Let and be analytic functions on the bounded domain that extend continuously to and satisfy on . Show that and have the same number of zeros in D, counting multiplicity.
The hint here says that neither nor has zeros on , so each has at most finitely many zeros in . Estimate shows for z near , so is continuously defined near . Increase in around any closed path near is zero.
We have covered Rouche's Theorem too. I do not see how to prove this right now. I need some pointers on where to go. Thanks.