Ahlfors-Beurling conformal invariant and relative capacity of compact sets

Abstract

Abstract. For a given domain D in the extended complex plane 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 relcap 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 relcap E coincides with the well-known half-plane capacity hcap 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.

Half-plane capacity and conformal radius

“…Even though the problems and results of [5] and our work are very different, both exploit the close relation between half-plane capacity and conformal radius. The behavior of the relative capacity under various symmetrizations and under some geometric transformations are also proved in [6], as well as the relation between relative capacity and Schwarzian derivative.…”

“…After a first version of this paper, we became aware of the papers [5] and [6]. It showed [5] that the half-plane capacity of a set is non-increasing under various notions of symmetrizations.…”

“…Since both sides of (6) are harmonic functions on D \ B, by the maximum principle it suffices to show (6) for z in the boundary of this region. If z ∈ ∂(D \ B) and z / ∈ D n , then ω n (z) = 0 and (6) is trivial in this case.…”

Производная Шварца И Покрытие Дуг Пучка Окружностей Голоморфными Функциями

“…Условие (1.1) является необходимым условием для получения оценок производной Шварца на границе и его геометрический смысл показан, например, в работах [2] и [3]. Неравенства для шварциана при более тонких (чем принадлежность орициклу) геометрических ограничениях на область ( ) представлены в работах [3] и [5]. Эти неравенства тесно связаны с оценками так называемой half-plane емкости [6].…”