A class of inequalities for intersection-closed set systems

Let $N$ be a finite set and $\mathcal{F}$, an intersection-closed family of subsets. Frankl conjectured that there always exists an element in $N$ which is contained in at most half the number of sets in $\mathcal{F}$ unless $\mathcal{F}=\{E\}$. We prove the validity of a class of inequalities which imply Frankl’s conjecture.