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Research Article The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 838529 10 pages doi 2009 838529 Research Article The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications Yuming Chu and Yupei Lv Department of Mathematics Huzhou Teachers College Huzhou 313000 China Correspondence should be addressed to Yuming Chu chuyuming2005@ Received 2 April 2009 Accepted 20 May 2009 Recommended by A. Laforgia We prove that the Hamy symmetric function Fn x r - 1 i1 i2 ir n n Xi Ý r is Schur harmonic convex for X e Rf. As its applications some analytic inequalities including the well-known Weierstrass inequalities are obtained. Copyright 2009 Y. Chu and Y. Lv. 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. 1. Introduction Throughout this paper we use Rn to denote the n-dimensional Euclidean space over the field of real numbers and Rf x x1zx2 . . xn e R n Xi 0 i 1 2 . n . For x x1 x2 . xn y y1 y2 . yn e Rf and a 0 we denote by x y x1 y1 x2 y2 . xn yn xy x1y1 x2y2 . xnyn ax ax1 ax2 . axn 1 _ 1 1 1 x x1 x2 xn For x x1 x2 . xn e R the Hamy symmetric function 1-3 was defined as Fn x r Fn xb x2 . xn r Ã. 1 i1 i2 --- ir n r nxi 7 1 1 r r 1 2 . n. 2 Journal of Inequalities and Applications Corresponding to this is the rth order Hamy mean ơn x r ơn X1 X2 x r Fn x r where n n n - r r . Hara et al. 1 established the following refinement of the classical arithmetic and geometric means inequality Gn x ơn x n ơn x n - 1 ơn x 2 ơn x 1 An x Here An x 1 n2 1 xi and Gn x n 1 xiý n denote the classical arithmetic and geometric means respectively. The paper 4 by Ku et al. contains some interesting inequalities including the fact that ơn x r r is log-concave the more results can also be found in the book 5 by Bullen. In 2 the Schur convexity of Hamy s