The boundary distortion and extremal problems in certain classes of univalent functions

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Abstract

We consider the class S1(r), 0 < r < 1, of functions f(z) = rz + a2z2 + ... that are regular and univalent in the unit disk U and have \f(z)\ < 1. We obtain sharp estimates for the 1-measure of the sets {θ : \f(eiθ)\ = 1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {0 : |f(eiθ )| > M}, M > 1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles.

Original languageEnglish
Pages (from-to)1341-1358
Number of pages18
JournalJournal of Mathematical Sciences
Volume79
Issue number5
DOIs
StatePublished - 1996

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