Let a and b be distinct complex numbers on the unit circle. Find the maximum possible value of \(\left|\dfrac{a-b}{1-\overline{a}b}\right|\).
Thank you!
The maximum value is 1/2. (You can find this value by letting a and b appraoch 1.)
+118692 idk,
but if a=1 and b=-1 than it would be 2/2 = 1
I suspect that is the biggest but I do not really know. I am just guessing.
+33653 Since they are both on the unit circle, a and b differ only in angle. Without loss of generality we can set the angle of b to be zero, so that b = 1.
Then \(|\frac{a-b}{1-\bar{a} b}|=|\frac{a-1}{1-\bar{a}}|\)
If we let \(a=e^{i\theta}\) then \(|\frac{a-1}{1-\bar{a}}|=|\frac{e^{i\theta}-1}{1-e^{-i\theta}}|=|e^{i\theta}|=1\)
So, as long as \(\theta \ne 0\) the ratio is 1
