Ahlfors-Beurling conformal invariant and relative capacity of compact sets

Dubinin, Vladimir; Vuorinen, Matti

doi:10.1090/S0002-9939-2014-12125-3

Ahlfors-Beurling conformal invariant and relative capacity of compact sets
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by Vladimir N. Dubinin and Matti Vuorinen PDF

Proc. Amer. Math. Soc. 142 (2014), 3865-3879 Request permission

Abstract:

For a given domain $D$ in the extended complex plane $\overline {\mathbb C}$ with an accessible boundary point $z_0 \in \partial D$ and for a subset $E \subset {D},$ relatively closed w.r.t. $D,$ we define the relative capacity $\textrm {rel cap}{ }E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $r(D\setminus E,z)/r(D, z)$ when $z$ approaches the point $z_0 .$ 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 $z_0=\infty$, the capacity $\textrm {rel cap}{ }E$ coincides with the well-known half-plane capacity $\textrm {hcap}{ }E .$ 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.

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