Supersaturation in union-closed families of sets

Let $k$ and $n$ be positive integers such that $2 \leq k \leq n+1$.

We prove that the number of $k$-chains in a union-closed family with universe $[n]$ and size $m$ is minimized when its member sets are largest possible.

We also show that, whenever the minimum is nonzero and $m>n$, there are no other minimizing families.