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arXiv · 1206.4814

Log-concavity for series in reciprocal gamma functions and applications

S. I. Kalmykov·D. B. Karp
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Abstract

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.

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S. I. Kalmykov, D. B. Karp. 2012-06-21. Log-concavity for series in reciprocal gamma functions and applications. https://arxiv.org/abs/1206.4814

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