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