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Zbl 1187.26013
Chu, Yu-Ming; Xia, Wei-Feng
Two sharp inequalities for power mean, geometric mean, and harmonic mean.
(English)
[J] J. Inequal. Appl. 2009, Article ID 741923, 6 p. (2009). ISSN 1029-242X/e

Summary: For $p\in\Bbb R$, the power mean of order $p$ of two positive numbers $a$ and $b$ is defined by $M_p(a,b)= ((a^p+b^p)/2)^{1/p}$, $p\ne 0$, and $M_p(a,b)= \sqrt{ab}$, $p=0$. In this paper, we establish two sharp inequalities as follows: $(2/3)G(a,b)+(1/3)H(a,b)\ge M_{-1/3}(a,b)$ and $(1/3)G(a,b)+ (2/3)H(a,b)\ge M_{-2/3}(a,b)$ for all $a,b>0$. Here $G(a,b)= \sqrt{ab}$ and $H(a,b)= 2ab/(a+b)$ denote the geometric mean and harmonic mean of $a$ and $b$, respectively.
MSC 2000:
*26E60 Means

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