Abstract
We prove that the complete elementary symmetric function
and the function
are Schur-convex functions in
, where
are nonnegative integers,
,
. For which, some inequalities are established by use of the theory of majorization. A problem given by K. V. Menon (Duke Mathematical Journal 35 (1968), 37–45) is also solved.
Similar content being viewed by others
References
Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F.. Volume 30. Springer, Berlin; 1961:xii+198.
Detemple DW, Robertson JM: On generalized symmetric means of two variables. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1979, (634–677):236–238.
Guan K: Inequalities of generalized-order symmetric mean. Journal of Chongqing Teachers College (Natural Science Edition) 1998,15(3):40–43.
Hardy GH, Littlewood JE, Pólya G: Some simple inequalities satisfied by convex functions. Messenger of Mathematics 1929, 58: 145–152.
Kuang JC: Applied Inequalities. 2nd edition. Human education Press, Changsha; 1993:xxvi+794.
Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering. Volume 143. Academic Press, New York; 1979:xx+569.
Menon KV: Inequalities for symmetric functions. Duke Mathematical Journal 1968, 35: 37–45. 10.1215/S0012-7094-68-03504-7
Mitrinović DS: Analytic Inequalities, Die Grundlehren der mathematischen Wisenschaften. Volume 1965. Springer, New York; 1970:xii+400.
Roberts AW, Varberg DE: Convex Functions, Pure and Applied Mathematics. Volume 57. Academic Press, New York; 1973:xx+300.
Shi HN: Refinement and generalization of a class of inequalities for symmetric functions. Mathematics in Practice and Theory 1999,29(4):81–84.
Zhang X-M: Optimization of Schur-convex functions. Mathematical Inequalities & Applications 1998,1(3):319–330.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Guan, K. Schur-convexity of the complete elementary symmetric function. J Inequal Appl 2006, 67624 (2006). https://doi.org/10.1155/JIA/2006/67624
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/JIA/2006/67624