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Sharp bounds for Seiffert mean in terms of contraharmonic mean. (English) Zbl 1231.26034
Summary: We find the greatest value \(\alpha\) and the least value \(\beta\) in \((1/2, 1)\) such that the double inequality \(C(\alpha a + (1 - \alpha) b, \alpha b + (1 - \alpha)a) < T(a, b) < C(\beta a + (1 - \beta)b, \beta b + (1 - \beta)a)\) holds for all \(a, b > 0\) with \(a \neq b\). Here, \(T(a, b) = (a - b)/[2 \arctan ((a - b)/(a + b))]\) and \(C(a, b) = (a^2 + b^2)/(a + b)\) are the Seiffert and contraharmonic means of \(a\) and \(b\), respectively.

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