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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some harmonic $n$-slit mappings

Author: Michael J. Dorff
Journal: Proc. Amer. Math. Soc. 126 (1998), 569-576
MSC (1991): Primary 30C55, 30C45
MathSciNet review: 1422862
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Abstract: The class $S_{\scriptscriptstyle H}$ consists of univalent, harmonic, and sense-preserving functions $f$ in the unit disk, $\Delta $, such that $f=h+\overline{g}$ where $h(z)=z+\sum _2^\infty a_kz^k$, $g(z)=\sum _1^\infty b_kz^k$. $S_{\scriptscriptstyle H}^{\scriptscriptstyle O}$ will denote the subclass with $b_1=0$. We present a collection of $n$-slit mappings $(n \geq 2)$ and prove that the $2$-slit mappings are in $S_{\scriptscriptstyle H}$ while for $n \geq 3$ the mappings are in $S_{\scriptscriptstyle H}^{\scriptscriptstyle O}$. Finally we show that these mappings establish the sharpness of a previous theorem by Clunie and Sheil-Small while disproving a conjecture about the inner mapping radius.

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Michael J. Dorff
Affiliation: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, Missouri 65409-0020

Received by editor(s): April 19, 1996
Received by editor(s) in revised form: August 23, 1996
Additional Notes: This work represents part the author’s Ph.D. thesis at the University of Kentucky
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society