalities for the ratio of Gamma functions.