关于, Neuman-Sándor, 平均的两个最佳不等式

杨月英; 马萍

doi:10.3969/j.issn.1000-5641.2018.04.003

关于, Neuman-Sándor, 平均的两个最佳不等式

doi: 10.3969/j.issn.1000-5641.2018.04.003

杨月英,
马萍

湖州职业技术学院机电与汽车工程学院, 浙江湖州 313000

基金项目:

湖州职业技术学院教改课题 2016xj26

浙江广播电视大学科学研究课题 XKT-17G26

详细信息

作者简介:
杨月英, 女, 副教授, 研究方向为解析不等式.E-mail:2004002@hzvtc.net.cn

中图分类号: O178
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- 文章访问数: 129
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- 被引次数: 0
出版历程
- 收稿日期: 2017-03-27
- 刊出日期: 2018-07-25

Two optimal inequalities for Neuman-Sándor means

Mechanic Electronic and Automobile Engineering College, Huzhou Vocational & Technical College, Huzhou Zhejiang 313000, China

摘要

摘要: 运用实分析方法, 研究了Neuman-Sándor平均$M(a, b)$与第二类反调和平均$D(a, b)$和调和根平方平均$\overline{H}(a, b)$ (及调和平均$H(a, b)$)凸组合的序关系.发现了最大值$\lambda_{1}, \lambda_{2}\in(0, 1)$和最小值$\mu_{1}, \mu_{2}\in(0, 1)$使得双边不等式 $\lambda_{1}D(a,b)+(1-\lambda_{1})\overline{H}(a,b) <M(a,b)<\mu_{1}D(a,b)+(1-\mu_{1})\overline{H}(a,b), \\ \lambda_{2}D(a,b)+(1-\lambda_{2})H(a,b)<M(a,b)<\mu_{2}D(a,b)+(1-\mu_{2})H(a,b)$ 对所有$a, b>0$且$a\neq b$成立.
- Neuman-Sándor平均 /
- 反调和平均 /
- 根平方平均 /
- 调和平均 /
- 不等式
Abstract: This paper deals with the inequalities involving Neuman-Sándor means using methods of real analysis. The convex combinations of the second contra-harmonic mean $D(a, b)$ and the harmonic root-square mean $\overline{H}(a, b)$ (or harmonic mean $H(a, b)$) for the Neuman-Sándor mean $M(a, b)$ are discussed. We find the maximum values $\lambda_{1}, \lambda_{2}\in(0, 1)$ and the minimum values $\mu_{1}, \mu_{2}\in(0, 1)$ such that the two-sided inequalities $\lambda_{1}D(a,b)+(1-\lambda_{1})\overline{H}(a,b) <M(a,b)<\mu_{1}D(a,b)+(1-\mu_{1})\overline{H}(a,b), \\ \lambda_{2}D(a,b)+(1-\lambda_{2})H(a,b)<M(a,b)<\mu_{2}D(a,b)+(1-\mu_{2})H(a,b)$ hold for all $a, b>0$ with $a\neq b$.
- Neuman-Sándor mean /
- contra-harmonic mean /
- root-square mean /
- harmonic mean /
- inequalities

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参考文献(14)

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