Inequalities for the gamma function with applications to permanents

The best known upper bound on the permanent of a 0-1 matrix relies on the knowledge of the number of nonzero entries per row. In certain applications only the total number of nonzero entries is known. In order to derive bounds in this situation we prove that the function f : (?1; 1) ! R, deened by f(x) := log ?(x+1) x , is concave, strictly increasing and satisses an analogue of the famous Bohr-Mollerup theorem. For further discussion of such bounds we derive some inequalities for this function. 1 Characterization of log ?(x+1) x The famous theorem of Bohr and Mollerup ((2] pp. 276) characterizes the ?-function by the convexity of its logarithm. For the function f : (?1; 1) ! R, deened by f(x) := log ?(x+1) x , we will prove a similar characterization by its concavity. Let Z ? denote the set of all negative integers. At rst we derive expansions for the function f : C n Z ? ! C, deened by f(z) := log ?(z+1) z , and its derivatives from the following well-known expansions ((2] pp. 274):