I posted this question before, and after some clarification I'll try again (since the first post was badly formulated).
If the real part of a holomorphic function is bounded, does that mean that the imaginary part also has to be bounded?
2026-03-27 08:41:09.1774600869
Real and imaginary part of holomorphic function
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No, consider $i\log z$ in the the right half plane.