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Two sharp inequalities for power mean, geometric mean, and harmonic mean. (English) Zbl 1187.26013
Summary: For $$p\in\mathbb R$$, the power mean of order $$p$$ of two positive numbers $$a$$ and $$b$$ is defined by $$M_p(a,b)= ((a^p+b^p)/2)^{1/p}$$, $$p\neq 0$$, and $$M_p(a,b)= \sqrt{ab}$$, $$p=0$$. In this paper, we establish two sharp inequalities as follows: $$(2/3)G(a,b)+(1/3)H(a,b)\geq M_{-1/3}(a,b)$$ and $$(1/3)G(a,b)+ (2/3)H(a,b)\geq M_{-2/3}(a,b)$$ 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:
 2.6e+61 Means
Full Text:
References:
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