关于幂平均、调和平均和指数平均的最佳不等式

数学物理学报 ›› 2011, Vol. 31 ›› Issue (5): 1377-1384.

关于幂平均、调和平均和指数平均的最佳不等式

史明宇¹, 褚玉明², 蒋月评³

1.河北大学数学与计算机学院 |河北保定 071002|2.湖州师范学院数学系 |浙江 |湖州 313000|3.湖南大学数学与计量经济学院 |长沙 410082

收稿日期:2009-11-15 修回日期:2010-12-06 出版日期:2011-10-25 发布日期:2011-10-25
基金资助:
国家自然科学基金(11071069)、浙江省自然科学基金(Y7080106)和浙江高等学校创新团队基金

Optimal Inequalities Related to the Power, Harmonic and Identric Means

SHI Ming-Yu¹, CHU Yu-Ming², JIANG Yue-Ping³

1.College of Mathematics and Computer Science, Hebei University, Hebei Baoding 071002;
2.Department of Mathematics, Huzhou Teachers College, Zhejiang Huzhou 313000;
3.College of Mathematics and Econometrics, |Hunan University, Changsha 410082

Received:2009-11-15 Revised:2010-12-06 Online:2011-10-25 Published:2011-10-25
Supported by:
国家自然科学基金(11071069)、浙江省自然科学基金(Y7080106)和浙江高等学校创新团队基金

摘要/Abstract

摘要：

对任意 a, b>0 , 该文建立了如下两个不等式: H^α(a, b)I^1-α(a, b)≥M_2-5α/3(a, b) 对 α∈(2/5,1), 成立及H^α(a, b)I^1-α(a, b)≤M_2-5α/3(a, b)对α∈(1/25, 4/25). 成立. 其中 H(a, b), I(a, b)和M_p(a, b) 分别表示两个正数a 与b 的调和平均、指数平均和p 阶幂平均.

关键词: 幂平均, 指数平均, 调和平均

Abstract:

For any a, b>0 , the authors present the following two optimal inequalities: H^α(a, b)I^1-α(a, b)≥M_2-5α/3(a, b) for α∈(2/5,1), and H^α(a, b)I^1-α(a, b)≤M_2-5α/3(a, b) for α∈(1/25, 4/25). Here, H(a, b), I(a, b) and M_p(a, b) denote the harmonic, identric and power means of order of p of two positive numbers a and b, respectively.

Key words: Power mean, Identric mean, Harmonic mean

中图分类号:

26E60

史明宇, 褚玉明, 蒋月评. 关于幂平均、调和平均和指数平均的最佳不等式[J]. 数学物理学报, 2011, 31(5): 1377-1384.

SHI Ming-Yu, CHU Yu-Ming, JIANG Yue-Ping. Optimal Inequalities Related to the Power, Harmonic and Identric Means[J]. Acta mathematica scientia,Series A, 2011, 31(5): 1377-1384.

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