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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