The number of generators and orders of Abelian subgroups of finite p-groups

Let $f$ ($F$) be the smallest function such that every finite $p$-group, all of whose Abelian subgroups are generated by at most n elements (all of whose Abelian subgroups have orders at most $p^n$, has at most $f(n)$ generators (has order not exceeding $p^{F(n)}$). It is established that the functions $f$ and $F$ have quadratic order of growth.