  We prove some sufficient conditions implying $l^p$ inequalities of the form
$||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in
certain positive real intervals. Our sufficient conditions are strictly weaker
than the usual majorization relation. The conditions are expressed in terms of
certain homogeneous symmetric polynomials in the entries of the vectors. These
polynomials include the elementary symmetric polynomials as a special case. We
also give a characterization of the majorization relation by means of symmetric
polynomials.
