Summary: 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 \[ \left| zf'(z)-f(z)\right| <\frac{1}{2}\quad (z\in \Delta). \] The class \(\Omega\) was introduced recently by Z. Peng and G. Zhong [Acta Math. Sci., Ser. B, Engl. Ed. 37, No. 1, 69–78 (2017; Zbl 1389.30072)]. Also let \(\mathcal{U}\) denote the class of functions \(f\) analytic and normalized in \(\Delta\) and satisfying the condition \[ \bigg| \left( \frac{z}{f(z)}\right)^2f'(z)-1\bigg| <1\quad (z\in \Delta ). \] 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-Szegö problem for the elements of \(\Omega\), the Toeplitz matrices for \(\Omega\) are also discussed in this article.