Towards a general distortion theory for univalent functions: Teichmuller spaces and coefficient problems of complex analysis

2025/07/26 by Krushkal, Samuel L.
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.19767

Abstract

Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental intrinsic features of holomorphy and of conformality. This paper surveys the results obtained by a new approach involving deep features of Teichmuller spaces. This approach was recently suggested by the author. The paper also contains some new results generalizing the classical coefficient conjectures and presents open problems.

Citations

• A link between covering and coefficient theorems for holomorphic functions
• The Grunsky operator and quasiconformality: old and new
• Teichmuller balls and biunivalent holomorphic functions
• Complex geodesics and variational calculus for univalent functions
• Strengthened Grunsky and Milin inequalities
• Distortion theorems for higher order Schwarzian derivatives of univalent functions
• Higher Schwarzian operators and combinatorics of the Schwarzian derivative
• A proof of the Bieberbach conjecture

Related

• Milin's coefficients, complex geometry of Teichmüller spaces and variational calculus for univalent functions
• A general coefficient theorem for univalent functions
• A short geometric proof of the Zalcman and Bieberbach conjectures
• A link between covering and coefficient theorems for holomorphic functions
• Teichmuller balls and biunivalent holomorphic functions