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

Abstract

Our main results are: (I) Let α ≠ 0 be a real number. The function (Γ ◦ exp)α is convex on \({\mathbf{R}}\) if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x0 = 1.4616... denotes the only positive zero of \({\psi = \Gamma'/\Gamma}\). (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 $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$ If there are a number b and a sequence of positive real numbers (an) \({(n \in \mathbf{N})}\) with \({{\rm lim}_{n\to\infty} a_n =0}\) such that for every n the function \({(f \circ {\rm exp})^{a_n}}\) is Jensen convex on (b, ∞), then f is the gamma function.