On subordination of some analytic functions

We define $\mathscr V(\alpha,\beta)$ ($alpha<1$ and $\beta>1$), the new subclass of analytic functions with bounded positive real part,
$$ \mathscr V(\alpha,\beta)^=\Bigl\{f\in\mathscr A\colon\alpha<\operatorname{Re}\Bigl\{\Bigl(\frac z{f(z)}\Bigr)^2f'(z)\Bigr\}<\beta\Bigr\}, $$
and study some properties of $\mathscr V(\alpha,\beta)$. We also study the class $\mathscr U(\gamma)$ ($\gamma>0$):
$$ \mathscr U(\gamma):=\Bigl\{f\in\mathscr A\colon\Bigl|\Bigl(\frac z{f(z)}\Bigr)^2f'(z)-1\Bigr|<\gamma\Bigr\}, $$
where $\mathscr A$ is the class of normalized functions.