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

We consider necessary and sufficient conditions for the convexity of a function x 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. © Birkhäuser Verlag, Basel, 1998.