Complex Analysis: Conformal Inequalities and the Bieberbach Conjecture

Abstract

Analytic Functions Definitions Jordan Contour Cauchy-Riemann Equations Line Integrals Cauchy's Theorems Harmonic Functions Piecewise Bounded Functions Metric Spaces Equicontinuity Univalent Functions Conformal Mapping Some Theorems Implications Analytic Continuation Schwarz Reflection Principle Bilinear Transformation Poisson's Formula Area Principle Area Theorems Bieberbach Conjecture Lebedev-Milin's Area Theorem Koebe's Theorem Grunsky Inequalities Polynomial Area Theorem Distortion Theorems Criterion for Univalency Schwarzian Derivative Lowner Theory Caratheodory's Kernel Theorem Lowner's Theorem Lowner-Kufarev Equation Applications Slit Mappings Higher-Order Coefficients Variation Method Fourth Coefficient Grunsky Matrix Higher-Order Coefficients Subclasses of Univalent Functions Basis Classes Functions with Positive Real Part Functions in Class S0 Typically Real Functions Starlike Functions Functions with Real Coefficients Functions in Class Sa Generalized Convexity Convex Functions Close-to-Convex Functions gamma-Spiral Functions Generalized Convexity Alpha-Convex Functions Coefficients Estimates Mean Modulus Hayman Index Conformal Inequalities Exponentiation of Inequalities Polynomials Orthogonal Polynomials Hypergeometric Functions Faber Polynomials De Branges Theorem Conjectures de Branges Theorem Alternate Proofs of de Branges Theorem de Branges and Weinstein Systems of Functions Epilogue: After de Branges Chordal Lowner Equations Lowner Curves Lowner Chains in Cn Multivariate Holomorphic Mappings Beurling Transforms 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.