华东师范大学学报(自然科学版) ›› 2017, Vol. ›› Issue (4): 41-51.doi: 10.3969/j.issn.1000-5641.2017.04.004

• 数学 • 上一篇    下一篇

Sándor-Yang平均关于一些二元平均凸组合的确界

徐会作   

  1. 温州广播电视大学 经管学院, 浙江 温州 325013
  • 收稿日期:2016-10-17 出版日期:2017-07-25 发布日期:2017-07-20
  • 作者简介:徐会作,男,讲师,研究方向为平均值理论、应用统计.E-mail:21888878@qq.com.
  • 基金资助:
    浙江广播电视大学科研课题(XKT-15G17)

Sharp bounds for Sándor-Yang means in terms of some bivariate means

XU Hui-zuo   

  1. School of Economics and Management, Wenzhou Broadcast and TV University, Wenzhou Zhejiang 325013, China
  • Received:2016-10-17 Online:2017-07-25 Published:2017-07-20

摘要: 运用精细化的实分析方法,研究了Sándor-Yang平均SQAa,b)、SQAa,b)与算术平均Aa,b)和二次平均Qa,b)凸组合以及算术平均Aa,b)和反调和平均Ca,b)凸组合的序关系.得到了关于Sándor-Yang平均SQAa,b)、SQAa,b)的四个精确双向不等式.

关键词: Schwab-Borchardt 平均, Sá, ndor-Yang 平均, 算术平均, 二次平均, 反调和平均

Abstract: This paper deals with the inequalities involving Sándor-Yang means derived from the Schwab-Borchardt mean using the method of real analysis. The convex com- binations of the arithmetic mean A(a,b) and quadratic Q(a,b) (or contra-harmonic mean C(a,b)) for the Sándor-Yang means SQA(a,b) and SQA(a,b) are disscused. The main results obtained are the sharp bounds of the two convex combinations, namely, the best possible parameters α1, α2, α3, α4, β1, β2, β3, β4 ∈ (0, 1), such that the double inequalities
α1Q(a,b) + (1-α1)A(a,b) < SQA(a,b) < β1Q(a,b) + (1-β1)A(a,b),
α2Q(a,b) + (1-α2)A(a,b) < SQA(a,b) < β2Q(a,b) + (1-β2)A(a,b),
α3C(a,b) + (1-α3)A(a,b) < SQA(a,b) < β3C(a,b) + (1-β3)A(a,b),
α4C(a,b) + (1-α4)A(a,b) < SQA(a,b) < β4C(a,b) + (1 -β4)A(a,b)
hold for all a, b > 0 and ab. Here A(a,b), Q(a,b) and C(a,b) denote respectively the classical arithmetic, quadratic, contra-harmonic means of a and b, SQA(a,b) and SQA(a,b) are two Sándor-Yang means derived from the Schwab-Borchardt mean.

Key words: Schwab-Borchardt mean, Sá, ndor-Yang mean, arithmetic mean, quadratic mean, contra-harmonic mean

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