Let $f$ holomorphic on $C$.
I'm looking for a counter exemple to : If $\sup_{|z|=r} |Re(f)| = O(r^{d})$ then $\sup_{|z|=r} |f| = O(r^{d})$
Actually, I'm wondering if I can find a entire function such as the growth of the real part is lower than the growth of the imaginary part.
Thank you for reading me :).
It is a beautiful fact about entire functions that there is no counterexample to your claim! This is the Borel—Carathéodory theorem. (In the formulation on that web page, take $R=2r$ to deduce the exact form of your claim.)