Necessary and Sufficient Conditions for Absolute Monotonicity of Functions Related to Gaussian Hypergeometric Functions

2025/09/20 by Zhao, Tiehong
#26A48 #33E05 #40A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.18199

Abstract

This paper systematically investigates the absolute monotonicity of two function families associated with the Gaussian hypergeometric function F(a, b; c; x) (where a,b,c∈ℝ₊): F_p(x)=(1-x)^pF(a,b;c;x) and G_p(x)=(1-x)^p exp(F(a,b;c;x)), as well as the logarithmic transform \lnF_p(x). Our primary goal is to establish necessary and sufficient conditions for the parameter p such that -F'_p, \pmG'_p and ±(\lnF_p)' are absolutely monotonic on (0,1). Additionally, we derive several results regarding the absolute monotonicity of their higher-order derivatives. As applications, we derive several new inequalities for the Gaussian hypergeometric function F(a,b;c;x). Most importantly, we develop a novel constructive approach based on Jurkat's criterion for power series ratios, which avoids limitations of cumbersome recursive/inductive methods in existing literature.

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