Open Access
July 2014 On some length problems for analytic functions
Mamoru Nunokawa, Janusz Sokół
Osaka J. Math. 51(3): 695-709 (July 2014).


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.


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Mamoru Nunokawa. Janusz Sokół . "On some length problems for analytic functions." Osaka J. Math. 51 (3) 695 - 709, July 2014.


Published: July 2014
First available in Project Euclid: 23 October 2014

zbMATH: 1304.05141
MathSciNet: MR3272613

Primary: 30C45
Secondary: 30C80

Rights: Copyright © 2014 Osaka University and Osaka City University, Departments of Mathematics

Vol.51 • No. 3 • July 2014
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