A sequence of holomorphic functions $f_n:D(0,1)\to D(0,1)\setminus\{0\}$ , satisfying $\sum_{n=1}^\infty|f_n(0)|<\infty$,

Question

By Harnack inequality, we have $\sum_{n=1}^{\infty}|f_n(z)|^3$ converges uniformly for all $|z|\leq\frac{1}{2}$.

The skill is in the following thread. For a sequence $\{f_j\}$ of holomorphic functions, what can we say given $\sum_{j=1}^\infty |f_j(0)|$ converges.

I would like to ask a further question. Could you give me an example such that the series $\sum_{n=1}^{\infty}|f_n(z)|^3$ diverges for any $z>\frac{1}{2}$?

$f_n$ not mapping to $0$ seems to make construction more difficult.

Thank you!