## The optimization for the inequalities of power means.(English)Zbl 1133.26324

Summary: Let $$M_n[t](a)$$ be the $$t$$-th power mean of a sequence a of positive real numbers, where $$a=(a_1,a_2,\dots, a_n)$$, $$n\geq 2$$, and $$\alpha,\lambda\in\mathbb{R}_{++}$$, $$m\geq 2,\sum_{j=1}^m\lambda_j=1$$, $$\min\{\alpha\}\leq \theta\leq \max\{\alpha\}$$. In this paper, we will state the important background and meaning of the inequality $$\prod_{j=1}^m \{M_n^{[\alpha_j]}(a)\}^{\lambda_j}\leq (\geq) M_n^{[\theta]}(a)$$; 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.

### MSC:

 2.6e+61 Means
Full Text:

### References:

 [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. [2] Department of Mathematics and Mechanics of Beijing University : Higher Algebra. People’s Education Press, Beijing; 1978. [3] Gardner, RJ, The brunn-Minkowski inequality, Bulletin of the American Mathematical Society. New Series, 39, 355-405, (2002) · Zbl 1019.26008 [4] Guan, K, Schur-convexity of the complete elementary symmetric function, Journal of Inequalities and Applications, 2006, 9 pages, (2006) · Zbl 1090.26009 [5] Kuang JC: Applied Inequalities. Hunan Education Press, Changsha; 2004. [6] Lai, L; Wen, JJ, Generalization for Hardy’s inequality of convex function, Journal of Southwest University for Nationalities (Natural Science Edition), 29, 269-274, (2003) [7] Leng, G; Zhao, C; He, B; Li, X, Inequalities for polars of mixed projection bodies, Science in China. Series A. Mathematics, 47, 175-186, (2004) · Zbl 1070.52006 [8] Lin, TP, The power mean and the logarithmic Mean, The American Mathematical Monthly, 81, 879-883, (1974) · Zbl 0292.26015 [9] Liu, Z, Comparison of some means, Journal of Mathematical Research and Exposition, 22, 583-588, (2002) · Zbl 1021.26015 [10] Macdonald IG: Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs. 2nd edition. The Clarendon Press, Oxford University Press, New York; 1995:x+475. · Zbl 0824.05059 [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. · Zbl 0437.26007 [12] Minc H: Permanents. Addison-Wesley, Massachusetts; 1988. · Zbl 0166.29904 [13] Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series). Volume 61. Kluwer Academic, Dordrecht; 1993:xviii+740. · Zbl 0771.26009 [14] Pečarić, JE; Svrtan, D, New refinements of the Jensen inequalities based on samples with repetitions, Journal of Mathematical Analysis and Applications, 222, 365-373, (1998) · Zbl 0912.26008 [15] Timofte, V, On the positivity of symmetric polynomial functions. I. general results, Journal of Mathematical Analysis and Applications, 284, 174-190, (2003) · Zbl 1031.05130 [16] Wang BY: An Introduction to the Theory of Majorizations. Beijing Normal University Press, Beijing; 1990. [17] Wang, ZL; Wang, XH, Quadrature formula and analytic inequalities-on the separation of power means by logarithmic Mean, Journal of Hangzhou University, 9, 156-159, (1982) [18] Wang, W-L; Wang, PF, A class of inequalities for the symmetric functions, Acta Mathematica Sinica, 27, 485-497, (1984) · Zbl 0561.26013 [19] Wang, W-L; Wen, JJ; Shi, HN, Optimal inequalities involving power means, Acta Mathematica Sinica, 47, 1053-1062, (2004) · Zbl 1121.26308 [20] Wei, Z; Qi, L; Birge, JR, A new method for nonsmooth convex optimization, Journal of Inequalities and Applications, 2, 157-179, (1998) · Zbl 0903.90131 [21] Wen, JJ, The optimal generalization of A-G-H inequalities and its applications, 12-16, (2004) [22] Wen JJ: Hardy means and their inequalities. to appear in Journal of Mathematics to appear in Journal of Mathematics · Zbl 0292.26015 [23] Wen, JJ; Wang, W-L; Lu, YJ, The method of descending dimension for establishing inequalities, Journal of Southwest University for Nationalities, 29, 527-532, (2003) [24] Wen, JJ; Xiao, CJ; Zhang, RX, Chebyshev’s inequality for a class of homogeneous and symmetric polynomials, Journal of Mathematics, 23, 431-436, (2003) · Zbl 1056.26015 [25] Wen, JJ; Zhang, RX; Zhang, Y, Inequalities involving the means of variance and their applications, Journal of Sichuan University. Natural Science Edition, 40, 1011-1018, (2003) · Zbl 1042.26511 [26] Zheng WX, Wang SW: An Introduction to Real and Functional Analysis (no.2). People’s Education Press, Shanghai; 1980.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.