# Maximum of an absolute value complex function

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 $|z|\leq1$.

• Hint: the maximum is achieved when $|z|=1$. – rfauffar Aug 16 '13 at 3:00
• Is $n\in\{0,1,2,\ldots\}$? – Lord Soth Aug 16 '13 at 3:01
• @LordSoth The problem itself doesn't say but from the book I think $n$ is any natural number. – The Fourth Man Aug 16 '13 at 3:12

Hint: By the triangle inequality, $|z^n + a | \leq |z^n| + |a| = 1+ |a|$.
• OK, let me see if I got it, for equality to hold it must mean $z^n=\lambda a$ for some real $\lambda$. Also, we need $|z|=1$ which means $\lambda=1/|a|$. This means $z$ is an $n-$th root of $a/|a|$. – The Fourth Man Aug 16 '13 at 15:28