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2017 >Issue 4: 41 - 51

DOI: https://doi.org/10.3969/j.issn.1000-5641.2017.04.004

Mathematics

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

  • XU Hui-zuo
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  • School of Economics and Management, Wenzhou Broadcast and TV University, Wenzhou Zhejiang 325013, China

Received date: 2016-10-17

  Online published: 2017-07-20

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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 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

Cite this article

XU Hui-zuo . Sharp bounds for Sándor-Yang means in terms of some bivariate means[J]. Journal of East China Normal University(Natural Science), 2017 , (4) : 41 -51 . DOI: 10.3969/j.issn.1000-5641.2017.04.004

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