Symmetric polynomials and $l^p$ inequalities for certain intervals of $p$

We prove some sufficient conditions implying $l^p$ inequalities of the form $\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}_p\le \mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }{\mathrm{\mid }}_p$ for vectors $x,y\in [0,\mathrm{\infty }{)}^n$ 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.