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Caratheodory convergence and harmonic measure

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 Added by Ilia Binder
 Publication date 2018
  fields
and research's language is English




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We give several new characterizations of Caratheodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.



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397 - D. Beliaev , S. Smirnov 2008
In this paper we rigorously compute the average multifractal spectrum of harmonic measure on the boundary of SLE clusters.
The Schur (resp. Caratheodory) class consists of all the analytic functions $f$ on the unit disk with $|f|le 1$ (resp. $Re f>0$ and $f(0)=1$). The Schur parameters $gamma_0,gamma_1,dots (|gamma_j|le 1)$ are known to parametrize the coefficients of functions in the Schur class. By employing a recursive formula for it, we describe the $n$-th coefficient of a Caratheodory function in terms of $n$ independent variables $gamma_1,dots, gamma_n$ with $|gamma_j|le 1.$ The mapping properties of those correspondences are also studied.
144 - V. N. Dubinin , M. Vuorinen 2010
Let $E$ be a continuum in the closed unit disk $|z|le 1$ of the complex $z$-plane which divides the open disk $|z| < 1$ into $nge 2$ pairwise non-intersecting simply connected domains $D_k,$ such that each of the domains $D_k$ contains some point $a_k$ on a prescribed circle $|z| = rho, 0 <rho <1, k=1,...,n,. $ It is shown that for some increasing function $Psi,$ independent of $E$ and the choice of the points $a_k,$ the mean value of the harmonic measures $$ Psi^{-1}[ frac{1}{n} sum_{k=1}^{k} Psi(omega(a_k,E, D_k))] $$ is greater than or equal to the harmonic measure $omega(rho, E^*, D^*),,$ where $E^* = {z: z^n in [-1,0] }$ and $D^* ={z: |z|<1, |{rm arg} z| < pi/n} ,.$ This implies, for instance, a solution to a problem of R.W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $inf_{E} max_{k=1,...,n} omega(a_k,E, D_k),$ for arbitrary points of the circle $|z| = rho ,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $|z| = rho ,.$
123 - Gennadi Henkin 2005
This article gives a complex analysis lighting on the problem which consists in restoring a bordered connected riemaniann surface from its boundary and its Dirichlet-Neumann operator. The three aspects of this problem, unicity, reconstruction and characterization are approached.
The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of any Kohn-Nirenberg domains are positive and complete.
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