New subclasses of Ozaka's convex functions

Let \mathcal{S}^{\ast}_{L}(\uplambda) and \mathcal{CV}_L(\uplambda) be the classes of functionsf, analytic in the unit disc \Updelta=\{z\colon|z|<۱\}, with thenormalization f(۰)=f'(۰)-۱=۰, which satisfies the conditions\begin{equation*}\frac{zf'(z)}{f(z)}\prec \left(۱+z\right)^{\uplambda}\quad\text{and}\quad \left(۱+\frac{zf''(z)}{f'(z)}\right)\prec \left(۱+z\right)^{\uplambda}\qquad \left(۰<\uplambda\le ۱ \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\{۱+\frac{zf''(z)}{f'(z)}\right\}<۱+\frac{\uplambda}{۲}\qquad\myp{z \in \Updelta}, \end{equation*}are considered.

Keywords:

Univalent functions , subordination , strongly starlike functions , Domain bounded by Sinusoidal spiral