华东师范大学学报（自然科学版） >

2018 , Vol. 2018 >Issue 4: 23 - 31

DOI: https://doi.org/10.3969/j.issn.1000-5641.2018.04.003

数学

关于,Neuman-Sándor,平均的两个最佳不等式

  • 杨月英 ,
  • 马萍
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  • 湖州职业技术学院 机电与汽车工程学院, 浙江 湖州 313000
杨月英,女,副教授,研究方向为解析不等式.E-mail:2004002@hzvtc.net.cn

收稿日期: 2017-03-27

  网络出版日期: 2018-07-19

基金资助

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）

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Two optimal inequalities for Neuman-Sándor means

  • YANG Yue-ying ,
  • MA Ping
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  • Mechanic Electronic and Automobile Engineering College, Huzhou Vocational & Technical College, Huzhou Zhejiang 313000, China

Received date: 2017-03-27

  Online published: 2018-07-19

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摘要

ZHAIYAO{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H（a，b）（及调和平均，H（a，b））凸组合的序关系.发现了最大值λ1，λ2∈（0，1）和最小值μ1，μ2∈（0，1）使得双边不等式
λ1D（a，b）+（1-λ1）H（a，b）<M（a，b）<μ1D（a，b）+（1-μ1）H（a，b），
λ2D（a，b）+（1-λ2） H（a，b）<M（a，b）<μ2D（a，b）+（1-μ2） H（a，b）
对所有a，b>0且a≠b成立.

关键词： Neuman-Sá; ndor平均; 反调和平均; 根平方平均; 调和平均; 不等式

本文引用格式

杨月英 , 马萍 . 关于,Neuman-Sándor,平均的两个最佳不等式[J]. 华东师范大学学报（自然科学版）, 2018 , 2018(4) : 23 -31 . DOI: 10.3969/j.issn.1000-5641.2018.04.003

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 H(a, b) (or harmonic mean H(a,b)) for the Neuman-Sándor mean M(a, b) are discussed. We find the maximum values λ1, λ2 ∈ (0, 1) and the minimum values μ1, μ2 ∈ (0, 1) such that the two-sided inequalities
λ1D(a, b) + (1-λ1)H(a, b) < M(a, b) < μ1D(a, b) + (1-μ1)H(a, b),
λ2D(a, b) + (1-λ2)H(a,b) < M(a, b) < μ2D(a, b) + (1-μ2)H(a,b)
hold for all a, b > 0 with a≠b.

Key words： Neuman-Sándor mean; contra-harmonic mean; root-square mean; harmonic mean; inequalities

参考文献

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