Log-concavity and Tur\'{a}n-type inequalities for the generalized hypergeometric function

TL;DR

This paper investigates the log-concavity and Turán-type inequalities of the generalized hypergeometric function, providing new proofs and conjectures about the zeros' reality using integral representations and properties of Meijer's G function.

Contribution

It introduces new methods to analyze the log-convexity and concavity of the generalized hypergeometric function and explores the implications for the zeros' distribution.

Findings

Proved log-convexity using integral representations and Meijer's G function.

Analyzed power series coefficients of the generalized Turánian.

Formulated a conjecture on the reality of zeros based on Laguerre inequalities.

Abstract

The paper studies logarithmic convexity and concavity of the generalized hypergeometric function with respect to simultaneous shift of several parameters. We use integral representations and properties of Meijer's $G$ function to prove log-convexity. When all parameters are shifted we use series manipulations to examine the power series coefficients of the generalized Tur\'{a}nian formed by the generalized hypergeometric function. In cases when all zeros of the generalized hypergeometric function are real, we further explore the consequences of the extended Laguerre inequalities and formulate a conjecture about reality of zeros.