Some extensions of the Open Door Lemma

TL;DR

This paper extends the Open Door Lemma to provide conditions under which analytic functions on the unit disk take values in a specified sector, broadening its applicability in complex analysis.

Contribution

The authors generalize the Open Door Lemma to include functions mapping into sectors, building on previous modifications for non-real initial points.

Findings

Extended the Open Door Lemma to sector-valued functions

Provided new sufficient conditions for analytic functions in the unit disk

Broadened the applicability of the original lemma

Abstract

Miller and Mocanu proved in their 1997 paper a greatly useful result which is now known as the Open Door Lemma. It provides a sufficient condition for an analytic function on the unit disk to have positive real part. Kuroki and Owa modified the lemma when the initial point is non-real. In the present note, by extending their methods, we give a sufficient condition for an analytic function on the unit disk to take its values in a given sector.