2012/06/21 by S. I. Kalmykov, Sergei Kalmykov, Kalmykov, S. I. +3
Mathematics · #26A51 #33C05 #33C15 #33C20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications #Mathematical functions and polynomials #math.CA #msc:26A51 #msc:33C05 #msc:33C15 #msc:33C20
paper · pdf · doi:10.48550/arxiv.1206.4814
14 pages, no figures
arxiv created 2012/06/21 · openalex publication_date 2012/06/21 · arxiv updated 2012/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.