  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.
