Conditions for a family to be normal

Question

What are the conditions that must satisfy $A$ and $B$ (with $A,B\subset\mathbb{C}$) such that the family $$\mathfrak{F}_{A,B} = \lbrace f(z)=az+b : a\in A, b\in B\rbrace , \ z\in\mathbb{C}$$ to be normal in $\mathcal{H}(\mathbb{C})$, where $f(z)$ is a entire function.

My idea is to apply Montel's theorem: for every $K\subset\mathbb{C}$ there is $M_{K}$ such that $|f(z)|\leq M_{K}$ for every $z\in K$. Then I obtain that the conditions are that $A$ and $B$ have to be bounded for which the family is normal.

This argument is correct?, or can you give me a hint of how to solve this of a more explicit way?

Answer 1

As Martin R pointed out, your argument is hard to judge without details. After adopting Montel's theorem, having deduced that it is necessary for $\mathfrak F_{A,B}$ to be normal that for each compact $K\subset\mathbb C$, $|f(z)|\leqslant M_K$ for some positive $M_K$ whenever $f\in\mathfrak F_{A,B}$ and $z\in K$, you can pick a compact $K_0$ that contains $0$ and $1$. Then consider $|f(0)|$ and $|f(1)|$ for each $f\in\mathfrak F_{A,B}$ and this gives the boundedness of $A$ and $B$.

Answer 2

Claim: The necessary and sufficient condition is that $A$ is bounded.

Proof:

Suppose $A$ is not bounded. Let $|a_n| \rightarrow \infty$ and let $b\in B$. Consider the sequence $f_n(z)=a_nz+b$. Clearly $|f_n(z)|$ is unbounded in a neighborhood of the origin. Yet, $f_n(0)=b$ so $f_n \not\rightarrow \infty$ in a neighborhood of the origin, thus $F$ is not normal.

On the other hand, suppose $A$ is bounded by $M<\infty$. Let |f(z)=az+b|, so |f'(z)=a|. Now apply Marty's theoream, where we see that $\dfrac{|f'(z)|}{1+|f(z)|^2} \leq \dfrac{a}{1+|f(z)|^2} \leq \dfrac{M}{1+|f(z)|^2} \leq M $. Marty's theorem then states that $F$ is normal.