Sufficient conditions and radii problems for a starlike class involving a differential inequality

Bull. Korean Math. Soc. 2020 Vol. 57, No. 6, 1409-1426 https://doi.org/10.4134/BKMS.b191074 Published online August 3, 2020 Printed November 30, 2020

Anbhu Swaminathan, Lateef Ahmad Wani Indian Institute of Technology Roorkee; Indian Institute of Technology Roorkee

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.