$O \subset \mathbb{C}$ region with ball $\overline{D(a, r)} \subset O$, $f$ analytic on $O$ s.t. $|f|$ constant on $\partial D(a, r)$.

Asked Aug 13 '19 at 16:46

Active Aug 13 '19 at 16:46

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Let $O \subset \mathbb{C}$ be a region containing the closure of a ball $B := D(a, r)$. Assume $f$ is analytic on $O$ with $|f|$ constant on $\partial B$. Show that $f$ has a zero in $B$.

I feel like the Argument Principle is useful here, but I'm not sure how to use it. I can assume $f$ has no zero in $B$ and conclude that $f'/f$ is analytic on $B$. We have $\int_{\partial B} f'/f = 0$ by Argument Princ. and really by the fact that $f'/f$ is analytic.

The other option is max modulus principle, but I'm not quite sure.

asked Aug 13 '19 at 16:46

Freddie

1,759

What if $|f|=0$ on the boundary? What if $|f|>0$? Try both cases. – Sean Nemetz Aug 13 '19 at 16:49
Where is the maximum of $|\frac{1}{f}|$ achieved if $f$ has no zero in B? – Conrad Aug 13 '19 at 16:54
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Or this one: Analytic $f$ with $|f|$ constant on $\partial D$, I need to show that it has at least one $0$ inside $D$ – Martin R Aug 13 '19 at 16:56
@MartinR Thanks Martin I was looking for a duplicate but couldn't find one. I'll close the question. – Freddie Aug 13 '19 at 17:10

$O \subset \mathbb{C}$ region with ball $\overline{D(a, r)} \subset O$, $f$ analytic on $O$ s.t. $|f|$ constant on $\partial D(a, r)$.

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