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.