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**poincare4223** Let $\displaystyle f(z)$ and $\displaystyle g(z)$ be analytic functions on the bounded domain $\displaystyle D$ that extend continuously to $\displaystyle \partial D$ and satisfy $\displaystyle |f(z)+g(z)|<|f(z)|+|g(z)|$ on $\displaystyle \partial D$. Show that $\displaystyle f(z)$ and $\displaystyle g(z)$ have the same number of zeros in D, counting multiplicity.

The hint here says that neither $\displaystyle f(z)$ nor $\displaystyle g(z)$ has zeros on $\displaystyle \partial D$, so each has at most finitely many zeros in $\displaystyle D$. Estimate shows $\displaystyle \frac{f(z)}{g(z)} \not \in (-\infty, 0]$ for z near $\displaystyle \partial D$, so $\displaystyle \text{Log}(f(z)/g(z))$ is continuously defined near $\displaystyle \partial D$. Increase in $\displaystyle \text{arg} f(z)/g(z)$ around any closed path near $\displaystyle \partial D$ 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.