Harmonic mappings in the exterior of the unit disk

Magdalena Gregorczyk; Jarosław Widomski

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2010)

  • Volume: 54, Issue: 1
  • ISSN: 0365-1029

Abstract

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In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition ∑ n = 1 ∞ n p ( | a n | + | b n | ) ≤ 1 . We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.

How to cite

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Magdalena Gregorczyk, and Jarosław Widomski. "Harmonic mappings in the exterior of the unit disk." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 54.1 (2010): null. <http://eudml.org/doc/289822>.

@article{MagdalenaGregorczyk2010,
abstract = {In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition$\sum _\{n=1\}^\{\infty \}n^\{p\}(|a_\{n\}|+|b_\{n\}|)\le 1$. We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.},
author = {Magdalena Gregorczyk, Jarosław Widomski},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Harmonic mapping; meromorphic; quasiconformal extension; radius of convexity; radius of univalence},
language = {eng},
number = {1},
pages = {null},
title = {Harmonic mappings in the exterior of the unit disk},
url = {http://eudml.org/doc/289822},
volume = {54},
year = {2010},
}

TY - JOUR
AU - Magdalena Gregorczyk
AU - Jarosław Widomski
TI - Harmonic mappings in the exterior of the unit disk
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2010
VL - 54
IS - 1
SP - null
AB - In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition$\sum _{n=1}^{\infty }n^{p}(|a_{n}|+|b_{n}|)\le 1$. We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.
LA - eng
KW - Harmonic mapping; meromorphic; quasiconformal extension; radius of convexity; radius of univalence
UR - http://eudml.org/doc/289822
ER -

References

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  1. Hengartner W., Schober G., Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1-31. 
  2. Jahangiri, Jay M., Harmonic meromorphic starlike functions, Bull. Korean Math. Soc. 37 (2000), No. 2, 291-301. 
  3. Jahangiri, Jay M., Silverman H., Meromorphic univalent harmonic functions with 
  4. negative coefficients, Bull. Korean Math. Soc. 36 (1999), No. 4, 763-770. 
  5. Lehto O., Virtanen K. I., Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin-Heidelberg-New York, Second Edition, 1973. 
  6. Pommerenke Ch., Univalent Functions, Vandenhoeck & Ruprecht in Gottingen, 1975. 
  7. Sheil-Small T., Complex Polynomials, Cambridge University Press, 2002. 

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