The Union-Closed Sets Conjecture for Small Families

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\le 2(m+\frac{m}{{\log }_2(m)-{\log }_2{\log }_2(m)})$ where $m$ denotes the number of elements in $\mathcal{A}$.