On extensions of the Open Door Lemma

Abstract

Abstract The Open Door Lemma is the well-known famous result due to Miller and Mocanu that provides a sufficient condition for an analytic function on the unit disk to have positive real part. This lemma has already been corrected by Kuroki and Owa for non-real initial point. Also, Li and Sugawa extended the Open Door Lemma in a such way that an analytic function takes its values in the given symmetric sector. Now, by the same methods, we provide a sufficient condition for an analytic function to take its values in the asymmetric arbitrary sector.