Determine whether the family $\mathcal{F}=\{f \text{ holomorphic in } \mathbb{D} : f(0)=1 \text{ and } \mathfrak{R}f(z)>0 \}$ is normal

Question

Let $\mathcal{F}=\{f \text{ holomorphic in } \mathbb{D} : f(0)=1 \text{ and } \mathfrak{R}f(z)>0 \: \forall z \in \mathbb{D}\}$. I want to determine whether this is a family of normal functions. In sight of Montel's theorem, I know that this is equivalent to being uniformly bounded on compact sets of $\mathbb{D}$, and I also know that this condition in families of holomorphic functions implies equicontinuity.

Let me show you my work so far. I know that the Möbius transformation $$T(z)=-\frac{z+1}{z-1}$$ maps the unit disc onto the right half plane (that is, $\mathfrak{R}f(z)>0$) and also $T(0)=1$. I was trying to prove that this transformation cannot be uniformly continuous when $r$ goes to $1^-$. This would mean that the family would't be uniformly bounded on compact sets and therefore, it wouldn't be normal. However I am not able to finish my argument. I have made other attemps but they resulted in failure. I have been struggling with this for hours now. I have also contemplated the possibility that the family is actually normal, but this doesn't look good to me since the modulus of the functions doen't have any upper restrictions. Can someone help me?

Answer 1

The family is normal. There are many different ways to prove it:

1. Define $\mathcal{G}:=\{\exp(-f):f\in\mathcal{F}\}$. $\mathcal{G}$ is normal by Montel's theorem. To prove now that $\mathcal{F}$ is normal, let $f_n$ be a sequence of functions in $\mathcal{F}$: we can extract a subsequence $f_{n_k}$ such that $\exp(-f_{n_k})$ converges locally uniformly to a holomorphic $g$ and it is easy to see by Hurwitz's theorem that $g\neq 0$. This implies that we can take its logarithm $h:=\ln(g)$. Since $\exp(f_{n_k}- h)\to 1$, there exists a sequence of integers $m_k$ such that $f_{n_k}+2\pi i m_k \to h$. Evaluating this in $0$ we get $1+2\pi i m_k\to h(0)$, so $m_k$ is bounded and we can extract a subsequence $j_k$ such that $m_{j_k}$ is convergent (it's actually definitely constant) and it follows that $f_{n_{j_k}}\to h$ locally uniformly, proving the claim.
2. Let $\tilde{\mathcal{F}}:=\{f:\text{Re}(f)>0\}$ and let $P$ be the property $P:=\{(f,A):f\in H(A); \text{Re}(f)>_{|A}0\}$. Since $(f,\mathbb{C})\in P$ iff $f$ is constant, the Bloch-Robinson principle implies that $\tilde{\mathcal{F}}$ is a (meromorphically) normal family. Since $\mathcal{F}$ cannot cluster at the constant function $\infty$ and it clearly is a closed subset of $\tilde{\mathcal{F}}$, it must be a (holomorphically) normal family.