Some Applications of a Lemma Concerning Analytic Functions Having Positive Real Parts in the Unit Disk

TL;DR

This paper explores applications of a classical lemma related to analytic functions with positive real parts, generalizing subclasses of univalent functions and unifying various existing results in complex analysis.

Contribution

It introduces a new generalization of subclasses of univalent functions based on a lemma by Babalola and Opoola, unifying many previous results.

Findings

Unified several known results in univalent function theory

Generalized subclasses of univalent functions

Extended classical lemmas to broader contexts

Abstract

In this paper we give some applications of a lemma of Babalola and Opoola \cite{BO}, which is a classical extension of an earlier one by Lewandowski, Miller and Zlotkiewicz \cite{LMZ}. The applications were given via a new generalization of some well-known subclasses of univalent functions, and they unify many known results.