Abstract
An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085, 2006) states that the complex harmonic function \(r(z) - \overline{z}\), where \(r\) is a rational function of degree \(n \ge 2\), has at most \(5 (n - 1)\) zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form \(r(z) - \overline{z}\) no more than \(5 (n - 1) - 1\) zeros can occur. Moreover, we show that \(r(z) - \overline{z}\) is regular, if it has the maximal number of zeros.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, J.H., Evans, N.W.: The Chang-Refsdal lens revisited. Mon. Not. R. Astron. Soc. 369(1), 317–334 (2006). doi:10.1111/j.1365-2966.2006.10303.x. http://mnras.oxfordjournals.org/content/369/1/317.abstract
Balk, M.B.: Polyanalytic Functions. Mathematical Research, vol. 63. Akademie-Verlag, Berlin (1991)
Beardon, A.F.: Iteration of rational functions, Graduate Texts in Mathematics. Complex analytic dynamical systems, vol. 132. Springer, New York (1991). doi:10.1007/978-1-4612-4422-6
Duren, P., Hengartner, W., Laugesen, R.S.: The argument principle for harmonic functions. Am. Math. Mon. 103(5), 411–415 (1996). doi:10.2307/2974933
Khavinson, D., Neumann, G.: On the number of zeros of certain rational harmonic functions. Proc. Am. Math. Soc. (electronic) 134(4), 1077–1085 (2006). doi:10.1090/S0002-9939-05-08058-5
Khavinson, D., Neumann, G.: From the fundamental theorem of algebra to astrophysics: a “harmonious” path. Not. Am. Math. Soc. 55(6), 666–675 (2008)
Liesen, J.: When is the adjoint of a matrix a low degree rational function in the matrix? SIAM J. Matrix Anal. Appl. 29(4), 1171–1180 (2007). doi:10.1137/060675538
Luce, R., Sète, O., Liesen, J.: Sharp parameter bounds for certain maximal point lenses. Gen. Relat. Gravit. 46(5):1736 (2014). doi:10.1007/s10714-014-1736-9
Rhie, S.H.: n-point Gravitational Lenses with 5(n-1) Images. ArXiv Astrophysics e-prints (2003)
Sète, O., Luce, R., Liesen, J.: Perturbing rational harmonic functions by poles. Comput. Methods Funct. Theory 1–27 (2014). doi:10.1007/s40315-014-0083-x
Sheil-Small, T.: Complex polynomials, Cambridge Studies in Advanced Mathematics, vol. 75. Cambridge University Press, Cambridge (2002). doi:10.1017/CBO9780511543074
Wegert, E.: Visual complex functions. An introduction with phase portraits. Birkhäuser/Springer Basel AG, Basel (2012). doi:10.1007/978-3-0348-0180-5
Wegert, E., Semmler, G.: Phase plots of complex functions: a journey in illustration. Not. Am. Math. Soc. 58(6), 768–780 (2011)
Acknowledgments
We would like to thank Dmitry Khavinson for pointing out reference [1] to us. We are grateful to the anonymous referee for the careful reading of the manuscript, and many valuable remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dmitry Khavinson.
The work of R. Luce was supported by Deutsche Forschungsgemeinschaft, Cluster of Excellence “UniCat”.
Rights and permissions
About this article
Cite this article
Luce, R., Sète, O. & Liesen, J. A Note on the Maximum Number of Zeros of \(r(z) - \overline{z}\) . Comput. Methods Funct. Theory 15, 439–448 (2015). https://doi.org/10.1007/s40315-015-0110-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-015-0110-6