On power deformations of univalent functions

Yong Chan Kim, Toshiyuki Sugawa

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

For an analytic function f (z) on the unit disk {pipe}z{pipe} < 1 with f (0) = f′(0) - 1 = 0 and f (z) ≠ 0, 0 < {pipe}z{pipe} < 1, we consider the power deformation f c(z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator → f c maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), {pipe}z{pipe} < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.

Original languageEnglish
Pages (from-to)231-240
Number of pages10
JournalMonatshefte fur Mathematik
Volume167
Issue number2
DOIs
Publication statusPublished - 2012 Aug 1

Keywords

  • Spirallike function
  • Univalent function
  • Variability region

ASJC Scopus subject areas

  • Mathematics(all)

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