类对称函数的Schur凸性和不等式

数学物理学报 ›› 2012, Vol. 32 ›› Issue (1): 80-89.

类对称函数的Schur凸性和不等式

龙波涌^1,2, 褚玉明¹

1.湖州师范学院数学系浙江湖州 313000;
2.$安徽大学数学科学学院合肥 230039

收稿日期:2009-05-14 修回日期:2011-05-29 出版日期:2012-02-25 发布日期:2012-02-25
基金资助:
国家自然科学基金(11071069)和浙江省高等学校创新团队基金(T200924)资助

The Schur Convexity and Inequalities for a Class of Symmetric Functions

LONG Bo-Yong^1,2, CHU Yu-Ming¹

1.Department of Mathematics, Huzhou Teachers College, Zhejiang Huzhou 313000;
2.College of Mathematics Science, Anhui University, Hefei 230039

Received:2009-05-14 Revised:2011-05-29 Online:2012-02-25 Published:2012-02-25
Supported by:
国家自然科学基金(11071069)和浙江省高等学校创新团队基金(T200924)资助

摘要/Abstract

摘要：

对x = (x₁, x₂,···, x_n) ∈ (0,1)ⁿ 和 r ∈ {1, 2,···, n} 定义对称函数
F_n(x, r) = F_n(x₁, x₂,···, x_n; r) =∏_{1≤i1<i2<···<ir≤n} ∑ _j=1^r(1+x_i₃/1- x_i₃)^1/r,

其中i₁, i₂, ···, i_r 是整数. 该文证明了F_n(x, r) 是(0,1)ⁿ 上的Schur凸、Schur乘性凸和Schur调和凸函数. 作为应用,利用控制理论建立了若干不等式.

关键词: Schur凸, Schur乘性凸, Schur调和凸

Abstract:

For x = (x₁, x₂,···, x_n) ∈ (0,1) and r ∈ {1, 2,···, n}, the symmetric function F_n(x, r) is defined by
F_n(x, r) = F_n(x₁, x₂,···, x_n; r) =∏_{1≤i1<i2<···<ir≤n} ∑ _j=1^r(1+x_i₃/1- x_i₃)^1/r,
where i₁, i₂, ···, i_r are integers. In this paper, it is proved that Fn(x,r) is Schur convex, Schur multiplicatively convex and Schur harmonic convex on (0,1)ⁿ. As applications, some inequalities are established by use of the theory of majorization.

Key words: Schur convex, Schur multiplicatively convex, Schur harmonic convex

中图分类号:

05E05

龙波涌, 褚玉明. 类对称函数的Schur凸性和不等式[J]. 数学物理学报, 2012, 32(1): 80-89.

LONG Bo-Yong, CHU Yu-Ming. The Schur Convexity and Inequalities for a Class of Symmetric Functions[J]. Acta mathematica scientia,Series A, 2012, 32(1): 80-89.

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