  Euler's gamma function is logarithmically convex on positive semi-axis.
Additivity of logarithmic convexity implies that the function sum of gammas
with non-negative coefficients is also log-convex. In this paper we investigate
the series in reciprocal gamma functions, 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 the sequence of coefficients
times factorial is log-concave and the sum is discrete Wright log-concave if
the coefficents are log-concave. 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.
