On the improvement of Mocanu’s conditions

Abstract

We estimate | Arg{p(z)}| for functions of the form p(z) = 1 + a 1 z + a 2 z 2 + a 3 z 3 + · · · in the unit disc D = {z : |z| < 1} under several assumptions. By using Nunokawa's lemma, we improve a few of Mocanu's results obtained by differential subordinations. Some applications for strongly starlikeness and convexity are formulated. MSC: Primary 30C45; secondary 30C80