  Extending a result from the paper of D. Khavinson and G. Swiatek, we show
that the rational harmonic function $\bar{r(z)} - z$, 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.
