On the average rank of LYM-sets

Let S be a finite set with some rank function r such that the Whitney numbers w i = I { x e S I r ( x ) = i}l are log-concave. Given k, me N so that Wk-1 < Wk <~ Wk+m, set W = w k + w k + 1 + "'" + Wk + ~" Generalizing a theorem of Kleitman and Milner, we prove that every F ~ S with cardinality IFI/> Whas average rank at least ( k w k + ... + (k + m)wk+m)/W, provided the normalized profile vector (x 1 ..... xn) of F satisfies the following LYM-type inequality: x o + xl + .-. + x, ~< m + 1.