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

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adil Khan, M., Niezgoda, M., Pečarić, J.E.: On a refinement of the majorisation type inequality. Demonstr. Math. 44, 49–57 (2011)

    MathSciNet  MATH  Google Scholar 

  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. 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. Hunter, D.B.: The positive-definiteness of the complete symmetric functions of even order. Math. Proc. Camb. Philos. Soc. 82(2), 255–258 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ivelić Bradanović, S., Latif, N., Pečarić, J.: On an upper bound for Sherman’s inequality. J. Inequal. Appl. 1, 165 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ivelić Bradanović, S., Pečarić, J.: Generalizations of Sherman’s inequality. Period. Math. Hung. 74, 197–219 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lemos, R., Soares, G.: Some log-majorizations and an extension of a determinantal inequality. Linear Algebra Appl. 547, 19–31 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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. Marshall, A.W., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and Its Application, 2nd edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  12. Merca, M.: An infinite sequence of inequalities involving special values of the Riemann zeta function. Math. Ineq. Appl. 21(1), 17–24 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Mutica, M., Duica, L.: Elderly schizophrenic patients? Clinical and social correlations. Eur. Neuropsychopharmacol. 26, S512 (2016)

    Article  Google Scholar 

  14. Newton, I.: Arithmetica universalis: sive de compositione et resolutione arithmetica liber Cantabrigi: typis academicis, Londini, impensis Benj. Tooke (1707)

  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)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Niezgoda, M.: Linear maps preserving group majorization. Linear Algebra Appl. 330, 113–127 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Niezgoda, M.: Majorization and refined Jensen-Mercer type inequalities for self-adjoint operators. Linear Algebra Appl. 467, 1–14 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pales, Z., Radacsi, E.: Characterizations of higher-order convexity properties with respect to Chebyshev systems. Aequat. Math. 90, 193–210 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Popoviciu, T.: Notes sur le fonctions convexes d’ ordre superieur (IX). Bull. Math. Soc. Roum. Sci. 43, 85–141 (1941)

    MATH  Google Scholar 

  21. Popoviciu, T.: Les Fonctions Convexes. Hermann, Paris (1944)

    MATH  Google Scholar 

  22. Rovenţa, I.: Schur-convexity of a class of symmetric functions. Ann. Univ. Craiova Math. Comput. Sci. Ser. 37(1), 12–18 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Tao, T.: https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-poHrBsitive-definiteness-of-the-even-degree-complete-homogeneous-symmetric-polyHrBnomials/. August 2017

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Authors and Affiliations

  1. Department of Mathematics, University of Craiova, Craiova, 200585, Romania

    Ionel Rovenţa

  2. Department of Applied Mathematics, University of Craiova, 200585, Craiova, Romania

    Laurenţiu Emanuel Temereancă

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  1. Ionel Rovenţa
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  2. Laurenţiu Emanuel Temereancă
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Correspondence to Ionel Rovenţa.

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Dedicated to the 70th Anniversary of Professor Constantin P. Niculescu.

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Rovenţa, I., Temereancă, L.E. A Note on the Positivity of the Even Degree Complete Homogeneous Symmetric Polynomials. Mediterr. J. Math. 16, 1 (2019). https://doi.org/10.1007/s00009-018-1275-9

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  • DOI: https://doi.org/10.1007/s00009-018-1275-9

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