Analytic and geometric function theory

Geometric function theory examines how analytic functions—those expressible as convergent power series in the complex plane—distort and reshape geometric objects, with particular attention to functions that are univalent, meaning they map distinct points to distinct points without folding or overlapping. Conformal mappings, which preserve angles locally, connect pure mathematical structure to practical problems in fluid dynamics, electrostatics, and cartography, while the broader study of harmonic and quasiconformal mappings extends these ideas to settings where strict conformality is relaxed. A central thread of active research involves bounding the coefficients of power series expansions for classes of univalent functions, a program energized by the 1985 proof of the Bieberbach conjecture and still generating sharp estimates for specific subclasses. Current work also investigates how subordination—a way of comparing the ranges of two analytic functions—organizes these classes structurally, and how special functions such as the Gaussian hypergeometric function interact with univalence and convexity criteria.

Works

39,436

Total citations

249,424

Keywords

Geometric Function TheoryComplex AnalysisAnalytic FunctionsUnivalent FunctionsConformal MappingCoefficient Estimates