This question is part of my assignment on complex analysis course and I was unable to solve it.
Find maximum and minimum values of $|\frac{z-\alpha}{1-\bar{\alpha} z} | $ on $|z|\leq 1$ . ( where $\alpha$ is such that |$\alpha$ |<1 and $\alpha$ is fixed.)
By maximum and minimum modulus principle it is clear that Maximum and minimum will be at |z|=1.
But I have no idea on how to find such valuevalues due to the complicated nature of function f.
So, it's my humble request to help me.
On
The map given by $f(z)=\frac{z-\alpha}{1-\overline{\alpha}z}$ for $|\alpha|<1$ is an automorphism of unit disc i.e. f maps unit disc onto unit disc, therefore, $|f(z)|=1$ for all $|z|=1$. You can use the following link to read a detailed discussion: Automorphisms of the unit disk
Actually, the minimum is $0$, which is taken when $z=\alpha$.
On the other hand, if $|z|=1$, then$$\left|1-\overline\alpha z\right|=\left|\overline zz-\overline\alpha z\right|=\left|\overline z-\overline\alpha\right||z|=|z-\alpha|$$and therefore$$\left|\frac{z-\alpha}{1-\overline\alpha z}\right|=1$$and therefore the maximum is $1$.