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Sharp one-parameter mean bounds for Yang mean. (English) Zbl 1400.26071
Summary: We prove that the double inequality \(J_\alpha(a, b) < U(a, b) < J_\beta(a, b)\) holds for all \(a, b > 0\) with \(a \neq b\) if and only if \(\alpha \leq \sqrt{2} /(\pi - \sqrt{2}) = 0.8187 \cdots\) and \(\beta \geq 3 / 2\), where \(U(a, b) = (a - b) / [\sqrt{2} \arctan((a - b) / \sqrt{2 a b})]\), and \(J_p(a, b) = p(a^{p + 1} - b^{p + 1}) / [(p + 1)(a^p - b^p)](p \neq 0, - 1)\), \(J_0(a, b) = (a - b) /(\log a - \log b)\), and \(J_{- 1}(a, b) = a b(\log a - \log b) /(a - b)\) are the Yang and \(p\)th one-parameter means of \(a\) and \(b\), respectively.

MSC:
26E60 Means
26D07 Inequalities involving other types of functions
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