Sharp bounds for Sándor-Yang means in terms of some bivariate means
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摘要: 运用精细化的实分析方法,研究了Sándor-Yang平均SQA(a,b)、SQA(a,b)与算术平均A(a,b)和二次平均Q(a,b)凸组合以及算术平均A(a,b)和反调和平均C(a,b)凸组合的序关系.得到了关于Sándor-Yang平均SQA(a,b)、SQA(a,b)的四个精确双向不等式.
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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 inequalitiesα1Q(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 a≠b. 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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