×

A note on the positivity of the even degree complete homogeneous symmetric polynomials. (English) Zbl 1404.05215

Summary: 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 [T. Tao, “Schur convexity and positive definiteness of the even degree complete homogeneous symmetric polynomials”, https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-positive-definiteness-of-the-even-degree-complete-homogeneous-symmetric-polynomials/]. 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.

MSC:

05E05 Symmetric functions and generalizations
26B25 Convexity of real functions of several variables, generalizations
26D05 Inequalities for trigonometric functions and polynomials
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramovich, S.; Ivelić, S.; Pečarić, JE, Improvement of Jensen-Steffensen’s inequality for superquadratic functions, Banach J. Math. Anal., 4, 159-169, (2010) · Zbl 1195.26037 · doi:10.15352/bjma/1272374678
[2] Adil Khan, M.; Niezgoda, M.; Pečarić, JE, On a refinement of the majorisation type inequality, Demonstr. Math., 44, 49-57, (2011) · Zbl 1223.26021
[3] Ciobotea, D.; Duica, L., Visual impairment in the elderly and its influence on the quality of life, J. Res. Soc. Interv., 5, 66-74, (2016)
[4] Duica, L.; Antonescu, E., Clinical and biochemical correlations of aggression in Young patients with mental disorders, J. Chem., 69, 1544-1549, (2018)
[5] Hunter, DB, The positive-definiteness of the complete symmetric functions of even order, Math. Proc. Camb. Philos. Soc., 82, 255-258, (1977) · Zbl 0369.42015 · doi:10.1017/S030500410005386X
[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) · doi:10.1186/s13660-018-1692-0
[7] Ivelić Bradanović, S.; Latif, N.; Pečarić, J., On an upper bound for Sherman’s inequality, J. Inequal. Appl., 1, 165, (2016) · Zbl 1345.26035 · doi:10.1186/s13660-016-1091-3
[8] Ivelić Bradanović, S.; Pečarić, J., Generalizations of Sherman’s inequality, Period. Math. Hung., 74, 197-219, (2017) · Zbl 1399.26042 · doi:10.1007/s10998-016-0154-z
[9] Lemos, R.; Soares, G., Some log-majorizations and an extension of a determinantal inequality, Linear Algebra Appl., 547, 19-31, (2018) · Zbl 1390.15062 · doi:10.1016/j.laa.2018.02.015
[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) · doi:10.1098/rstl.1729.0011
[11] Marshall, A.W., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and Its Application, 2nd edn. Springer, Berlin (2011) · Zbl 1219.26003 · doi:10.1007/978-0-387-68276-1
[12] Merca, M., An infinite sequence of inequalities involving special values of the Riemann zeta function, Math. Ineq. Appl., 21, 17-24, (2018) · Zbl 1401.11122
[13] Mutica, M.; Duica, L., Elderly schizophrenic patients? Clinical and social correlations, Eur. Neuropsychopharmacol., 26, s512, (2016) · doi:10.1016/S0924-977X(16)31536-X
[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) · Zbl 1100.26002 · doi:10.1007/0-387-31077-0
[16] Niculescu, CP; Rovenţa, I., An approach of majorization in spaces with a curved geometry, J. Math. Anal. Appl., 411, 119-128, (2014) · Zbl 1325.39016 · doi:10.1016/j.jmaa.2013.09.038
[17] Niezgoda, M., Linear maps preserving group majorization, Linear Algebra Appl., 330, 113-127, (2001) · Zbl 0991.47021 · doi:10.1016/S0024-3795(01)00257-9
[18] Niezgoda, M., Majorization and refined Jensen-Mercer type inequalities for self-adjoint operators, Linear Algebra Appl., 467, 1-14, (2015) · Zbl 1331.47023 · doi:10.1016/j.laa.2014.10.040
[19] Pales, Z.; Radacsi, E., Characterizations of higher-order convexity properties with respect to Chebyshev systems, Aequat. Math., 90, 193-210, (2016) · Zbl 1351.26026 · doi:10.1007/s00010-015-0377-8
[20] Popoviciu, T., Notes sur le fonctions convexes d’ ordre superieur (IX), Bull. Math. Soc. Roum. Sci., 43, 85-141, (1941) · Zbl 0060.14909
[21] Popoviciu, T.: Les Fonctions Convexes. Hermann, Paris (1944) · Zbl 0060.14911
[22] Rovenţa, I., Schur-convexity of a class of symmetric functions, Ann. Univ. Craiova Math. Comput. Sci. Ser., 37, 12-18, (2010) · Zbl 1224.26056
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.