Analytic and geometric function theory

Geometric function theory examines how analytic functions — those expressible locally as convergent power series in the complex plane — distort and transform geometric shapes, with particular attention to functions that are univalent, meaning they map distinct inputs to distinct outputs without folding or overlapping. Conformal mappings, which preserve local angles, sit at the center of the theory and connect pure complex analysis to problems in fluid dynamics, electrostatics, and the geometry of surfaces. A long-standing thread of investigation concerns coefficient estimates: bounding the size of the coefficients in a function's power series expansion, a program that culminated in the proof of the Bieberbach conjecture in 1985 but continues to generate refined questions about subclasses of mappings, including harmonic and quasiconformal variants where the classical tools require significant extension. Current active directions include understanding subordination chains — ways of comparing functions through inclusion of their image domains — and exploiting connections between hypergeometric functions and extremal problems in the theory of univalent mappings.

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Keywords

Geometric Function TheoryComplex AnalysisAnalytic FunctionsUnivalent FunctionsConformal MappingCoefficient Estimates