complex analysis - maximize AC*BC when AB is a unit circle diameter and C is inside the unit circle

Question

We're given a diameter AB in the unit circle and we need to find a point C in the closed ball of B(0,1) that maximize the multiplication |AC|*|BC|.

Write $A=e^{i\theta}$ so $B=-e^{i\theta}$ and let $C=z$ when $|z|<=1$

define $f:B(0,1)\rightarrow\mathbb{C}$
$f(z)=e^{2i\theta}-z^2$

$|f|=|AC||BC|$ and By the maximum principle we know that $|f|$ has its maximum on $|S(0,1)|$

So i got this far, but I have no idea how to continue.

Thanks in advance!

Why don't you take $A=1$ and $B=-1$? It's the same problem up to rotation. Then let $z = e^{i\theta}$. — user180040, Oct 04 '14 at 16:45
good idea. so I found out that the maximum is 2 with z=i or z=-i, how do I make sure they're the only points? thanks again :) — Chiptus, Oct 04 '14 at 16:50
$|z-1||z+1| = |z^2 - 1| = |e^{2i\theta} - 1|$. What's the furthest point from $1$ on the unit circle? — user180040, Oct 04 '14 at 16:51

score 1 · Answer 1 · answered Oct 04 '14 at 16:45

1

Hint: $|z-a|\cdot|z-(-a)| = |z^2-a^2| \le |z^2| + |a^2|$.

answered Oct 04 '14 at 16:45

1 Answers1