On Geometrical Properties of Certain Analytic functions

TL;DR

This paper explores geometric properties of a class of analytic functions defined via subordination, establishing growth theorems, Koebe domains, and sharp inequalities, with applications to specific subclasses and Bohr radii.

Contribution

It introduces a new class of analytic functions and derives growth theorems, Koebe domains, and sharp bounds, extending previous results and defining new subclasses based on geometric conditions.

Findings

Established growth theorem under geometric conditions on aa

Obtained sharp inequalities for Koebe domain

Calculated sharp Bohr radii for specific subclasses

Abstract

We introduce the class of analytic functions $$\mathcal{F}(\psi):= \left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right) \prec \psi(z),\; \psi(0)=0 \right\},$$ where $\psi$ is univalent and establish the growth theorem with some geometric conditions on $\psi$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $\alpha$ and $\beta$ for the functions in the classes $\mathcal{BS}(\alpha):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/{(1-\alpha z^2)},\; \alpha\in [0,1) \}$ and $\mathcal{S}_{cs}(\beta):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/({(1-z)(1+\beta z)}),\; \beta\in [0,1) \}$, respectively which improves the earlier known bounds. The sharp Bohr-radii for the classes $S(\mathcal{BS}(\alpha))$ and $\mathcal{BS}(\alpha)$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.