I have the following problem:
Let $\{b_n\}_{n\geq 1}$ a sequence of complex numbers such that $|b_n|<r$ for all $n\geq 1$ and $\lim_{n\to\infty}|b_n|=r$.
Exists an analytic function $f$ in $|z|<r$ such that $f(b_n)$ is a zero of order $n$?
I think that the answer is no and I know that the zeros of an analytic function are isolated points. But, if is it the right way I don't know how to continue.
What happens in the case that $f$ is meromorphic and $f(b_n)$ is a pole of order $n$?
Any hint or idea will be appreciated !