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I can prove parts $a$ and $b$ of this question using the Cauchy Riemann equations. However, I can't see how to prove part $c$. Does anyone know how to do it?

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up vote 3 down vote accepted

Just follow the hint: if $|f(z)|=c$ then $|f(z)|^2=c^2$. Now, proceed by cases if $c=0$ then $f$ is the constant zero. So let us assume that $c\ne 0$. It follows that $\overline {f(z)}=\frac c {f(z)}$ hence $ \overline {f(z)}$ is analytic as is a quotient of analytic functions and the denominator is never zero. By (b) we conclude that $f$ is constant.

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