Equilibrium points of logarithmic potentials on convex domains

Langley, J.

doi:10.1090/S0002-9939-07-08791-6

Equilibrium points of logarithmic potentials on convex domains
HTML articles powered by AMS MathViewer

by J. K. Langley

Proc. Amer. Math. Soc. 135 (2007), 2821-2826

DOI: https://doi.org/10.1090/S0002-9939-07-08791-6

Published electronically: February 7, 2007

PDF | Request permission

Abstract:

Let $D$ be a convex domain in $\mathbb {C}$. Let $a_k > 0$ be summable constants and let $z_k \in D$. If the $z_k$ converge sufficiently rapidly to $\zeta \in \partial D$ from within an appropriate Stolz angle, then the function $\sum _{k=1}^\infty a_k /( z - z_k )$ has infinitely many zeros in $D$. An example shows that the hypotheses on the $z_k$ are not redundant and that two recently advanced conjectures are false.

References

Albert Baernstein II, Proof of Edrei’s spread conjecture, Proc. London Math. Soc. (3) 26 (1973), 418–434. MR 374429, DOI 10.1112/plms/s3-26.3.418
Ludwig Bieberbach, Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt, Die Grundlehren der mathematischen Wissenschaften, Band 66, Springer-Verlag, Berlin-New York, 1965 (German). Zweite umgearbeitete und erweiterte Auflage. MR 0176133
J. Borcea, Equilibrium points of logarithmic potentials induced by positive charge distributions I: generalised de Bruijn-Springer relations, Trans. Amer. Math. Soc., to appear.
J. Clunie, A. Erëmenko, and J. Rossi, On equilibrium points of logarithmic and Newtonian potentials, J. London Math. Soc. (2) 47 (1993), no. 2, 309–320. MR 1207951, DOI 10.1112/jlms/s2-47.2.309
Alexandre Eremenko, Jim Langley, and John Rossi, On the zeros of meromorphic functions of the form $f(z)=\sum ^\infty _{k=1}a_k/(z-z_k)$, J. Anal. Math. 62 (1994), 271–286. MR 1269209, DOI 10.1007/BF02835958
A. A. Gol′dberg and I. V. Ostrovskiĭ, Raspredelenie znacheniĭ meromorfnykh funktsiĭ, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0280720
W.K. Hayman, Meromorphic functions, Oxford at the Clarendon Press, 1964.
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
James K. Langley and John Rossi, Meromorphic functions of the form $f(z)=\sum ^\infty _{n=1}a_n/(z-z_n)$, Rev. Mat. Iberoamericana 20 (2004), no. 1, 285–314. MR 2076782, DOI 10.4171/RMI/390
J. K. Langley and John Rossi, Critical points of certain discrete potentials, Complex Var. Theory Appl. 49 (2004), no. 7-9, 621–637. MR 2088052, DOI 10.1080/02781070410001732142
Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330, DOI 10.1007/978-3-662-06842-7
G. Valiron, Lectures on the general theory of integral functions, Chelsea, New York, 1949.