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 Loewner 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 -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.
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