Further results for a subclass of univalent functions related with differential equation

Let $\Omega$ denote the class of functions $f$ analytic in the open unit disc $\Delta$, normalized by the condition $f(0)=f'(0)-1=0$ and satisfying the inequality \begin{equation*} \left|zf'(z)-f(z)\right|<\frac{1}{2}\quad(z\in\Delta). \end{equation*} The class $\Omega$ was introduced recently by Peng and Zhong (Acta Math Sci {\bf37B(1)}:69--78, 2017). Also let $\mathcal{U}$ denote the class of functions $f$ analytic and normalized in $\Delta$ and satisfying the condition \begin{equation*} \left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<1\quad(z\in\Delta). \end{equation*} In this article, we obtain some further results for the class $\Omega$ including, an extremal function and more examples of $\Omega$, inclusion relation between $\Omega$ and $\mathcal{U}$, the radius of starlikeness, convexity and close--to--convexity and sufficient condition for function $f$ to be in $\Omega$. Furthermore, along with the settlement of the coefficient problem and the Fekete--Szeg\"{o} problem for the elements of $\Omega$, the Toeplitz matrices for $\Omega$ are also discussed in this article.