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Zbl 1223.26049
Chu, Yuming; Long, Boyong
Sharp inequalities between means. (English)
[J] Math. Inequal. Appl. 14, No. 3, 647-655 (2011). ISSN 1331-4343

Summary: For $p\in\Bbb R$, the $p$-th power mean $M_p(a,b)$, arithmetic mean $A(a,b)$, geometric mean $G(a,b)$, and harmonic mean $H(a,b)$ of two positive numbers $a$ and $b$ are defined by $$M_p(a,b)=\cases\left(\frac{a^p+b^p}{w}\right),\quad p\ne 0,\\ \sqrt{ab},\quad p=0\endcases,$$ $A(a,b)=(a+b)/2$, $G(a,b)=\sqrt{ab}$, and $H(a,b)=2ab/(a+b)$, respectively. In this paper, we answer the questions: For $\alpha\in(0,1)$, what are the greatest values $p$, $r$ and $m$, and the least values $q$, $s$ and $n$, such that the inequalities $M_p(a,b)\le A^\alpha(a,b)G^{1-\alpha}(a,b)\le M_q(a,b)$, $M_r(a,b)\le G^\alpha(a,b)H^{1-\alpha}(a,b)\le M_s(a,b)$ and $M_m(a,b)\le A^\alpha(a,b)H^{1-\alpha}(a,b)\le M_n(a,b)$ hold for all $a,b>0$?

MSC 2000:: *26E60 Means

Keywords: power mean; arithmetic mean; geometric mean; harmonic mean

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