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

Bull. Korean Math. Soc. Published online August 3, 2020

ANBHU SWAMINATHAN and LATEEF AHMAD WANI
INDIAN INSTITUTE OF TECHNOLOGY ROORKEE, ROORKEE-247667, UTTARAKHAND, INDIA

Abstract : Let $\mathcal{A}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+\sum_{k=n+1}^\infty a_kz^k,n\in\mathbb{N}$ defined on the open unit disk $\mathbb{D}$, 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*}
In this paper, we make use of differential subordination technique to obtain sufficient conditions for the class $\Omega_n$. Writing $\Omega:=\Omega_1$, we obtain inclusion properties of $\Omega$ with respect to functions which map $\mathbb{D}$ onto 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.