Equivalence of compact and uniform convergence for sequences of holomorphic functions

Question

Let $D\subset \mathbb{C}$ be a domain and let $(f_n)$ be a sequence of holomorphic functions on $D$. I would like to know if the following equivalence holds: $(f_n)$ converges compactly on $D$ if and only if $(f_n)$ converges uniformly on $D$. I am especially interested into the case $D=B_R(0)$ for some $R>0$, but would also like to know what topological properties of $D$ (maybe simply connectedness, convexity) are needed such that the above equivalence holds. References to literature are also appreciated.

Answer 1

On locally compact spaces, uniform convergence on compacts is equivalent to local uniform convergence. It is not equivalent to uniform convergence: $z^n$ for example converges to $0$ on the unit disc but the convergence is not uniform.

One can prove that, for every domain $D\subseteq \mathbb{C}$, there's a sequence of holomorphic functions on $D$ such that $f_n$ converges to $0$ uniformly on compact subsets of $D$ and yet the convergence is not uniform on $D$: for $\mathbb{C}$, an example is $\frac{z}{n}$, as marlasca23 noted in the comments to the question. If $D\subsetneq \mathbb{C}$, take $z_0\in\partial D$ and let $f_n:=\frac{1}{n(z-z_0)}$. So the equivalence you mention never holds.