Peng and Zhong (Acta Math Sci {\bf37B(1)}:69--78, 2017) introduced and studied a new subclass of analytic functions as follows: \begin{equation*} \Omega:=\left\{f\in \mathcal{A}:\left|zf'(z)-f(z)\right|<\frac{1}{2}, z\in \Delta\right\}, \end{equation*} where $\mathcal{A}$ is the class of analytic and normalized functions and $\Delta$ is the open unit disc on the complex plane. The class $\Omega$ is a subclass of the starlike univalent functions. In this paper, we obtain some new results for the class $\Omega$ and improve some results that earlier obtained by Peng and Zhong.查看全文