Log-convexity and log-concavity for series in gamma ratios and applications

Abstract

Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of parameter. Four types of such power series are considered where parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of $q$-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.