This problem is from the qualifying exam for complex analysis.
Let $\mathcal{A}$ be a family of functions holomorphic in the unit disk $\mathbb{D}=\{z: |z|<1\}$. Denote $\mathcal{A'}= \{f':f\in \mathcal{A}\}$. Assume $\mathcal{A'}$ is normal and $\mathrm{sup}\{|f(0)|:f\in \mathcal{A}\}<\infty$.
Prove that $\mathcal{A}$ is normal.
My attempt: To prove that the given family of functions is normal, I'd like to apply Montel's theorem, which states that if a family of holomorphic functions on an open subset of $\mathbb{C}$ is uniformly bounded on compact subsets, then the family is normal. Therefore, I only have to prove that $\mathcal{A}$ is uniformly bounded on every compact subset of $\mathbb{D}$.
Otherwise, I just have to directly apply the definition of normal family.
However, I have no idea for either way. Does anyone have ideas?
Best Answer
Let $(f_n)_{n\in\mathbb N}$ be a sequence of elements of $\mathcal A$; you want to prove that it has a subsequence which converges uniformly. Take a subsequence $(f_{n_k})_{k\in\mathbb N}$ such that $(f_{n_k}')_{k\in\mathbb N}$ converges uniformly to a function $g$. The sequence $\bigl(f_{n_k}(0)\bigr)_{k\in\mathbb N}$ is bounded and therefore it has a convergent subsequence. We can assume, without loss of generality, that $\lim_{k\to\infty}f_{n_k}(0)=l$. Now, define $f$ as the primitive of $g$ such that $f(0)=l$. Then $(f_{n_k})_{k\in\mathbb N}$ converges uniformly to $f$.