Convex icebergs and sectorial starlike functions.

Roger Barnard, Matthew Lochman, Alexander Solynin

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the complex plane $\mathbb{C}$ as a space filled by two different media, separated by the real axis $\mathbb{R}$. For a planar body (connected compact set) $E$ in $\mathbb{C}$, the sets $E_+=E\cap\{z:\Im z>0\}$ and $E_-=E\backslash\{z:\Im z>0\}$ will be considered as a ``visible'' part of $E$ and an ``invisible'' part of $E$, respectively. In a recent paper of Barnard, Pearce and Solynin, the authors discussed the problem of estimating the maximum draft, $\min\{\Im z:z\in E_-\}$, from characteristics of the whole body $E$ and characteristics of its ``visible'' part $E_+$. Their study was focused on the general case of compact sets, i.e. they worked with sets $E$ which are not necessarily convex. In this paper, we investigate the maximal draft of $E$ as a function of the logarithmic capacity of $E$ and the area of $E_+$ under the more natural, yet also more difficult, assumption that $E$ is convex. As a tool in our proof, we consider extremal problems for the related cla
Original languageEnglish
Pages (from-to)635-682
JournalComputational Methods and Function Theory
StatePublished - Apr 2013

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