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If f is holomorphic in an open subset $G \subset \mathbb{C}$, and if $f'(a)\neq0$ for some $a \in G$, then there exists $r>0$ such that \begin{eqnarray}|f'(z)-f'(a)|<|f'(a)|,\end{eqnarray} for $z \in D(a,r)$ ($D$ for 'disk' with centre $a$, radius $r$).

The above is what I intend to prove. I've tried to use Cauchy's integral formulae, i.e \begin{eqnarray}2\pi i f'(z)=\int_{\partial D(a,r)}\frac{f(w)}{(w-z)^2} \, dw,\end{eqnarray} or \begin{eqnarray}2\pi i f'(z)=\int_{\partial D(a,r)}\frac{f'(w)}{w-z} \, dw,\end{eqnarray}

but I don't get anywhere? If someone would give a hint I'd appreciate it.

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    I don't see a question. Perhaps you intended to ask how to prove the statement in the first sentence?2012-09-21
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    Yes of course. My bad :)2012-09-21
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    Probably you can differentiate the power series for $f$ centered at $a$ and estimate the remainder when $z$ is very close to $a$.2012-09-21
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    I will try to do that. Thanks.2012-09-21

3 Answers 3