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