Some characterizations of the Euler gamma function (Q888116)

The author proves a general Bohr-Mollerup type theorem on the characterization of the Euler gamma function. A consequence of his result states as follows: If \(f:(0,\infty)\to (0,\infty)\) satisfies \(f(x+1)= xf(x)\) for \(x>0\), with \(f(1)= 1\), and if for every positive integer \(n\), the function \(x\to (f(x^n))^{1/n}\) is convex, then \(f\) is the gamma function.