On Applications of Extended Jack's Lemma

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.