1
$\begingroup$

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 $|z|\leq1$.

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

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

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

$\endgroup$
  • 1
    $\begingroup$ 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 $a/|a|$. $\endgroup$ – The Fourth Man Aug 16 '13 at 15:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.