Log-concavity for series in reciprocal gamma functions and applications

Abstract

Euler's gamma function Γ(x) is logarithmically convex on (0, ∞). Additivity of logarithmic convexity implies that the function x→∑fkΓ(x+k) is also log-convex (assuming convergence) if the coefficients are non-negative. In this paper, we investigate the series ∑fkΓ(x+k)−1, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the sum is log-concave if and is discrete Wright log-concave if . We conjecture that the latter condition is in fact sufficient for the log-concavity of the sum. We exemplify our general theorems by deriving known and new inequalities for the modified Bessel, Kummer and generalized hypergeometric functions and their parameter derivatives.