Sharp bounds for Seiffert mean in terms of contraharmonic mean (Q655603)

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.