Previous |  Up |  Next

Article

Keywords:
geometric mean; power mean; Hermitian matrix; permanent of a complex; simplex; arithmetic-geometric inequality
Summary:
There are many relations involving the geometric means $G_n(x)$ and power means $[A_n(x^{\gamma })]^{1/\gamma }$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\geq A_n(x^{\gamma })$ and $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\leq A_n(x^{\gamma })$ with parameters $\lambda \in \Bbb R$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated.
References:
[1] Bullen, P. S., Mitrinovic, D. S., Vasic, P. M.: Means and Their Inequalities. Reidel, Dordrecht (1988). MR 0947142 | Zbl 0687.26005
[2] Wang, W. L., Wen, J. J., Shi, H. N.: Optimal inequalities involving power means. Acta Math. Sin. 47 (2004), 1053-1062 Chinese. MR 2128070
[3] Pečarić, J. E., Wen, J. J., Wang, W. L., Tao, L.: A generalization of Maclaurin's inequalities and its applications. Math. Inequal. Appl. 8 (2005), 583-598. MR 2174887
[4] Wang, W. L., Lin, Z. C.: A conjecture of the strengthened Jensen's inequality. Journal of Chengdu University (Natural Science Edition) 10 (1991), 9-13 Chinese.
[5] Chen, J., Wang, Z.: Proof of an analytic inequality. J. Ninbo Univ. 5 (1992), 12-14 Chinese.
[6] Chen, J., Wang, Z.: The converse of an analytic inequality. J. Ninbo Univ. 2 (1994), 13-15 Chinese.
[7] Wen, J. J., Zhang, Z. H.: Vandermonde-type determinants and inequalities. Applied Mathematics E-Notes 6 (2006), 211-218. MR 2231746 | Zbl 1157.15316
[8] Wen, J. J., Wang, W. L.: Chebyshev type inequalities involving permanents and their application. Linear Alg. Appl. (2007), 422 295-303. MR 2299014
[9] Wen, J. J., Shi, H. N.: Optimizing sharpening for Maclaurin inequality. Journal of Chengdu University (Natural Science Edition) 19 (2000), 1-8 Chinese.
[10] Wen, J. J., Wang, W. L.: The inequalities involving generalized interpolation polynomial. Computer and Mathematics with Applications 56 (2008), 1045-1058 [Online: http://dx.doi.org/10.1016/j.camwa.2008.01.032] DOI 10.1016/j.camwa.2008.01.032 | MR 2435283
[11] Wen, J. J., Gao, C. B.: Geometric inequalities involving the central distance of the centered 2-surround system. Acta. Math. Sin. 51 (2008), 815-832 Chinese. MR 2454021 | Zbl 1174.26015
Partner of
EuDML logo