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

Monotonicity patterns and functional inequalities for classical and generalized Wright functions