On the number of zeros of certain rational harmonic functions

Abstract

Extending a result from the paper of D. Khavinson and G. Swiatek, 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 of S. H. Rhie concerning the maximum number of lensed images due to an n-point gravitational lens.

9 pages, 2 figures; revision discusses applications to gravitational lensing and notes that a result of S. H. Rhie settles the question of sharpness of the bound