On Geometrical Properties of Certain Analytic functions

Let \(\psi\) be univalent, we introduce a newly defined class of analytic functions, given by $$\begin{aligned} {\mathcal {F}}(\psi ):= \left\{ f\in {A}}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \psi (z),\ (0)=0 \right\} . \end{aligned}$$Note that functions in this may not univalent. Further, establish the growth theorem with some geometrical conditions on and obtain Koebe domain related sharp inequalities. As an application, for complete range \(\alpha\) \(\beta\), classes \({{\mathcal {B}}}{{\mathcal {S}}}(\alpha \{f\in {A}} : ({zf'(z)}/{f(z)})-1 {z}/{(1-\alpha z^2)},\ \alpha \in [0,1) \}\) \({\mathcal {S}}_{\hbox {cs}}(\beta {z}/({(1-z)(1+\beta z)}),\ \beta \}\), respectively, which fact improves earlier known bounds. The Bohr radii \(S({{\mathcal ))= \{g=\sum _{k=1}^{\infty }b_k z^k g\prec f,\ {{\mathcal ) )\) are also obtained. A few examples as well certain basis geometry discussed.