A note on a subclass of bazilevi\v{c} functions

TL;DR

This paper introduces a new subclass of Bazilevi{} functions, explores its Hardy space membership, provides a necessary condition for a special case, and derives a sharp coefficient estimate.

Contribution

It defines a new subclass of Bazilevi{} functions, analyzes its properties, and establishes a sharp coefficient bound for a specific case.

Findings

Identified the Hardy space for the subclass of Bazilevi{} functions.

Provided a necessary condition for a particular subclass case.

Derived a sharp coefficient estimate for functions in ().

Abstract

In this artcle, we introduce and investigate a subclass of Bazilevi{\v{c}} functions, denoted by $\mathcal{B}_{\varphi_{A,B}}(\alpha^{(m)})$. We determine the Hardy space to which this subclass of Bazilevi{\v{c}} functions belong to. Additionally, we provide a necessary condition for a particular case of this subclass. Finally, we obtain a sharp coefficient estimate for the functions associated with $\mathcal{B}_1(\alpha).$