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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:
26E60 Means
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