On the number of zeros of certain rational harmonic functions

Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function 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.