Functional estimates and integral inequalities for the Fox-Wright function

In this paper, our aim is to show some mean value inequalities for the Fox-Wright function. In order to prove our main results, we present some monotonicity, convexity and concavity properties for some classes of functions related to the Fox-Wright function, which are in fact equivalents to the corresponding Turan type inequalities for this function. As a direct consequences, it deduces some new results involving some special functions, such as the generalized hypergeometric function, the four parameters Wright function and three parameter Mittag-Leffler type function. At the end of this paper, several integrals inequalities associated to the Fox-Wright function are established. As applications, new functional inequalities (such as Turan-type inequalities) for the incomplete Fox-Wright and incomplete generalized hypergeometric functions are established.