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Asymptotic tracts of harmonic functions II

Published online by Cambridge University Press:  20 January 2009

K. F. Barth  and
D. A. Brannan
K. F. Barth
Affiliation:
Syracuse UniversityDepartment of MathematicsSyracuseNew York 13244, U.S.A.
D. A. Brannan
Affiliation:
The Open UniversityDepartment of Pure MathematicsWalton HallMilton Keynes MK7 6AA, United Kingdom
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Abstract

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An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,

and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.

The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 35 - 52
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

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