On certain subclasses of close-to-convex functions related with the second-order differential subordination

Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*} \mathcal{R}(\alpha,\beta):=\left\{ f\in \mathcal{A}: {\rm Re}\left\{f'(z)+\frac{1+e^{i\alpha}}{2}zf''(z)\right\}>\beta,\, |z|<1\right\} \end{equation*} and \begin{equation*} \mathcal{L}_\alpha(b):=\left\{f\in\mathcal{A}:\left|f'(z) +\frac{1+e^{i\alpha}}{2}zf''(z)-b\right|< b,\, |z|<1 \right\}, \end{equation*} where $-\pi<\alpha\leq \pi$, $0\leq \beta<1$ and $b>1/2$. We show that if $f\in \mathcal{R}(\alpha,\beta)$, then ${\rm Re}\{f'(z)\}$ and ${\rm Re}\{f(z)/z\}$ are greater than $\beta$, and if $f\in\mathcal{L}_\alpha(b)$, then $0<{\rm Re}\{f'(z)\}<2b$. Also, some another interesting properties of the class $\mathcal{L}_\alpha(b)$ are investigated. Finally, the radius of univalence of 2-th section sum of $f\in \mathcal{R}(\alpha,\beta)$ is obtained.