Some characterizations of the Euler gamma function

Abstract

Assume that $f:(0,\infty)\rightarrow (0,\infty ) $ is bounded from above on a set of positive Lebesgue measures or on a set of the second category with the Baire property and satisfies the functional equation $f(x+1)=xf(x)$ for $x>0$ and $f(1)=1$. We prove that, if there is a positive sequence $(p_{n})$, $\lim_{n\rightarrow \infty }p_{n}=\infty$, such that for every $n\in \mathbb{N}$, the function $x\mapsto \log ( x^{p_{n}})$ is Jensen convex in the interval $(1,\infty ) $; or there are two positive sequences $(p_{n})$ and $(q_{n})$, $\lim_{n\rightarrow \infty }p_{n}=\infty$, $% \lim_{n\rightarrow \infty }q_{n}=0$ such that, for every $n\in \mathbb{N}$, the function $x\mapsto [f(x^{p_{n}})] ^{q_{n}}$ is Jensen convex in the interval $(1,\infty)$, then $f$ is the Euler gamma function.