Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Valence of complex-valued planar harmonic functions

Author: Genevra Neumann
Journal: Trans. Amer. Math. Soc. 357 (2005), 3133-3167
MSC (2000): Primary 30C99, 26B99; Secondary 31A05, 26C99
Published electronically: December 2, 2004
MathSciNet review: 2135739
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The valence of a function $f$ at a point $w$is the number of distinct, finite solutions to $f(z) = w$. Let $f$ be a complex-valued harmonic function in an open set $R \subseteq \mathbb{C} $. Let $S$ denote the critical set of $f$and $C(f)$ the global cluster set of $f$. We show that $f(S) \cup C(f)$ partitions the complex plane into regions of constant valence. We give some conditions such that $f(S) \cup C(f)$has empty interior. We also show that a component $R_0 \subseteq R \backslash f^{-1} (f(S) \cup C(f))$ is an $n_0$-fold covering of some component $\Omega_0 \subseteq \mathbb{C}\backslash (f(S) \cup C(f))$. If $\Omega_0$ is simply connected, then $f$ is univalent on $R_0$. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for $C^1$ functions on open sets in $\mathbb{R} ^2$ are first stated in that form and then applied to the case of planar harmonic functions. If $f$ is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of $\mathbb{C}\backslash (f(S) \cup C(f))$sharing a common boundary arc in $f(S) \backslash C(f)$.

References [Enhancements On Off] (What's this?)

  • [AL88] Y. Abu-Muhanna and A. Lyzzaik, A geometric criterion for decomposition and multivalence, Math. Proc. Cambridge Phil. Soc. 103 (1988), 487-495. MR 89e:30010
  • [Bal91] Mark Benevich Balk, Polyanalytic Functions, Mathematical research, volume 63, Akademie Verlag GmbH (1991). MR 93k:30076
  • [BC55] Marcel Brelot and Gustave Choquet, Polynômes harmoniques et polyharmoniques, Second colloque sur les équations aux dérivées partielles, Bruxelles, 1954, pp. 45-66. Georges Thone, Liège; Masson $\&$ Cie (1955). MR 16:1108e
  • [BHN99] Daoud Bshouty, Walter Hengartner, and M. Naghibi-Beidokhti, p-valent harmonic mappings with finite Blaschke dilatations, XII-th Conference on Analytic Functions (Lublin, 1998), Ann. Univ. Mariae Curie-Sklodowska Sect. A, 53 (1999), 9-26. MR 2001j:30016
  • [BHS95] Daoud Bshouty, Walter Hengartner, and Tiferet Suez, The exact bound on the number of zeros of harmonic polynomials, J. Anal. Math. 67 (1995), 207-218. MR 97f:30025
  • [CL66] E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge University Press (1966). MR 38:325
  • [DK97] Peter Duren and Dmitry Khavinson, Boundary correspondence and dilatation of harmonic mappings, Complex Variables Theory Appl. 33 (1997), 105-111. MR 98m:30039
  • [KS03] Dmitry Khavinson and Grzegorz Swiatek, On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), 409-414.
  • [Lew36] Hans Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692.
  • [Lyz92] Abdallah Lyzzaik, Local properties of light harmonic mappings, Canad. J. Math. 44 (1992), 135-153. MR 93e:30048
  • [Mun75] James R. Munkres, Topology: A First Course, Prentice-Hall, Inc. (1975). MR 57:4063
  • [Neu03] Genevra Chasanov Neumann, Valence of harmonic functions, Ph.D. dissertation, University of California, Berkeley. 2003.
  • [OS86] M. Ortel and W. Smith, A covering theorem for continuous locally univalent maps of the plane, Bull. London Math. Soc. 18 (1986), 359-363. MR 88b:30013
  • [Smi71] Kennan T. Smith, Primer of Modern Analysis, Bogden & Quigley, Inc. (1971). MR 84m:26002
  • [Sto56] S. Stöilow, Leçons sur les principes topologiques de la theórie des fonctions analytiques, deuxième edition, Gauthier-Villars (1956). MR 18:568b
  • [ST00] T. J. Suffridge and J. W. Thompson, Local behavior of harmonic mappings, Complex Variables Theory Appl. 41 (2000), 63-80. MR 2001a:30019
  • [Wil94] Alan Stephen Wilmshurst, Complex harmonic mappings and the valence of harmonic polynomials, D.Phil. thesis, University of York, England. 1994.
  • [Wil98] A. S. Wilmshurst, The valence of harmonic polynomials, Proc. Amer. Math. Soc. 126 (1998), 2077-2081. MR 98h:30029

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30C99, 26B99, 31A05, 26C99

Retrieve articles in all journals with MSC (2000): 30C99, 26B99, 31A05, 26C99

Additional Information

Genevra Neumann
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Keywords: Planar harmonic functions, $C^1$ functions in $\mathbb{R}^2$, regions of constant valence
Received by editor(s): September 17, 2003
Published electronically: December 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society