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On bounded univalent functions that omit two given values

Dimitrios Betsakos (1999)

Colloquium Mathematicae

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 D 0 bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0

On supports of dynamical laminations and biaccessible points in polynomial Julia sets

Stanislav K. Smirnov (2001)

Colloquium Mathematicae

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.

Pertti Mattila (1996)

Publicacions Matemàtiques

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 &lt; ∞, 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.

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