Is the set of derivatives of analytic functions of the unit disk equicontinuous?

Question

Let $F$ be the set of analytic functions from the (open) unit disk $\mathbb{D} $ to itself. Let $G$ be the set of derivative functions of $F$. I'm trying to show that $G$ is not an equicontinuous family of functions by showing that $G$ contains a sequence of functions that doesn't contain a subsequence that converges to a continuous function, thereby demonstrating that the conclusion of the Arzelà–Ascoli theorem fails for $G$. Since $G$ is a uniformly pointwise bounded family of functions, we would be done.

My suggestion for such a candidate sequence of functions is as follows: Let $f_n(z) = e^{inp} z$ where $p$ is a positive irrational number and $n \geq 1$. I conjecture that the sequence of derivatives $f'_n(z) = e^{inp} $ would do the job, but not sure how to prove it.

Answer 1

Cauchy's integral formula for the second derivative $$ f''(a) = \frac{2!}{2 \pi i} \int_{|z|=r} \frac{f(z)}{(z-a)^3} dz \, . $$ for $|a| < r < 1$ implies that the family $\{ f'' \mid f \in F \}$ of second derivatives is uniformly bounded on each closed disk $K = \{ z: |z-a| \le \varepsilon \} \subset \Bbb D$.

It follows that the family $G$ of first derivatives is uniformly equicontinuous on $K$, and in particular equicontinuous at each point $a \in \Bbb D$.

In your example $f'_n(z) = e^{inp}$ there is a subsequence converging to the constant $1$, because $np$ comes arbitrarily close to multiples of $2 \pi$.

Answer 2

By the compactness of the unit circle, there is a subsequence $n_k$ such that $e^{in_k p}$ converges to some $e^{i\theta},$ so your trial balloon is fast losing altitude. I would suggest looking at Montel's theorem.