Ein verallgemeinerter transfiniter Durchmesser im Zusammenhang mit einer quasikonformen Normalabbildung
Equality cases for condenser capacity inequalities under symmetrization
It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.
Extremal Length and Univalent Functions, I. The Angular Derivative.
Extremal metrics and modulus
We give a new proof of Beurling’s result related to the equality of the extremal length and the Dirichlet integral of solution of a mixed Dirichlet-Neuman problem. Our approach is influenced by Gehring’s work in space. Also, some generalizations of Gehring’s result are presented.
Extremal plurisubharmonic functions in
Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map.
Isoperimetric inequalities for capacities in the plane.
Itération des polynômes et propriétés d'orthogonalité
J. E. McMillan's Area theorem
Length of curves under conformal mappings.
Lindelöf-type theorems for quasiconformal and quasiregular mappings
Logarithmic capacity is not subadditive – a fine topology approach
In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.
On a Problem Concerning Harmonic Measure.
On an extension of the definition of transfinite diameter and some applications.
On bounded univalent functions that omit two given values
Let a,b ∈ z: 0<|z|<1 and let S(a,b) be the class of all univalent functions f that map the unit disk into {a,bwith f(0)=0. We study the problem of maximizing |f’(0)| among all f ∈ S(a,b). Using the method of extremal metric we show that there exists a unique extremal function which maps onto a simply connnected domain bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0
On certain harmonic measures on the unit disk
On extremal problems related to inverse balayage.
On some applications of harmonic measure in the geometric theory of analytic functions
On supports of dynamical laminations and biaccessible points in polynomial Julia sets
We use Beurling estimates and Zdunik's theorem to prove that the support of a lamination of the circle corresponding to a connected polynomial Julia set has zero length, unless f is conjugate to a Chebyshev polynomial. Equivalently, except for the Chebyshev case, the biaccessible points in the connected polynomial Julia set have zero harmonic measure.
On the analytic capacity and curvature of some Cantor sets with non-σ-finite length.
We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying ∫01 r−3 h(r)2 dr < ∞, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results.