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Suppose a holomorphic function $f$ from the open right halfplane into itself satisfies $$ |f(z)| \le M |z|.$$ My feeling is that this growth condition condition should (maybe via the Pick-Schwarz type Lemmata) have some heavy consequence for $f$.

Concrete questions are: does $f$ maybe send the right half plane in fact in some smaller sector $S_\theta = \{ z: \arg(z) \le \theta < \pi/2\}$ ? More generally, can one relate $\Re(f(z))$ with $|f(z)|$?

Thank you, Eric

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The identity map $f(z)=z$ does not map the halfplane into a smaller sector. Also, $f(z)=z+ic$ has the same real part but different modulus. –  user31373 Aug 8 '12 at 12:17

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