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

## TL;DR

This paper explores a specific class of univalent functions defined by a differential inequality, providing new results including extremal functions, inclusion relations, geometric properties, coefficient bounds, and Toeplitz matrix analysis.

## Contribution

It introduces new properties, examples, and bounds for the class , including extremal functions, geometric radii, and coefficient problems, expanding understanding of this subclass of univalent functions.

## Key findings

- Identified extremal functions within 
- Established inclusion relations between  and 
- Determined radii for starlikeness, convexity, and close-to-convexity

## Abstract

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.