Bilinear Maps Complex Analysis

[Sistine]

Homework Statement

Find a function $$g$$ analytic in $$|z|\leq 2$$, with $$g(2/3)=0$$ and $$|g(z)|= 1$$ on $$|z|=2$$

Homework Equations

Bilinear maps

$$B_{\alpha}(z)=\frac{z-\alpha}{1-\overline{\alpha}z}$$


$$|B_{\alpha}(z)|=1$$ on $$|z|=1$$

The Attempt at a Solution

I tried using the maximum modulus theorem but I did not manage to find such a function.

 

[Dick]

Why are you messing around with the maximum modulus theorem when your relevant equation is a bilinear map? Use the bilinear map. It maps the unit circle |z|=1 to the unit circle. Modify it so it maps the circle |z|=2 to |z|=1. Now you still have an 'a' in the function. Fix 'a' so that g(2/3)=0.