Representations and inequalities for generalized hypergeometric functions

TL;DR

This paper introduces a unified integral representation for generalized hypergeometric functions, enabling new insights into their properties such as monotonicity, convexity, and bounds, with applications to Bessel-type functions.

Contribution

It provides a unified integral representation for generalized hypergeometric functions and derives new properties and bounds using positivity and series representations.

Findings

Established complete monotonicity and log-convexity in upper parameters.

Proved monotonicity of ratios and new bounds for hypergeometric functions.

Derived two-sided inequalities for Bessel-type hypergeometric functions.

Abstract

We find an integral representation for the generalized hypergeometric function unifying known representations via generalized Stieltjes, Laplace and cosine Fourier transforms. Using positivity conditions for the weight in this representation we establish various new facts regarding generalized hypergeometric functions, including complete monotonicity, log-convexity in upper parameters, monotonicity of ratios and new proofs of Luke's bounds. Besides, we derive two-sided inequalities for the Bessel type hypergeometric functions by using their series representations.