Extending a result from [ ] , 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 5 n - 5 complex zeros .
Applications to gravitational lensing are discussed .
In particular , this result settles a conjecture [ ] 
concerning the maximum number of lensed images due to an n -point gravitational lens .