On certain subclasses of close-to-convex functions related with the second-order differential subordination (1901.02670v2)

Abstract: 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.