Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

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 $\partial B(z_0,\varrho)$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.