I wanna tackle the following exercise:
Let $\mathcal{F}$ be the class of all functions $f \in \mathcal{H}(\mathbb{D})$ satisfying $f(0)=1$ and $\Re(f) > 0$. Show that $\mathcal{F}$ is a normal family.
My work so far:
I defined a möbius transformation $\varphi:\mathcal{A} \to \mathbb{D},\ z \mapsto \frac{z-1}{z+1}$ where $\mathcal{A} = \{z \in \mathbb{C} : \Re(z) > 0 \}$ with the property $\varphi(1) = 0$. Then I defined a new map $g= \varphi\ \circ \ f$ on and onto the unit disk $\mathbb{D}$. This new map $g$ has the property $g(0) = \varphi(f(0))=0$ and therefore satisfies the conditions for Schwarz' lemma, which further implied
$\vert g \vert = \vert \varphi(f) \vert \leq 1 \ \ \forall z \in \mathbb{D}$.
From Montel's theorem we now have a normal family of functions given by $g$.
My question is if there exists any kind of theorem or simple statement that would allow me to transfer the normality of the family $\varphi(f)$ to the family $\mathcal{F}$? I tried using the propreties of the möbius transformation as a conformal mapping but nothing led to a satisfying result so far.
Any help is appreciated.