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Euler's gamma function is logarithmically convex on positive semi-axis.Additivity of logarithmic convexity implies that the function sum of gammaswith non-negative coefficients is also log-convex. In this paper we investigatethe series in reciprocal gamma functions, where each term is clearlylog-concave. Log-concavity is not preserved by addition, so that non-negativityof the coefficients is now insufficient to draw any conclusions about the sum.We demonstrate that the sum is log-concave if the sequence of coefficientstimes factorial is log-concave and the sum is discrete Wright log-concave ifthe coefficents are log-concave. We conjecture that the latter condition is infact sufficient for the log-concavity of the sum. We exemplify our generaltheorems by deriving known and new inequalities for the modified Bessel, Kummerand generalized hypergeometric functions and their parameter derivatives.

Log-concavity for series in reciprocal gamma functions and applications