Complex Analysis: Conformal Inequalities and the Bieberbach Conjecture

1st Edition

Prem K. Kythe

Chapman and Hall/CRC
Published December 22, 2015
Reference - 343 Pages - 33 B/W Illustrations
ISBN 9781498718974 - CAT# K25479
Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics

USD$115.00

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Features

  • Explores the impact of the Bieberbach conjecture on creative thinking in mathematics during the last century
  • Covers novel techniques for solving problems in the complex plane
  • Proves the de Branges theorem and presents Weinstein’s alternative proof
  • Establishes that a Bieberbach conjecture does not exist in the case of several complex variables
  • Provides proofs of all non-elementary theorems
  • Includes 90 end-of-chapter exercises with complete solutions
  • Contains an extensive bibliography for further study of the topics
  • Offers more details about mappings, parametrized curves, Green’s theorems, two-dimensional potential flows, and subordination principle in the appendices

Summary

Complex Analysis: Conformal Inequalities and the Bieberbach Conjecture discusses the mathematical analysis created around the Bieberbach conjecture, which is responsible for the development of many beautiful aspects of complex analysis, especially in the geometric-function theory of univalent functions. Assuming basic knowledge of complex analysis and differential equations, the book is suitable for graduate students engaged in analytical research on the topics and researchers working on related areas of complex analysis in one or more complex variables.

The author first reviews the theory of analytic functions, univalent functions, and conformal mapping before covering various theorems related to the area principle and discussing Löwner theory. He then presents Schiffer’s variation method, the bounds for the fourth and higher-order coefficients, various subclasses of univalent functions, generalized convexity and the class of α-convex functions, and numerical estimates of the coefficient problem. The book goes on to summarize orthogonal polynomials, explore the de Branges theorem, and address current and emerging developments since the de Branges theorem.