On some length problems for analytic functions

Abstract

Let $\mathcal{A}$ be the class of functions \begin{equation*} f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n} \end{equation*} which are analytic in the unit disk $\mathbb{D}=\{z\colon \lvert z\rvert <1\}$. Let $C(r)$ be the closed curve which is the image of the circle $\lvert z\rvert =r<1$ under the mapping $w=f(z)$, $L(r)$ the length of $C(r)$, and let $A(r)$ be the area enclosed by the curve $C(r)$. It was shown in [13] that if $f\in\mathcal{A}$, $f$ is starlike with respect to the origin, and for $0\leq r<1$, $A(r)<A$, an absolute constant, then \begin{equation} \label{eq0.1} L(r)=\mathcal{O}\left(\log\frac{1}{1-r}\right) \quad\text{as}\quad r\to 1. \end{equation} It is the purpose of this work to prove, using a modified methods than that in [13], a strengthened form of (0.1) for Bazilevi\u{c} functions, strongly starlike functions and for close-to-convex functions.