Sufficient Conditions and Radius Problems for a starlike Class Involving a Differential Inequality

TL;DR

This paper establishes sufficient conditions for a class of analytic functions involving differential inequalities to be starlike, explores their properties, and solves radius problems with sharp estimates using differential subordination techniques.

Contribution

It introduces new sufficient conditions for the class ng involving differential inequalities, and analyzes their geometric properties and radius problems.

Findings

Derived sufficient conditions for ng using differential subordination.

Established inclusion relations with parabolic starlike and uniformly starlike classes.

Obtained sharp radius estimates and provided graphical illustrations.

Abstract

Let $\mathcal{A}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+\sum_{k=n+1}^\infty a_kz^k,n\in\mathbb{N}$ and let \begin{align*} \Omega_n:=\left\{f\in\mathcal{A}_n:\left|zf'(z)-f(z)\right|<\frac{1}{2},\; z\in\mathbb{D}\right\}. \end{align*} We make use of differential subordination technique to obtain sufficient conditions for the class $\Omega_n$, and then employ these conditions to construct functions which involve double integrals and members of $\Omega_n$. We also consider a subclass $\widehat{\Omega}_n\subset\Omega_n$ and obtain subordination results for members of $\widehat{\Omega}_n$ besides a necessary and sufficient condition. Writing $\Omega_1=\Omega$, we obtain inclusion properties of $\Omega$ with respect to functions defined on certain parabolic regions and as a consequence, establish a relation connecting the parabolic starlike class $\mathcal{S}_p$ and the uniformly starlike $UST$. Various radius problems for the class $\Omega$ are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.