# 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} }

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…

## 5 Citations

### Norms on complex matrices induced by complete homogeneous symmetric polynomials

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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

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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

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- 2021

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

- MathematicsLinear and Multilinear Algebra
- 2022

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…

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

- MathematicsBanach Journal of Mathematical Analysis
- 2021

## References

SHOWING 1-10 OF 13 REFERENCES

### Two inequalities for the complete symmetric functions

- MathematicsMathematical 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

- Computer Science, MathematicsNIPS
- 2017

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

- MathematicsCanadian Journal of Mathematics
- 1957

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

- MathematicsProceedings 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

- Mathematics, Computer Science
- 2005

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

- MathematicsProceedings 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

- Mathematics
- 1952

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

- Mathematics
- 1987

- 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

- MathematicsThe Mathematical Gazette
- 1896

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…