  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.
