[Math] Maximum of an absolute value complex function

Question

I'm working my way through Marsden's Basic Complex Analysis book and I can't solve this problem. It's problem 23 of section 1.2 if that helps.

Let $a$ be a complex number, find the maximum of $|z^n+a|$ for those $z$ with $|z|\leq1$.

Answer 1

Hint: By the triangle inequality, $|z^n + a | \leq |z^n| + |a| = 1+ |a|$.

Can equality hold? If so, when does it hold?