Question on radius of convergence

Question

Can anyone help me with the following problem:

I have a solid geometric picture of what is going on in my head, but I can't seem to turn that into a proof.

Answer 1

First, we see that if $R(z_0) = \infty$ for some $z_0$, then $f$ is entire and hence $R(z) = \infty$ everywhere.

So, assume that $R(z_1) < \infty$ (and hence $R(z_2) < \infty$ by the first remark).

Suppose $R(z_1) > R(z_2)+|z_1-z_2|$. In particular, choose $\rho$ such that $R(z_2) < \rho < R(z_1)-|z_1-z_2|$. This means that $B(z_2, R(z_2)) \subset B(z_2, \rho) \subset B(z_1, R(z_1))$, and the containment is strict.

To see this, if $x \in B(z_2, R(z_2))$, then $x \in B(z_2, \rho) $ and if $x \in B(z_2,\rho)$, we have $|z_1-x| \le |z_1-z_2|+|z_2-x| < |z_1-z_2|+\rho < R(z_1)$.

Hence $f$ is analytic on $B(z_2, \rho)$, which contradicts $R(z_2) < \rho$.

Switching the roles of $z_1,z_2$ finishes the proof.