Here is what I am trying to prove:
Let $A$ and $B$ be two set families, then $G(A \bigcup B,m)\le G(A,m)G(B,m)$ where $G$ is the growth function.
I think the above result holds iff both $G(A,m)$ and $G(B,m)$ are greater than or equal to $2$.
Counterexample:
Let $A$ and $B$ contain single sets which are disjoint. Then $G(A \bigcup B,1)=2$, while $G(A,1)=G(B,1)=1$.
Is my reasoning correct?