2011/12/19 by В. Н. Дубинин, Vladimir N. Dubinin, Dubinin, Vladimir N. +3
Mathematics · #30C85 #60J67 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Probability (math.PR) #math.CV #math.PR #msc:30C85 #msc:60J67
paper · pdf · doi:10.48550/arxiv.1112.4245
13 pages, 6 figures
openalex publication_date 2011/12/19 · arxiv created 2012/12/26 · arxiv updated 2012/12/27 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
For a given domain D in the extended complex plane \mathbb C with an accessible boundary point z0 ∈ ∂ D and for a subset E ⊂ D, relatively closed w.r.t. D, we define the relative capacity \rc E as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant r(D∖ E,z)/r(D, z) when z approaches the point z0. Here r(G,z) denotes the inner radius at z of the connected component of the set G containing the point z. The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain D is the upper half plane and z0=∞ the capacity \rc E coincides with the well-known half-plane capacity \hc E. Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization and under some other geometric transformations. Some applications to bounded holomorphic functions of the unit disk are given.