Chapman and Hall/CRC
343 pages | 33 B/W Illus.
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.
Piecewise Bounded Functions
Schwarz Reflection Principle
Lebedev–Milin’s Area Theorem
Polynomial Area Theorem
Criterion for Univalency
Carathéodory’s Kernel Theorem
Subclasses of Univalent Functions
Functions with Positive Real Part
Functions in Class S0
Typically Real Functions
Functions with Real Coefficients
Functions in Class Sa
Exponentiation of Inequalities
De Branges Theorem
de Branges Theorem
Alternate Proofs of de Branges Theorem
de Branges and Weinstein Systems of Functions
Epilogue: After de Branges
Chordal Löwner Equations
Löwner Chains in Cn
Multivariate Holomorphic Mappings
Appendix A: Mappings
Appendix B: Parametrized Curves
Appendix C: Green’s Theorems
Appendix D: Two-Dimensional Potential Flows
Appendix E: Subordination Principle
Exercises appear at the end of each chapter.