Theory of certain Non-Univalent Analytic functions

We investigate the non-univalent function’s properties reminiscent of the theory of univalent starlike functions. Let the analytic function $\psi (z)={\sum }_{i=1}^{\mathrm{\infty }}{A}_i{z}^i$, $A_1\mathrm{\ne }0$ be univalent in the unit disk. Non-univalent functions may be found in the class $\mathcal{F}(\psi )$ of analytic functions $f$ of the form $f(z)=z+{\sum }_{k=2}^{\mathrm{\infty }}{a}_k{z}^k$ satisfying $(z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)-1)\prec \psi (z)$. Such functions, like the Ma and Minda classes of starlike functions, also have nice geometric properties. For these functions, growth and distortion theorems have been established. Further, we obtain bounds for some sharp coefficient functionals and establish the Bohr and Rogosinki phenomenon for the class $\mathcal{F}(\psi )$. Non-analytic functions that share properties of analytic functions are known as Poly-analytic functions. Moreover, we compute Bohr and Rogosinski’s radius for Poly-analytic functions with analytic counterparts in the class $\mathcal{F}(\psi )$ or classes of Ma-Minda starlike and convex functions.