  We investigate the non-univalent function's properties reminiscent of the
theory of univalent starlike functions. Let the analytic function
$\psi(z)=\sum_{i=1}^{\infty}A_i z^i$, $A_1\neq0$ 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 $f(z)=z+\sum_{k=2}^{\infty}a_k z^k$
satisfying $({zf'(z)}/{f(z)}-1) \prec \psi(z)$. 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.
