On a class of univalent functions defined by a differential inequality

Abstract

For $0<\lambda\le 1$, let $\mathcal{U}(\lambda)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<\lambda$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-called theory of star functions, we also solve the integral mean problem for functions in $\mathcal{U}(\lambda)$. We also obtain the estimate of the Fekete-Szeg\"{o} functional and the pre-Schwarzian norm of certain nonlinear integral transform of functions in $\mathcal{U}(\lambda)$. Further, for the class of meromorphic functions which are defined in $\Delta:=\{\zeta\in\mathbb{\widehat{C}}:|\zeta|>1\}$ and associated with the class $\mathcal{U}(\lambda)$, we obtain a sufficient condition for a function $g$ to be an extreme point of this class.