Convex icebergs and sectorial starlike functions

Roger W. Barnard, Matthew H. Lochman, Alexander Yu Solynin

Research output: Contribution to journalArticle

Abstract

We consider the complex plane ℂ as a space filled by two different media, separated by the real axis ℝ. For a planar body (connected compact set) E in C, the sets E+=E∩{z:>0} and E-=E\{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{:z∈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 class of sectorial starlike functions.

Original languageEnglish
Pages (from-to)635-682
Number of pages48
JournalComputational Methods and Function Theory
Volume13
Issue number4
DOIs
StatePublished - 2013

Keywords

  • Convex function
  • Iceberg-type problem
  • Logarithmic capacity
  • Omitted area problem
  • Starlike function

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