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Let $G\subset \mathbb C$ be a convex set and $f$ is an holomorphic function. I've proved that if $|f'(z)|<1$ for all $z\in G$ then $|f(z)-f(w)|<|z-w|$ for all $z,w\in G$.

Now I'm trying to find an example of an holomorphic function and an open connected set (which is not a convex set) that satifies $|f'(z)|<1$ for all $z\in G$ but not $|f(z)-f(w)|<|z-w|$

Thanks.

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You can obtain large values of $|f(z) - f(w)|$ for small $|z-w|$ if e.g. your domain is an annulus with a line removed, e.g. $\mathcal{D} = \{z | 1 < |z| < 2, \arg z \ne \pi\}$. Find a function that is holomorphic in this domain and that cannot be extended because it has a jump across that line. Tweak that function until it satisfies $|f'(z)| \le 1$ for all $z \in \mathcal{D}$. Or tweak the domain a bit.

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    Check the derivative and check the behavior of the function across the segment on the negative real axis where the annulus has been cut.2012-12-13