New concavity and convexity results for symmetric polynomials and their ratios

@article{Sra2018NewCA,
  title={New concavity and convexity results for symmetric polynomials and their ratios},
  author={Suvrit Sra},
  journal={Linear and Multilinear Algebra},
  year={2018},
  volume={68},
  pages={1031 - 1038}
}
  • S. Sra
  • Published 27 March 2018
  • Mathematics
  • Linear and Multilinear Algebra
ABSTRACT We prove ‘power’ generalizations of Marcus–Lopes style (including McLeod and Bullen) concavity inequalities for elementary symmetric polynomials, and similar generalizations to convexity inequalities of McLeod and Baston for complete homogeneous symmetric polynomials. We also present additional concavity results for elementary symmetric polynomials, of which the main result is a concavity theorem that yields a well-known log-convexity 1972 result of Muir for positive definite matrices… 

Norms on complex matrices induced by complete homogeneous symmetric polynomials

We introduce a remarkable new family of norms on the space of n×n$n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and

Some New Methods for Generating Convex Functions

We present some new methods for constructing convex functions. One of the methods is based on the composition of a convex function of several variables which is separately monotone with convex and

New applications to combinatorics and invariant matrix norms of an integral representation of natural powers of the numerical values

ABSTRACT. Let ∨A be the k-th symmetric tensor power of A ∈ Mn(C). In [23], we have expressed the normalized trace of ∨A as an integral of the k-th powers of the numerical values of A over the unit

On an integral representation of the normalized trace of the k-th symmetric tensor power of matrices and some applications

S n of C with respect to the normalized Euclidean surface measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric

References

SHOWING 1-10 OF 13 REFERENCES

Two inequalities for the complete symmetric functions

  • V. Baston
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1978
In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequality where ht denotes the complete symmetric function of order t. In this note we

Elementary Symmetric Polynomials for Optimal Experimental Design

This work revisits the classical problem of optimal experimental design under a new mathematical model grounded in a geometric motivation and introduces models based on elementary symmetric polynomials, which capture "partial volumes" and offer a graded interpolation between the widely used A-optimal design and D-Optimal design models.

Inequalities for Symmetric Functions and Hermitian Matrices

The purpose of this paper is to present two concavity results for symmetric functions and apply these to obtain inequalities connecting the characteristic roots of the non-negative Hermitian (n.n.h.)

Inequalities concerning the inverses of positive definite matrices

  • W. Muir
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 1974
Much has been written on inequalities concerning positive definite matrices, but a new insight may be gained by examining inequalities from the standpoint of the inverse matrix. The standard

Low rank approximations of symmetric polynomials and asymptotic counting of contingency tables

It is shown that the complete symmetric polynomial of a fixed degree in n variables can be epsilon-approximated coefficient-wise by a sum of powers of O(log n) linear forms, from which it follows that if the row sums are bounded by a constant fixed in advance, there is a deterministic approximation algorithm to compute the logarithmic asymptotic of the number of tables.

On four Inequalities in Symmetric Functions

  • J. McLeod
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 1959
Let us denote by α the set of n real numbers α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in α1, …, αn, and by ck(α) and hk(α) the elementary and

DRESHER'S INEQUALITY

Abstract : This note gives an elementary proof of the Dresher inequality, based on the Minkowski inequality and an inequality due to Radon. The Radon inequality is gotten easily by transforming the

Handbook of means and their inequalities

- Preface to 'Means and their Inequalities'. Preface to the Handbook. Basic References. - Notations. 1. Referencing. 2. Bibliographic References. 3. Symbols for some Important Inequalities. 4.

Mathematical Notes

if o>>2, c, a have common factor, .". only possible value of to is 2. = 2m, c -a = 2n and c — m + n, a = m-n. And 6 = c a? = imn .-. ran is an exact square, .-. m, n are both exact squares or have a