Application of maximum principle in a circle

Question

Let $C(0,1) \subset \mathbb C$ be the unit circle. The question is to compute $$ \max_{z \in C(0,1)} \vert z^2+1 \vert.$$ Apparently one can use the maximum principle to do so but I really don't see how. Can someone help me ?

Answer 1

Estimate $|z^2+1|\le|z|^2+1\le 1+1=2$ and note that $z=1$ gives equality.

Answer 2

Hint $$|z^2+1|^2 = (z^2+1)\overline{(z^2+1)}$$ Where $\overline w$ is complex conjugate of $w$ and use that $.^2$ is monotonic increasing. Now you can expand and investigate right hand side with your favourite technique.