I'm trying to answer the following question:
Let $D$ denote the open unit disc. Does there exist a continuous function $f:\overline{D}\rightarrow\mathbb{C}$ that is analytic on $D,$ and whose restriction to $\partial\overline{D}$ is the function $f|_{\partial\overline{D}}(z)=({\text{Re}}(z))^3$?
My instinct is no. Supposing there were, I would like to use the [maximum/minimum] modulus principle to state that $|f|$ does not attain its [maximum/minimum] value within $D.$ Because $|f|$ on the boundary has maximum value $1$ and minimum value $0,$ this implies that $0<|f(z)|<1$ for all $z\in D.$ Are we then able to find some $z\in D$ that has $f(z)=0,$ obtaining a contradiction?
I'm a bit stuck. Any hint/guidance is appreciated. Thanks!
Best Answer
$|\operatorname{Im} f|$ attains its maximum on $\partial D$. But $|\operatorname{Im} f|=0$ on $\partial D$, hence $f$ takes on only real values and is therefore constant, a contradiction.
(To prove that $|\operatorname{Im} f|$ attains its maximum on $\partial D$ just copy the proof of the maximum modulus theorem using the open mapping theorem)