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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!

 Jan 23, 2021
 #1
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-1

The maximum value is 1/2.  (You can find this value by letting a and b appraoch 1.)

 Jan 23, 2021
 #2
avatar+112523 
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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.

 Jan 24, 2021
 #3
avatar+31703 
+2

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

 Jan 24, 2021
edited by Alan  Jan 24, 2021
 #4
avatar+112523 
0

Thanks Alan,

 

How did you get this step?

 

\(|\frac{e^{i\theta}-1}{1-e^{-i\theta}}|=|e^{i\theta}|\)

Melody  Jan 24, 2021
 #5
avatar+31703 
+2

As follows:

 

Alan  Jan 25, 2021
 #6
avatar+112523 
0

Thanks Alan :)

Melody  Jan 26, 2021

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