Let f be analytic in an open neigbourhood containing the closed ball $\overline{B}(0,R)$ and $f(0)=0$. If $Re(f(z)) \leq M $ on $\partial{B}(0,R)$ then $Re(f(z)) \leq M$ in $\overline{B}(0,R)$. I'm not sure how to tackle this, any proofs?
2026-03-30 10:11:47.1774865507
Boundedness of real part of analytic function on the boundary implies boundedness in the disk
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