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Suppose that $f(z)=u+iv$ is entire, and the harmonic function $u(x,y)$ has an upper bound. Then how to show that $u(x,y)$ must be constant throughout the plane?

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Hint: $g(z):=e^{f(z)}=e^{u(z)}e^{iv(z)}$ is entire and bounded. So by Liouville...?

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