The optimization for the inequalities of power means
- 561 Downloads
Let Open image in new window be the Open image in new window th power mean of a sequence Open image in new window of positive real numbers, where Open image in new window , and Open image in new window . In this paper, we will state the important background and meaning of the inequality Open image in new window ; a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.
KeywordsReal Number Open Problem Mathematical Analysis Positive Real Number Symmetrical Polynomial
- 1.Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications (East European Series). Volume 31. D. Reidel, Dordrecht; 1988:xx+459.Google Scholar
- 2.Department of Mathematics and Mechanics of Beijing University : Higher Algebra. People's Education Press, Beijing; 1978.Google Scholar
- 5.Kuang JC: Applied Inequalities. Hunan Education Press, Changsha; 2004.Google Scholar
- 6.Lai L, Wen JJ: Generalization for Hardy's inequality of convex function. Journal of Southwest University for Nationalities (Natural Science Edition) 2003,29(3):269–274.Google Scholar
- 11.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.Google Scholar
- 16.Wang BY: An Introduction to the Theory of Majorizations. Beijing Normal University Press, Beijing; 1990.Google Scholar
- 17.Wang ZL, Wang XH: Quadrature formula and analytic inequalities-on the separation of power means by logarithmic mean. Journal of Hangzhou University 1982,9(2):156–159.Google Scholar
- 21.Wen JJ: The optimal generalization of A-G-H inequalities and its applications. Journal of Shaanxi Normal University 2004, 12–16.Google Scholar
- 22.Wen JJ: Hardy means and their inequalities. to appear in Journal of Mathematics to appear in Journal of MathematicsGoogle Scholar
- 23.Wen JJ, Wang W-L, Lu YJ: The method of descending dimension for establishing inequalities. Journal of Southwest University for Nationalities 2003,29(5):527–532.Google Scholar
- 26.Zheng WX, Wang SW: An Introduction to Real and Functional Analysis (no.2). People's Education Press, Shanghai; 1980.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.