Rogosinski's lemma for univalent functions, hyperbolic Archimedean\n spirals and the Loewner equation

2013/11/04 by Oliver Roth, Roth, Oliver, Sebastian Schleißinger +1
Mathematics · #30C55 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1311.0677

openalex publication_date 2013/11/04 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We describe the region \V(z₀) of values of f(z₀) for all\nnormalized bounded univalent functions f in the unit disk mathbbD at a\nfixed point z₀ \∈ mathbbD. The proof is based on identifying\n\V(z₀) as the reachable set of the radial Loewner differential\nequation. We also prove an analogous result for the upper half-plane using the\nchordal Loewner equation.\n

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