  We prove that the union-closed sets conjecture is true for separating
union-closed families $\mathcal{A}$ with
  $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$
where $m$ denotes the number of elements in $\mathcal{A}$.
