Hey guys.
I have this problem, I need to show that it's true and I don't have a clue.
I tried to do like alpha = x+yi but it got me nowhere, any ideas?
Thanks.
For $\displaystyle |\alpha| < 1, \alpha \in \mathbb{C}$ define $\displaystyle f(z) = \frac{z-\alpha}{1 - \overline{\alpha} z}$.
It is defined for $\displaystyle |z|\leq 1$, this is because $\displaystyle |1 - \overline{\alpha}z| \geq \bigg||1| - |\overline{\alpha}||z|\bigg| \geq 1 - |\alpha| > 0$.
Therefore, on the closed disk $\displaystyle |z|\leq 1$ the function $\displaystyle f$ is a continous function that is analytic on the open disk $\displaystyle |z|<1$. By the maximum modulos theorem theorem the largest value that $\displaystyle |f|$ attains happens on the boundary of the disk i.e. on $\displaystyle |z|=1$.
We know that $\displaystyle |f|^2 = f ~ \overline{f}$, therefore, if $\displaystyle |z|= 1$ we have:
$\displaystyle |f(z)|^2 = \frac{z-\alpha}{1-\overline{\alpha}z} \cdot \overline{ \frac{z-\alpha}{1-\overline{\alpha} z} } = \frac{z-\alpha}{1-\overline{\alpha} z} \cdot \frac{\overline{z}-\overline{\alpha}}{1-\alpha \overline{z}} = 1$.
Therefore, $\displaystyle |f| \leq 1$ for all $\displaystyle |z|\leq 1$ and furthermore $\displaystyle |f| = 1$ if and only if $\displaystyle |z|=1$.