Some harmonic $n$-slit mappings
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- by Michael J. Dorff PDF
- Proc. Amer. Math. Soc. 126 (1998), 569-576 Request permission
Abstract:
The class $S_H$ consists of univalent, harmonic, and sense-preserving functions $f$ in the unit disk, $\Delta$, such that $f=h+\overline {g}$ where $h(z)=z+\sum _2^\infty a_kz^k$, $g(z)=\sum _1^\infty b_kz^k$. $S_H^O$ 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 $S_H$ while for $n \geq 3$ the mappings are in $S_H^O$. 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.References
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Additional Information
- Michael J. Dorff
- Affiliation: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, Missouri 65409-0020
- MR Author ID: 613817
- ORCID: 0000-0001-7724-4423
- Email: mdorff@umr.edu
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 569-576
- MSC (1991): Primary 30C55, 30C45
- DOI: https://doi.org/10.1090/S0002-9939-98-04105-7
- MathSciNet review: 1422862