New subclasses of Ozaka's convex functions

Let $\mathcal{S}^{\ast}_{L}(\uplambda)$ and $\mathcal{CV}_L(\uplambda)$ be the classes of functions$f$, analytic in the unit disc $\Updelta=\{z\colon|z|<1\}$, with thenormalization $f(0)=f'(0)-1=0$, which satisfies the conditions\begin{equation*}\frac{zf'(z)}{f(z)}\prec \left(1+z\right)^{\uplambda}\quad\text{and}\quad \left(1+\frac{zf''(z)}{f'(z)}\right)\prec \left(1+z\right)^{\uplambda}\qquad \left(0<\uplambda\le 1 \right),\end{equation*}where $\prec$ is the subordination relation, respectively. The classes$\mathcal{S}^{\ast}_{L}(\uplambda)$ and $\mathcal{CV}_L(\uplambda)$ are subfamilies of the known classes of strongly starlike and convex functions of order $\uplambda$. We consider  the relations between $\mathcal{S}^{\ast}_{L}(\uplambda)$, $\mathcal{CV}_L(\uplambda)$ and other classes geometrically defined. Also, we obtain the sharp radius of convexity for functions belonging to $\mathcal{S}^{\ast}_{L}(\uplambda)$ class. Furthermore,  the norm of pre-Schwarzian derivatives  and univalency of functions $f$ which satisfy the condition\begin{equation*}\Re\left\{1+\frac{zf''(z)}{f'(z)}\right\}<1+\frac{\uplambda}{2}\qquad\myp{z \in \Updelta}, \end{equation*}are considered.