Let $f_n(z) = z^n$.
(a) We consider $f_n$’s as functions on $Ω = {|z| > 1}$. Show that $\{f_n\}_{n≥1}$ is not normal for the distance in $\mathbb{C}$, but it converges for the chordal distance on $\mathbb{C}^*$ uniformly on compact sets.
My attempt: Let $K$ be a compact set in $Ω$. We can put $r = \max|z|$ for $z \in Ω$.
Then $\max|f_n(z)-1| = \max|z^n - 1| = r^n -1$, $r^n$ goes to infity as $n$ goes to infinity, therefore $\{f_n\}_{n≥1}$ is not normal for the distance in $\mathbb{C}$.
If we look at the chordal distance on the other hand we get that
$$\max \frac{2|f_n(z)|-0}{\sqrt{1+|f_n(z)|^2}\sqrt{1+|0|^2}} = \max \frac {2|z|^n}{\sqrt{1+|z|^{2n}}} = \frac{2r^n}{\sqrt{1+r^{2n}}}.$$ This goes to zero as $n$ goes to infinty since denominator goes to $∞$ quicker than the numerator.
Therefore $f_n$ converges uniformly on compact sets for the chordal distance on $\mathbb{C}^*$.
(b) $f_n$ are functions on $\mathbb{C}$ show that $\{f_n\}$ is not spherically normal.
My approach: my attempt was to look at the spherical derivatives $f_n^{ ♯}(z)$ and show that they are not uniformly bounded.
I was wondering if someone could check my approaches because I'm pretty sure they are wrong but I don't really get how else to approach these questions, thank you so much in advance!