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 $f(z)=z+{\sum }_{k=n+1}^{\mathrm{\infty }}{a}_k{z}^k,n\in \mathbb{N}$ 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 ${\mathrm{\Omega }}_n$, and then employ these conditions to construct functions which involve double integrals and members of ${\mathrm{\Omega }}_n$. We also consider a subclass ${\widehat{\mathrm{\Omega }}}_n\subset {\mathrm{\Omega }}_n$ and obtain subordination results for members of ${\widehat{\mathrm{\Omega }}}_n$ besides a necessary and sufficient condition. Writing ${\mathrm{\Omega }}_1=\mathrm{\Omega }$, we obtain inclusion properties of $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.