A minimal area problem for nonvanishing functions

R. W. Barnard, C. Richardson, A. Yu Solynin

Research output: Contribution to journalArticle

Abstract

The minimal area covered by the image of the unit disk is found for nonvanishing univalent functions normalized by the conditions f(0) = 1, fʹ(0) = α. Two different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral representation can be employed to obtain the solution upon a priori knowledge of the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an explicit solution.

Original languageEnglish
Pages (from-to)21-36
Number of pages16
JournalSt. Petersburg Mathematical Journal
Volume18
Issue number1
DOIs
StatePublished - 2007

Keywords

  • Minimal area problem
  • Nonvanishing analytic function
  • Symmetrization
  • Typically real function

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