Abstract. In this paper, we investigate the Schur m-power convexity of the generalized hamy symmetric function1 r for x ∈ R n + and r ∈ N with 1 r n , which generalizes some known results.Mathematics subject classification (2010): 05E05, 26B25.
We investigate the conditions under which the symmetric functions , (x, ) = ∏ 1≤ 1 < 2 <⋅⋅⋅< ≤ ((∑ =1 ) 1/ ) , = 1, 2, . . . , , are Schur -power convex for ∈ R ++ and > 0. As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving the th power mean and the arithmetic, the geometric, or the harmonic means are presented.Rovent,a proved that (4) is a Schur convex function on .) They proved that * (x) is Schur convex, Schur geometrically and harmonically convex on . Recently, Yang [7-9] generalized the notion of Schur convexity to Schur m-power convexity, which contains the Schur convexity, Schur geometrical convexity, and Schur harmonic convexity. Moreover, he discussed Schur m-power convexity of Stolarsky means [7], Gini means [8], and Daróczy means [9]. Wang and Yang showed that generalized Hamy symmetric function [10] and a class of symmetric functions [11] are Schur m-power convex. Now we define the more general dual form of symmetric function.
In this paper we present a geometric inequality for a finite number of points on an (n -1 )-dimensional sphere S~-I(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E ~. Classifications (1991): 51M16, 51M25.
Mathematics Subject
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