arXiv subjects

Ivo Klemes

Publications and source records attributed to Ivo Klemes.

4 recordsLinked to original sources

arXiv2013

More symmetric polynomials related to p-norms

It is known that the elementary symmetric polynomials $e_k(x)$ have the property that if $ x, y \in [0,\infty)^n$ and $e_k(x) \leq e_k(y)$ for all $k$, then $||x||_p \leq ||y||_p$ for all real $0\leq p \leq 1$, and moreover $||x||_p \geq ||y||_p$ for $1\leq p \leq 2$ provided $||x||_1 =||y||_1$. Previously the author proved this kind of property for $p>2$, for certain polynomials $F_{k,r}(x)$ which generalize the $e_k(x)$. In this paper we give two additional generalizations of this type, involving two other families of polynomials. When $x$ consists of the eigenvalues of a matrix $A$, we give a formula for the polynomials in terms of the entries of $A$, generalizing sums of principal $k \times k$ subdeterminants.

math.CA↗

arXiv2011

Polarization of an inequality

We generalize a previous inequality related to a sharp version of the Littlewood conjecture on the minimal $L_1$-norm of $N$-term exponential sums $f$ on the unit circle. The new result concerns replacing the expression $\log(1+t|f|^2)$ with $\log (\sum_{k=1}^K t_k|f_{k}|^2)$. The proof occurs on the level of finite Toeplitz matrices, where it reduces to an inequality between their polarized determinants (or "mixed discriminants").

math.CA↗

arXiv2011

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

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.

math.CA↗

arXiv2008

Symmetric polynomials, p-norm inequalities, and certain functionals related to majorization

We study a variant of the majorization relation. In particular we consider inequalities involving some Schur-concave symmetric polynomials related to the multinomial expansion. We also discuss how these topics were motivated by conjectures on sharp Lp inequalities between complex exponential sums conjectured by Hardy and Littlewood (still open problems), and why the usual majorization relation does not hold in that context.

math.CA↗