  Let $\phi$ be a normalized convex function defined on open unit disk
$\mathbb{D}$. For a unified class of normalized analytic functions which
satisfy the second order differential subordination $f'(z)+ \alpha z f''(z)
\prec \phi(z)$ for all $z\in \mathbb{D}$, we investigate the distortion theorem
and growth theorem. Further, the bounds on initial logarithmic coefficients,
inverse coefficient and the second Hankel determinant involving the inverse
coefficients are examined.
