Mapping properties of analytic functions on the unit disk

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Let f be analytic on the unit disk D with f (0) = 0. In 1989, D. Marshall conjectured the existence of the universal constant ro > 0such that f (roD) C Dm := {w : |w| <M} whenever the area, counting multiplicity, of a portion of f (D)over DM is <πM2. Recently, P. Poggi-Corradini (2007) proved this conjecture with an unspecified constant by the method of extremal metrics. In this note we show that such a universal constant ro exists for a much larger class consisting of analytic functions omitting two values of a certain doubly-sheeted Riemann surface. We also find a numerical value, ro = .03949 ., which is sharp for the problem in this larger class but is not sharp for Marshall's problem.

Original languageEnglish
Pages (from-to)577-585
Number of pages9
JournalProceedings of the American Mathematical Society
Issue number2
StatePublished - Feb 2008


  • Analytic function
  • Growth theorem
  • Hyperbolic metric


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