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

摘要

Our main results are: (I) Let α ≠ 0 be a real number. The function (Γ o{open} exp)~α is convex on Here, x_0 = 1. 4616... denotes the only positive zero of ψ=Γ′Γ. (II) Assume that a function f: (0, ∞) → (0, ∞) 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 ∈ N) with lim_(n→∞) = 0 such that for every n the function (f o{open} exp)~(an) is Jensen convex on (b, ∞), then f is the gamma function.