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A Note on the Positivity of the Even Degree Complete Homogeneous Symmetric Polynomials

  • Ionel RovenţaEmail author
  • Laurenţiu Emanuel Temereancă
Article
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Abstract

This article deals with the positivity of a nice family of symmetric polynomials, namely complete homogeneous symmetric polynomials. We are able to give a positive answer to a question arising in Tao (https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-positive-definiteness-of-the-even-degree-complete-homogeneous-symmetric-polynomials/, 2017). Our strategy follows two different ideas, one of them based on a Schur-convexity argument and the other one uses a method with divided differences. Several Newton’s type inequalities are also discussed.

Keywords

Symmetric polynomials divided differences and Schur-convexity 

Mathematics Subject Classification

Primary 26B25 Secondary 26D05 05E05 
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Notes

References

  1. 1.
    Abramovich, S., Ivelić, S., Pečarić, J.E.: Improvement of Jensen-Steffensen’s inequality for superquadratic functions. Banach J. Math. Anal. 4(1), 159–169 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Adil Khan, M., Niezgoda, M., Pečarić, J.E.: On a refinement of the majorisation type inequality. Demonstr. Math. 44, 49–57 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ciobotea, D., Duica, L.: Visual impairment in the elderly and its influence on the quality of life. J. Res. Soc. Interv. 5(4), 66–74 (2016)Google Scholar
  4. 4.
    Duica, L., Antonescu, E.: Clinical and biochemical correlations of aggression in Young patients with mental disorders. J. Chem. 69, 1544–1549 (2018)Google Scholar
  5. 5.
    Hunter, D.B.: The positive-definiteness of the complete symmetric functions of even order. Math. Proc. Camb. Philos. Soc. 82(2), 255–258 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ivelić Bradanović, S., Latif, N., Pečarić, D., Pečarić, J.: Sherman’s and related inequalities with applications in information theory. J. Inequal. Appl. 2018, 98 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ivelić Bradanović, S., Latif, N., Pečarić, J.: On an upper bound for Sherman’s inequality. J. Inequal. Appl. 1, 165 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ivelić Bradanović, S., Pečarić, J.: Generalizations of Sherman’s inequality. Period. Math. Hung. 74, 197–219 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Lemos, R., Soares, G.: Some log-majorizations and an extension of a determinantal inequality. Linear Algebra Appl. 547, 19–31 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Maclaurin, C.: A second letter to Martin Kolkes, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra. Phil. Trans. 36, 59–96 (1729)Google Scholar
  11. 11.
    Marshall, A.W., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and Its Application, 2nd edn. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  12. 12.
    Merca, M.: An infinite sequence of inequalities involving special values of the Riemann zeta function. Math. Ineq. Appl. 21(1), 17–24 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Mutica, M., Duica, L.: Elderly schizophrenic patients? Clinical and social correlations. Eur. Neuropsychopharmacol. 26, S512 (2016)CrossRefGoogle Scholar
  14. 14.
    Newton, I.: Arithmetica universalis: sive de compositione et resolutione arithmetica liber Cantabrigi: typis academicis, Londini, impensis Benj. Tooke (1707)Google Scholar
  15. 15.
    Niculescu, C.P., Persson, L.-E.: Convex Functions and their Applications. A Contemporary Approach, CMS Books in Mathematics, vol. 23. Springer, New York (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    Niculescu, C.P., Rovenţa, I.: An approach of majorization in spaces with a curved geometry. J. Math. Anal. Appl. 411(1), 119–128 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Niezgoda, M.: Linear maps preserving group majorization. Linear Algebra Appl. 330, 113–127 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Niezgoda, M.: Majorization and refined Jensen-Mercer type inequalities for self-adjoint operators. Linear Algebra Appl. 467, 1–14 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Pales, Z., Radacsi, E.: Characterizations of higher-order convexity properties with respect to Chebyshev systems. Aequat. Math. 90, 193–210 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Popoviciu, T.: Notes sur le fonctions convexes d’ ordre superieur (IX). Bull. Math. Soc. Roum. Sci. 43, 85–141 (1941)zbMATHGoogle Scholar
  21. 21.
    Popoviciu, T.: Les Fonctions Convexes. Hermann, Paris (1944)zbMATHGoogle Scholar
  22. 22.
    Rovenţa, I.: Schur-convexity of a class of symmetric functions. Ann. Univ. Craiova Math. Comput. Sci. Ser. 37(1), 12–18 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ionel Rovenţa
    • 1
    Email author
  • Laurenţiu Emanuel Temereancă
    • 2
  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania
  2. 2.Department of Applied MathematicsUniversity of CraiovaCraiovaRomania

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