A convexity property and a new characterization of Euler's gamma function

Our main results are: (I) Let alpha not equal 0 be a real number. The function (Gamma circle exp)(alpha) is convex on R if and only if alpha >= max(0<t<x0) (-1/t psi(t) - psi'(t)/psi(t)(2)) = 0.0258 .... Here, x(0) = 1.4616 ... denotes the only positive zero of psi = Gamma'/Gamma. (II) Assume that a function f : (0, infinity) -> (0, infinity) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies f(x + 1) = xf(x) for x > 0 and f(1) = 1. If there are a number b and a sequence of positive real numbers (a(n)) (n is an element of N) with lim(n ->infinity) a(n) = 0 such that for every n the function (f circle exp)(an) is Jensen convex on (b, infinity), then f is the gamma function.