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

Abstract

Let ${\mathcal{A}}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+{\sum }_{k=n+1}^{\infty }{a}_k{z}^k,n\in \mathbb{N}$ and let \begin{align*} Ω_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 ${\hat{\Omega }}_n\subset {\Omega }_n$ and obtain subordination results for members of ${\hat{\Omega }}_n$ 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.

20 pages