On a problem of H.-H. Kairies concerning Euler's Gamma function

Abstract

The Bohr-Mollerup theorem on the Euler $Γ$ function states: If $f : ℝ_+ → ℝ_+$ satisfies the functional equation $f(x+1) = xf(x)$ on $ℝ_+$, log ○ $f$ is convex on $(γ , +∞)$ for some $γ ≥ 0$ and $f(1) = 1$ then $f = Γ$. We give some partial answers to the question posed by H.-H. Kairies: By what other function can the logarithm be replaced in this statement.