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On classes of uniformly starlike functions

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Wiśniowska-Wajnryb A.

Annales Polonici Mathematici

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2013

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tom 108

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nr 1

11-19

EN

We geometrically define subclasses of starlike functions related to the class of uniformly starlike functions introduced by A. W. Goodman in 1991. We give an analytic characterization of these classes, some radius properties, and examples of functions in these classes. Our classes generalize the class of uniformly starlike functions, and many results of Goodman are special cases of our results.

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Non-standard automorphisms of branched coverings of a disk and a sphere

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Wajnryb B., Wiśniowska-Wajnryb A.

Fundamenta Mathematicae

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2012

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tom 218

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nr 1

1-11

EN

Let Y be a closed 2-dimensional disk or a 2-sphere. We consider a simple, d-sheeted branched covering π: X → Y. We fix a base point A₀ in Y (A₀ ∈ ∂Y if Y is a disk). We consider the homeomorphisms h of Y which fix ∂Y pointwise and lift to homeomorphisms ϕ of X-the automorphisms of π. We prove that if Y is a sphere then every such ϕ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber $π^{-1}(A₀)$ pointwise. If Y is a disk, we describe explicitly a small set of automorphisms of π which induce all allowable permutations of $π^{-1}(A₀)$. This complements our result in Fund. Math. 217 (2012), no. 2, where we found a set of generators for the group of isotopy classes of automorphisms of π which fix the fiber $π^{-1}(A₀)$ pointwise.

3

Lifting of homeomorphisms to branched coverings of a disk

63%

Wajnryb B., Wiśniowska-Wajnryb A.

Fundamenta Mathematicae

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2012

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tom 217

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nr 2

95-122

EN

We consider a simple, possibly disconnected, d-sheeted branched covering π of a closed 2-dimensional disk D by a surface X. The isotopy classes of homeomorphisms of D which are pointwise fixed on the boundary of D and permute the branch values, form the braid group Bₙ, where n is the number of branch values. Some of these homeomorphisms can be lifted to homeomorphisms of X which fix pointwise the fiber over the base point. They form a subgroup $L^π$ of finite index in Bₙ. For each equivalence class of simple, d-sheeted coverings π of D with n branch values we find an explicit small set generating $L^π$. The generators are powers of half-twists.

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Annales Polonici Mathematici

3 Wiśniowska-Wajnryb A.

2 Wajnryb B.

1 2013

2 2012

On classes of uniformly starlike functions

Non-standard automorphisms of branched coverings of a disk and a sphere

Lifting of homeomorphisms to branched coverings of a disk