For a Sperner family
A
$
{\sum\nolimits_i {{\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} \leqslant 1} }
$
{\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)}
$
{\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).
Mathematics Subject Classification (2000):
05D05
* Research supported by the Sonderforschungsbereich 343 
Diskrete Strukturen in der Mathematik
, University of Bielefeld.