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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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##### References:
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