I come across these question when I am studying George Cain Complex analysis.
Suppose $f$ is analytic on a connected open set $D$, and suppose f^{'}(z)=0 for all $z\in D$. Prove that $f$ is constant.
Suppose $f$ is analytic on the set $D$, and suppose $Ref$ is constant on $D$. Is $f$ necessarily constant on $D$? Explain.
Suppose $f$ is analytic on the set $D$ and suppose $|f(z)|$ is constant on D. Is $f$ necessarily constant on $D$? Explain.
No hint is found in the text. Please I need a hint and reference to prove the above statements.