THE MODULUS OF THE IMAGE ANNULI UNDER UNIVALENT HARMONIC MAPPINGS AND A CONJECTURE OF NITSCHE

The object of the paper is to show that if f is a univalent, harmonic mapping of the annulus A(r, 1) = {z : r < ∣z∣ < 1} onto the annulus A(R, 1), and if s is the length of the segment of the Grötzsch ring domain associated with A(r, 1), then R < s. This gives the first, quantitative upper...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2001-10, Vol.64 (2), p.369-384
Main Author: LYZZAIK, ABDALLAH
Format: Article
Language:English
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Summary:The object of the paper is to show that if f is a univalent, harmonic mapping of the annulus A(r, 1) = {z : r < ∣z∣ < 1} onto the annulus A(R, 1), and if s is the length of the segment of the Grötzsch ring domain associated with A(r, 1), then R < s. This gives the first, quantitative upper bound of R, which relates to a question of J. C. C. Nitsche that he raised in 1962. The question of whether this bound is sharp remains open.
ISSN:0024-6107
1469-7750
DOI:10.1112/S0024610701002460