On Applications of Extended Jack's Lemma

TL;DR

This paper explores a new class of analytic functions related to sector domains, utilizing an extended Jack's lemma to establish conditions for their inclusion, and connects these findings to existing results.

Contribution

It introduces the class ${\bf\mathcal{G}}(\alpha,\beta)$ and applies an extended Jack's lemma to derive new sufficient conditions for these functions.

Findings

Derived open-door lemma type conditions for ${\bf\mathcal{G}}(\alpha,\beta)$

Connected special cases to known results

Extended Jack's lemma application in sector domain analysis

Abstract

We introduce and study the class ${\bf\mathcal{G}}(\alpha,\beta)$ comprising analytic functions associated with a sector domain, where $\alpha,\beta\in(0,1]$. Using the extended version of Jack's lemma, we deduce Open-Door lemma type sufficient conditions for functions to be in ${\bf\mathcal{G}}(\alpha,\beta)$. Furthermore, we point out special cases of our results that align with known results.