0
$\begingroup$

Assume we have sets $\Omega_1 \subsetneq \Omega \subset \mathbb{C}$, both open and connected. Further, let $f_n$ and $f$ be holomorphic functions on $\Omega$ such that $f_n \to f$ uniformly on each compact subset of $\Omega_1$. Can we conclude anything about convergence on $\Omega$?

I have no intuition about the answer. Would anyone provide an idea for a proof or counterexample?

$\endgroup$

1 Answer 1

0
$\begingroup$

No. If $f_n(z)=1+z+z^2+\cdots+z^n$, then $(f_n)_{n\in\mathbb N}$ converges uniformly to $\frac1{1-z}$ on every compact subset of the open disc $D(0,1)$. However, $D(0,1)$ is the largest open subset $\Omega$ of $\mathbb C$ such that $(f_n)_{n\in\mathbb N}$ converges uniformly on every compact subset of $\Omega$.

$\endgroup$
1
  • $\begingroup$ Of course, thanks. $\endgroup$
    – agb
    Jul 27, 2017 at 17:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .