On a class of univalent functions

Abstract

Let $\mathcal{A}$ be the class of analytic functions in the unit disk $\mathbb{D}$ with the normalization $f\left(0\right)=f^{\prime }\left(0\right)-1=0$. Denote by $\mathcal{N}$ the class of functions $f\in \mathcal{A}$ which satisfy the condition

$$|-z^3{\left(\frac{z}{f\left(z\right)}\right)}^{‴}+f^{\prime }\left(z\right){\left(\frac{z}{f\left(z\right)}\right)}^2-1|\le 1\text{,}z\in \mathbb{D}\text{.}$$

We show that functions in $\mathcal{N}$ are univalent in $\mathbb{D}$ but not necessarily starlike. Also, we present the characterization formula, necessary and sufficient coefficient conditions for functions to be in the class $\mathcal{N}$.