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

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.