New Integral representations for the Fox-Wright functions and its applications

TL;DR

This paper develops new integral representations for Fox-Wright functions, explores their positivity and monotonicity properties, and applies these findings to inequalities and bounds related to special functions and series.

Contribution

It introduces novel integral representations and transforms for Fox-Wright functions, establishing conditions for complete monotonicity and deriving new inequalities and bounds.

Findings

New Laplace and Stieltjes transforms for Fox-Wright functions

Conditions for complete monotonicity of these functions

Extended inequalities and bounds for related special functions

Abstract

Our aim in this paper is to derive several new integral representations of the Fox-Wright functions. In particular, we give new Laplace and Stieltjes transform for this special functions under a special restriction on parameters. From the positivity conditions for the weight in these representations, we found sufficient conditions to be imposed on the parameters of the Fox-Wright functions that it be completely monotonic. As applications, we derive a class of function related to the Fox H-functions is positive definite and an investigation of a class of the Fox H-function is non-negative. Moreover, we extended the Luke's inequalities and we establish a new Tur\'an type inequalities for the Fox-Wright function. Finally, by appealing to each of the Luke's inequalities, two sets of two-sided bounding inequalities for the generalized Mathieu's type series are proved.