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Sharp inequalities between one-parameter and power means. (English) Zbl 1299.26069
Summary: For all \(a, b>0\) with \(a\neq b\), we prove that \(J_{p}(a, b)>M_{\frac{1+2p}{3}}(a, b)\) for \(p\in(-2, -\frac{1}{2})\cup(1, +\infty)\) and \(J_{p}(a, b)<M_{\frac{1+2p}{3}}(a, b)\) for \(p\in(-\infty,-2)\cup (-\frac{1}{2}, 1)\), and the parameter \(\frac{1+2p}{3}\) in either case is the best possible. Here, \(J_{p}(a, b)\) and \(M_{p}(a, b)\) are the one-parameter and power means of order \(p\) of two positive numbers \(a\) and \(b\), respectively.

MSC:
26E60 Means
26D20 Other analytical inequalities
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