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

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}(α,β):=\left\{ f\in \mathcal{A}: {\rm Re}\left\{f'(z)+\frac{1+e^{iα}}{2}zf''(z)\right\}>β,\, |z|<1\right\} \end{equation*} and \begin{equation*} \mathcal{L}_α(b):=\left\{f\in\mathcal{A}:\left|f'(z) +\frac{1+e^{iα}}{2}zf''(z)-b\right|< b,\, |z|<1 \right\}, \end{equation*} where $-\pi <\alpha \le \pi$, $0\le \beta <1$ and $b>1/2$. We show that if $f\in \mathcal{R}(\alpha ,\beta )$, then $\mathrm{Re}\{f^{\prime }(z)\}$ and $\mathrm{Re}\{f(z)/z\}$ are greater than $\beta$, and if $f\in {\mathcal{L}}_{\alpha }(b)$, then $0<\mathrm{Re}\{f^{\prime }(z)\}<2b$. 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.

9 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1809.03022