On the number of zeros of certain rational harmonic functions

Khavinson, Dmitry; Neumann, Genevra

doi:10.1090/S0002-9939-05-08058-5

On the number of zeros of certain rational harmonic functions
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by Dmitry Khavinson and Genevra Neumann PDF

Proc. Amer. Math. Soc. 134 (2006), 1077-1085 Request permission

Abstract:

Extending a result of Khavinson and Świa̧tek (2003) we show that the rational harmonic function $\overline {r(z)} - z$, where $r(z)$ is a rational function of degree $n > 1$, has no more than $5n - 5$ complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an $n$-point gravitational lens.

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