Functional estimates and integral inequalities for the Fox–Wright function

Abstract

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.