ISSN:0278-1077    

DOI:10.1080/17476938608814131

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

The problem of how the Fredholm eigenvalue δ(D) of a simply connected domainDvaries under “small” perturbations ofDhas been much studied. The difficulty consists in determining the sense in which such a perturbation is small. Here we consider a metric which arises in the definition of universal Teichmüller space. We show that, with this metric, the Fredholm eigenvalue δ(D) is a locally Lipschitz function ofD, ifDis a quasiconformal disk.

ISSN:0278-1077    

DOI:10.1080/17476938608814132

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

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 diskDcanformally onto G. We consider the connection betwen the repulsive fixed points ζ∈G of f and the fixed points δ∈∂Dof 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′(δ1where h(δ1)=ζ.

ISSN:0278-1077    

DOI:10.1080/17476938608814133

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Let Ω denote a simply connected region on the Riemann spherePsuch that Ω is convex relative to spherical geometry and Ω ≠P. The quantity (1 +|z|2λΩ(z) is called the spherical density of the hyperbolic metric λΩ(z)|dz|on Ω. Two sharp lower bounds for the spherical density are obtained. First (1 +|z|2)λΩ(z)⩾ cosecwith equality if and only if Ω is a hemisphere, wheredenotes the spherical distance from z to. Second, iffor all ζ∈Ω then λΩ(z)⩾π/4Ø These bounds lead to covering theorems for the classKsα of all univalent functionsfin the unit diskDsuch thatandf(D) is spherically convex. The Koebe setf∈Ks(α)) is the spherical disk about the origin with spherical radius (1 2) aresin α while the spherical Bloch constant for the familyKs(α)is απ4. For both covering results all extremal functions are identified.

ISSN:0278-1077    

DOI:10.1080/17476938608814134

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Certain geometrical properties of Jordan domains arise in the study of boundary behavior of Riemann maps. The property investigated here is the following wedge condition: there is a fixed wedge such that for each boundary point of the domain the wedge can be moved rigidly to have its vertex at the boundary point and its interior in the domain. Geometrical consequences of this property include that the boundary is rectifiable and has bounded arc length-interior distance ratio. It follows from work of Pommerenke that the domain is a Smirnov domain and that the derivative of a conformal mapping of the unit disk onto the domain satisfies the MuckenhouptAcondition.

ISSN:0278-1077    

DOI:10.1080/17476938608814135

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

A new proof by the method of module (method of extremal length) is given for the following result of Ostrowski: If a univalent function one<Rez,|Imz|<π/2 is semi-conformal atz=+x, thenasu→+x. Our method reveals that the conclusion holds for some quasiconformal maps.

ISSN:0278-1077    

DOI:10.1080/17476938608814136

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Letfbe a conformal mapping of a bounded Jordan domainDin the complex plane onto the unit disk. This paper examines the consequences for the local geometry ofDnear a boundary pointz0of the mappingf-or, to be more precise, of the homeomorphic extension of this mapping to the closure ofD—satisfying a Hölder condition atz0or, alternatively, of its inverse satisfying a Hölder condition at the pointf(z0). In particular, the compatibility of Hölder conditions with the presence of cusps in the boundary ofDis investigated.

ISSN:0278-1077    

DOI:10.1080/17476938608814137

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

LetRbe a Riemann surface with Green's functiong(z,a). LetaεRandp>0 be fixed. For a functionfanalytic onR, letaεRbe the least harmonic majorant of |f(z)|ponRandFor anfwithf(a) = 0. we define. whereg*(z,a) denotes the harmonic conjugate ofg(z,a). In this paper we show the equalityin the case for regular Riemann surfaces and the inequalityfor general Riemann surfaces. We also consider a higher dimensional case.

ISSN:0278-1077    

DOI:10.1080/17476938608814138

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Suppose that the boundary of a simply connected domain depends holomorphically on a complex parameter. Then the domain will depend on that parameter: one can ask how its normalized Riemann mapping function will depend on the parameter. Previously known results have required smoothness conditions on the boundary and have restricted the parameter to real values. In this paper we show that the Improved Lambda Lemma can be used to remove these requirements and restrictions.

ISSN:0278-1077    

DOI:10.1080/17476938608814139

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Nonconstant soluliuns of the partial differential equationwhereais analytic in the open unit disk and |a| <1, are orientation-preserving harmonic mappings. We consider the case whereais a finite Blaschke product, so that ‖a‖= 1, and describe the behavior at the boundary. Some applications include conditions under which the image is a polygon and a situation where a univalent solution does not exist that maps onto a prescribed domain with a normalization at three boundary points.

ISSN:0278-1077    

DOI:10.1080/17476938608814140

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor