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

Abstract

Let $Ω$ denote the class of functions $f$ analytic in the open unit disc $Δ$, 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\inΔ). \end{equation*} The class $Ω$ 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 $Δ$ and satisfying the condition \begin{equation*} \left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<1\quad(z\inΔ). \end{equation*} In this article, we obtain some further results for the class $Ω$ including, an extremal function and more examples of $Ω$, inclusion relation between $Ω$ and $\mathcal{U}$, the radius of starlikeness, convexity and close--to--convexity and sufficient condition for function $f$ to be in $Ω$. Furthermore, along with the settlement of the coefficient problem and the Fekete--Szegö problem for the elements of $Ω$, the Toeplitz matrices for $Ω$ are also discussed in this article.