$u+v$ is bounded $\Rightarrow$ $(u+v)^2=u^2+v^2+2uv$ is bounded $\Rightarrow$ $|f|^2+2uv$ is bounded.
What can I do further to reach the conclusion? or this is not the way to reach there?
Thank you.
2026-03-27 03:59:58.1774583998
Let $f=u+iv$ is an entire function. If $u+v$ is bounded then is $f$ constant or not? $(u=Ref(z)$ and $v=Imf(z))$
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Hint: One approach is to use the fact that if the real part of an entire function is bounded, then the function is constant. Can you construct from $f$ an entire function whose real part is $u+v$?