# 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…
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## References

SHOWING 1-10 OF 13 REFERENCES

• 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
• Computer Science, Mathematics
NIPS
• 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.
• Mathematics
Canadian 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.)
• 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
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.
• 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
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
- 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.
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