# Hurwitz's Theorem Proof Question

Hurwitz's Theorem

Can anyone explain in the proof of Hurwitz's Theorem on the wikipedia page, the line where it says $\frac{f_k'(z)}{f_k(z)}$ converges uniformly by Morera's Theorem? I do not see how that follows from Morera's Theorem.

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I agree ... no idea how one should apply Morera's Theorem. But you can prove it like that:

$$\left|\frac{f_k'(z)}{f_k(z)}-\frac{f'(z)}{f(z)}\right| \leq \left| \frac{f_k'(z)}{f_k(z)}-\frac{f'(z)}{f_k(z)} \right| + \left| \frac{f'(z)}{f_k(z)} - \frac{f'(z)}{f(z)}\right| \\ \leq \frac{2}{\delta} |f_k'(z)-f'(z)| + \|f'\|_{\partial B(z_0,\varrho)} \cdot \frac{2}{\delta^2} \left| f(z)-f_k(z) \right|$$

for all $z \in \partial B(z_0,\varrho)$. Since $f \to f_k$ (and therefore $f_k' \to f_k$) compactly we obtain $\frac{f_k'}{f_k} \to \frac{f}{f'}$ uniformly on $\partial B(z_0,\varrho)$

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