Techniques to Derive Estimates for Integral Means and Other Geometric Quantities Related to Conformal Mappings

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Abstract

We will describe few methods to derive estimates for integral means and for other asymptotic expressions related to conformal mappings. One method will start from classical inequalities for conformal mappings such as the Goluzin inequalities and the exponential Goluzin inequalities. Then the simple idea of approximating integrals with the aid of their Riemann sums will serve us to obtain such estimates. A second method is to start from a certain elementary identity proved by Hardy in 1915 and use it combined with distortion theorems in S to obtain more integrals estimates. Finally, the main result in the author’s Masters thesis which in fact was already known to Bendixon will give us a method to estimate the geometric distance from a point in the image of a conformal mapping to the boundary of this image. The estimate will be in terms of a rather arbitrary sequence in the domain of the definition that converges to the pre-image of the point in the image from which the distance is measured.

Original languageEnglish
Title of host publicationNew Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative
Subtitle of host publicationAdvances and Applications
PublisherSpringer Science and Business Media Deutschland GmbH
Pages341-354
Number of pages14
Volume286
ISBN (Electronic)978-3-030-76473-9
ISBN (Print)978-3-030-76472-2
DOIs
StatePublished - 1 Jan 2021

Publication series

NameOperator Theory: Advances and Applications
Volume286
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Conformal mappings
  • Distance to the boundary of the image
  • Goluzin inequalities
  • Integral means
  • Schlicht functions

ASJC Scopus subject areas

  • Analysis

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