2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69865Extending 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.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 boundComplex VariablesAstrophysics26C15 (Primary); 30D05, 83C99 (Secondary)text