A convexity property and a new characterization of Euler's gamma function (Q1942313)

The authors investigate the convexity property of the function \((\Gamma\circ \exp)^\alpha\), where \(\alpha \neq 0\) is a real number. In particular, the authors proved that the function \((\Gamma\circ \exp)^\alpha\) is convex if and only if \[ \alpha \geq \max_{0<t<x_ 0}\Big(-\frac1{t\psi (t)}-\frac{\psi\prime(t)}{\psi(t)^2}\Big)=0.0258\ldots, \] where \(x_ 0 =1.4616\ldots\) denotes the only positive zero of \(\psi=\frac{\Gamma'}{\Gamma}\). Furthermore, the authors give a new characterization of the gamma function: \(f:(0,\infty) \to (0,\infty)\) be bounded from above on a set of positive Lebesgue measure (or a set of the second category with the Baire property) and satisfy \[ f(x+1)=xf(x) \quad \text{for} \quad x>0\quad \text{and}\quad f(1)=1. \] Furthermore, if there exists a number \(b\) and a sequence of positive real numbers \((a_ n)\), \((n\in\mathbb{ N}) \) with \(\lim_{n\to\infty}a_n=0\) such that for every \(n\) the function \((f\circ \exp)^{a_n}\) is Jensen convex on \((b,\infty)\), then \(f\) is the gamma function. This result gives an improvement of the recent work of \textit{D. Gronau} and \textit{J. Matkowski} [Publ. Math. 63, No. 1--2, 105--113 (2003; Zbl 1027.33001)].