DUALITY IN SPACES OF LORCH ANALYTIC MAPPINGS

AbstractWe define Hardy spaces $${\mathcal {H}}^p$$ H p for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $${\mathcal {H}}^p$$ H p -theory for Quasiconformal Mappings (Pure Appl Math Q 7(1):19–50, 2011).

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Fixed points of quasi-nonexpansive mappings

Journal of the Australian Mathematical Society ◽

10.1017/s144678870001123x ◽

1972 ◽

Vol 13 (2) ◽

pp. 167-170 ◽

Cited By ~ 39

Author(s):

W. G. Dotson

Keyword(s):

Fixed Points ◽

Banach Spaces ◽

Linear Space ◽

Complex Plane ◽

Normed Linear Space ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

The Other ◽

Commuting Mappings ◽

Analytic Mappings

A self-mapping T of a subset C of a normed linear space is said to be non-expansive provided ║Tx — Ty║ ≦ ║x – y║ holds for all x, y ∈ C. There has been a number of recent results on common fixed points of commutative families of nonexpansive mappings in Banach spaces, for example see DeMarr [6], Browder [3], and Belluce and Kirk [1], [2]. There have also been several recent results concerning common fixed points of two commuting mappings, one of which satisfies some condition like nonexpansiveness while the other is only continuous, for example see DeMarr [5], Jungck [8], Singh [11], [12], and Cano [4]. These results, with the exception of Cano's, have been confined to mappings from the reals to the reals. Some recent results on common fixed points of commuting analytic mappings in the complex plane have also been obtained, for example see Singh [13] and Shields [10].

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Analytic Mappings

Principal Functions ◽

10.1007/978-1-4684-8038-2_7 ◽

1968 ◽

pp. 211-231

Author(s):

Burton Rodin ◽

Leo Sario

Keyword(s):

Analytic Mappings

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Remarks on the existence of analytic mappings

Kodai Mathematical Seminar Reports ◽

10.2996/kmj/1138846115 ◽

1970 ◽

Vol 22 (2) ◽

pp. 172-177

Author(s):

Mitsuru Ozawa

Keyword(s):

Analytic Mappings

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Growth Theorems for a Subclass of Strongly Spirallike Functions

Journal of Applied Mathematics ◽

10.1155/2014/608641 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Yan-Yan Cui ◽

Chao-Jun Wang ◽

Si-Feng Zhu

Keyword(s):

Unit Disk ◽

Complex Plane ◽

Starlike Functions ◽

Geometric Properties ◽

Strongly Starlike Functions ◽

Strongly Starlike ◽

Spirallike Functions ◽

Analytic Mappings

In this paper we consider a subclass of strongly spirallike functions on the unit diskDin the complex planeC, namely, strongly almost spirallike functions of typeβand orderα. We obtain the growth results for strongly almost spirallike functions of typeβand orderαon the unit diskDinCby using subordination principles and the geometric properties of analytic mappings. Furthermore we get the growth theorems for strongly almost starlike functions of orderαand strongly starlike functions on the unit diskDofC. These growth results follow the deviation results of these functions.

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Local complexity growth for iterations of real analytic mappings and semicontinuity moduli of the entropy

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700006350 ◽

1991 ◽

Vol 11 (3) ◽

pp. 583-602 ◽

Cited By ~ 10

Author(s):

Y. Yomdin

Keyword(s):

Local Complexity ◽

Real Analytic ◽

Analytic Mappings

AbstractWe consider some ways in which regularity of a mapping influences dynamics of its iterations and growth of various complexity-type invariants.

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