Polynomial Lym Inequalities

For a Sperner family A ⊂ 2^[n] let A _i denote the family of all i-element sets in A. We sharpen the LYM inequality ${\sum\nolimits_i {{\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} \leqslant 1} }$ by adding to the LHS all possible products of fractions ${\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)}$, with suitable coefficients. A corresponding inequality is established also for the linear lattice and the lattice of subsets of a multiset (with all elements having the same multiplicity).