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