Existence of analytic function having different values on $|z|=1$ and $|z|=2$

Question

Does there exist an entire function satisying $f(z)=z$ for $|z|=1$ and $f(z)=z^2$ for $|z|=2$?

I think no, but what argument would do here? Would it be maximum modulus or Liouville theorem? A counterexample should involve power series, I think? Any hints? Thanks beforehand.

Answer 1

If $f(z)=z$ for $|z|=1$ then $f(z)-z$ is an entire function whose zeros have a limit point. This implies $f(z)=z$ for all $z$. So no such function can exist.

Answer 2

We have

$$f'(0)= \frac{1}{2 \pi i}\int_{|z|=1} \frac{f(w)}{w^2} dw =\frac{1}{2 \pi i}\int_{|z|=1} \frac{w}{w^2} dw =\frac{1}{2 \pi i}\int_{|z|=1} \frac{1}{w} dw=1$$

and

$$f'(0)= \frac{1}{2 \pi i}\int_{|z|=2} \frac{f(w)}{w^2} dw =\frac{1}{2 \pi i}\int_{|z|=2} \frac{w^2}{w^2} dw =\frac{1}{2 \pi i}\int_{|z|=2} 1 dw=0.$$

This contradiction shows that no such function exists.