Inequalities for the gamma function with applications to permanents (Q1918538)

The theorem of Bohr-Mollerup (1922) characterizes the gamma-function through the functional equation and log-convexity. The authors show that \(f(x) = \log (\Gamma (x+1))/x\) can be characterized by the corresponding functional equation, concavity and \(f(1) = 0\). This result is used to prove some upper bounds for the permanent of a \(0-1\)-matrix, as e.g. \(\text{per} (A) \leq (\Gamma (\alpha /n + 1))^{n^2/ \alpha}\). For related bounds see also \textit{R. A. Brualdi}, \textit{J. L. Goldwasser} and \textit{T. S. Michael}, J. Comb. Theory, Ser. A 47, No. 2, 207-245 (1988; Zbl 0647.15002)].