Monotonicity patterns and functional inequalities for classical and generalized Wright functions

In this paper our aim is to present the completely monotonicity and convexity properties for the Wright function. As consequences of these results, we present some functional inequalities. Moreover, we derive the monotonicity and log-convexity results for the generalized Wright functions. As applications, we present several new inequalities (like Tur\'an type inequalities) and we prove some geometric properties for four--parametric Mittag--Leffler functions.