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I saw this question on my book (Complex Analysis/Donald & Newman):

Let $f(z)$ be an analytic function in the punctured plane $\{ z \mid z \neq 0 \}$ and assume that $|f(z)| \leq \sqrt{|z|} + \frac{1}{\sqrt{|z|}}$. Show that $f$ is constant.


How I can do it by Cauchy's formula?!

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    It might be a standard estimate of $f'(z)$ is zero using Cauchy's formula.2012-12-17

3 Answers 3