Sufficient conditions and radius problems for the Silverman class

TL;DR

This paper investigates the Silverman class of analytic functions and a related class Omega, establishing sufficient differential inequalities, inclusion relations with known subclasses, and radius estimates to understand their geometric properties.

Contribution

It introduces new differential inequality conditions for the Silverman class and the class Omega, and determines their inclusion relations and radius bounds within well-known subclasses.

Findings

Derived sufficient conditions via differential inequalities.

Established inclusion relations with subclasses of starlike functions.

Calculated radius estimates for the classes within known subclasses.

Abstract

For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}:\left|\dfrac{1-\alpha+\alpha zf''(z)/f'(z)}{zf'(z)/f(z)}-(1-\alpha)\right|<\lambda, z\in\mathbb{D}\right\}, \end{equation} the general form of Silverman class introduced by Tuneski and Irnak. For this class we derive some sufficient conditions in the form of differential inequalities. Further, we consider the class $\Omega,$ given by \begin{equation}\label{omega} \Omega=\left\{f\in\mathcal{A}:|zf'(z)-f(z)|<\dfrac{1}{2},\;z\in\mathbb{D}\right\}. \end{equation} For the above two classes, we establish inclusion relations involving some other well known subclasses of $\mathcal{S}^*$ and find radius estimates for different pairs involving these classes.