| 孙明保,张映辉,张再云,陈南博.一类对称函数的Schur 凸性及其应用[J].数学年刊A辑,2017,38(2):177~190 |
| 一类对称函数的Schur 凸性及其应用 |
| Schur Convexity of a Class of Symmetric Functions with Its Applications |
| Received:April 23, 2015 Revised:May 09, 2016 |
| DOI：10.16205/j.cnki.cama.2017.0014 |
| 中文关键词: Symmetric functions, Schur convex, Schur multiplicative convex, Schur harmonic convex, Theory of majorization |
| 英文关键词:Symmetric functions, Schur convex, Schur multiplicative convex, Schur harmonic convex, Theory of majorization |
| 基金项目:本文受到国家自然科学基金(No.11271118,No.10871061,No.11301172,No.11671101),
湖南省自然科学基金(No.12JJ3002,No.2016JJ2061),湖南省教育厅资助科研项目(No.11A043,No.15B102),
湖南省重点学科建设项目(No.201176)和湖南省高校科技创新团队支持计划(No.2014207)的资助. |
| Author Name | Affiliation | E-mail | | SUN Mingbao | Corresponding author. School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, Hunan, China. | sun_mingbao@163.com | | ZHANG Yinghui | School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, Hunan, China. | zhangyinghui1009@yahoo.com.cn | | ZHANG Zaiyun | School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, Hunan, China. | zhangzaiyun1226@126.com | | CHEN Nanbo | School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, Hunan, China. | flyingnb@126.com |
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| 中文摘要: |
| 对 $x=(x_1, \cdots, x_n)\in [0, 1)^n\cup (1, +\infty)^n$, 定义对称函数
$${F_n}(x,r)={F_n}(x_1, x_2, \cdots, x_n; r)=\sum_{i_1+i_2+\cdots+i_n=r}\Big({\frac{1+x_1}{1-x_1}}\Big)^{i_1}
\Big({\frac{1+x_2}{1-x_2}}\Big)^{i_2}\cdots\Big({\frac{1+x_n}{1-x_n}}\Big)^{i_n},$$
其中$r\in\mathbb{N},\ i_1, i_2, \cdots , i_n$ 为非负整数.
研究了${F_n}(x,r)$的Schur 凸性、Schur 乘性凸性和Schur 调和凸性.
作为应用, 用控制理论建立了一些不等式, 特别地, 给出了高维空间的一些新的几何不等式. |
| 英文摘要: |
| For $x=(x_1, \cdots, x_n)\in [0, 1)^n\cup (1, +\infty)^n$, the
symmetric function $F_n(x,r)$ is defined by
$$
{F_n}(x,r)={F_n}(x_1, x_2, \cdots, x_n; r)=\sum_{i_1+i_2+\cdots+i_n=r}\Big({\frac{1+x_1}{1-x_1}}\Big)^{i_1}
\Big({\frac{1+x_2}{1-x_2}}\Big)^{i_2}\cdots\Big({\frac{1+x_n}{1-x_n}}\Big)^{i_n},
$$
where $r\in\mathbb{N}$, and $i_1, i_2, \cdots , i_n$ are
non-negative integers. In this paper, the Schur convexity, Schur
multiplicative convexity and Schur harmonic convexity of
${F_n}(x,r)$ are investigated. As applications, the authors
establish some inequalities by use of the theory of majorization. In
particular, the authors give some new geometric inequalities in the
$n$-dimensional space. |
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