Abstract
Our aim in this paper is to derive several new integral representations for the Fox–Wright functions. In particular, we give new Laplace and Stieltjes transforms for this special function under some restrictions on parameters. From the positivity conditions on the weight in these representations, we found sufficient conditions to be imposed on the parameters of the Fox–Wright functions which allow us to conclude that it is completely monotonic. As applications, we derive a class of functions that are related to the Fox H-functions and are positive definite. Moreover, we extended the Luke's inequalities and we establish new Turán 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.
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