How to define a function using some given properties?

Hi.

I have some complex numbers

$z = r \cdot {e^{i\varphi }}$

$\varphi \in \left( {0;\frac{\pi }{4}} \right)$

$r \in \left( {0; + \infty } \right)$

and I need to define a function $w(z)$
which would map everything inside a circle

$\left| w \right| < 1$

it is also given that

$w(0) = 1$

$w({e^{i\frac{\pi }{8}}}) = 0$

How should I approach this? Many thanks.