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Optimal one-parameter mean bounds for the convex combination of arithmetic and logarithmic means. (English) Zbl 1333.26035
Summary: We find the greatest value \(p_1 = p_1(\alpha )\) and the least value \(p_2 = p_2(\alpha )\) such that the double inequality \(J_{p_1}(a,b) < \alpha A(a,b)+(1-\alpha )L(a,b)< J_{p_2}(a,b)\) holds for any \(\alpha \in (0,1)\) and all \(a,b>0\) with \(a\neq b\). Here, \(A(a,b)\), \(L(a,b)\) and \(J_p(a,b)\) denote the arithmetic, logarithmic and \(p\)th one-parameter means of two positive numbers \(a\) and \(b\), respectively.

MSC:
26E60 Means
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