2.I.4G

Let the function $f=u+i v$ be analytic in the complex plane $\mathbb{C}$ with $u, v$ real-valued.

Prove that, if $u v$ is bounded above everywhere on $\mathbb{C}$, then $f$ is constant.

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2.I.4G

Let the function $f=u+i v$ be analytic in the complex plane $\mathbb{C}$ with $u, v$ real-valued.

Prove that, if $u v$ is bounded above everywhere on $\mathbb{C}$, then $f$ is constant.