Ch. Pommerenke

The de Branges theorem on univalent functions

We present a simplified version of the de Branges proof of the Lebedev- Milin conjecture which implies the Robertson and Bieberbach conjectures. As an application of an analysis of the technique, it is shown that the method could not be used directly to prove the Bieberbach conjecture.

On the mean growth of the solutions of complex linear differential equations in the disk

Let w be a solution of the differential equation w(z)+q(z)w(z)=0 where q is analytic in the unit disk. We study first the mean square growth of w and give a condition for w to belong to the Hardy space H2 : the proof is based on Carleson measures Then it is shown that implies that w is of bounded characteristic: the proof uses ideas of Hille together with the Hardy Littlewood maximal theorem. This result is finally applied to the case that q is a Г-automorphic form of weight 2 for some Fuchsian group Г this case arises if we consider our differential equation in a multiply connected domain and then use uniformization.

On conformal mapping and iteration of rational functions

Let J(f) denote the Julia Fatou set of the rational function f and let G be a simply connected invariant component of its complement. Let h map the unit disk D canformally onto G. We consider the connection betwen the repulsive fixed points ζ∈G of f and the fixed points δ∈∂D of the finite Blaschkc product ϕ=hf h. We show in particular that h has an angular limit ζ=h(δ) at δ and give an inequality relating f″(ζ) and ϕ′(δ1 where h(δ1)=ζ.

Intrinsic rotations of simply connected regions, II

Let D be a simply connected bounded region in the plane. Let z0eD. Let F θ denote the conformal automorphism of D with . Let Θ be the set of θ e (0, 2π) such that F θ has a continuous extension to the closure of D. This paper investigates geometric and prime end properties of ðD which can be inferred from properties of Θ.

Hölder continuity of conformal maps with quasiconformal extension

If f is a bounded univalent function in the unit disk D with a K-quasiconformal extension to the plane. then it is known that f satisfies a Holder condition of the form |f(z2)−f(z1)|M|z2−z1|1−k(z1,z2∊D where k=(K−1)/(K+1). For two different normalizalions estimates of the Holder constant M are given such that M tends to 1 as K tends to 1.

On Prime Ends and Plane Continua

Let

Hyperbolically Convex Functions, Dimension and Capacity

f

The grunsky operator and coefficient differences

be a conformal map of the unit disk

On the stationary measures of critical branching processes

{\bb D}

CONFORMAL IMAGES OF BOREL SETS

onto the domain