On the number of zeros of certain rational harmonic functions

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.