Sharp generalized Seiffert mean bounds for Toader mean (Q642718)

Summary: For \(p \in [0, 1]\), the generalized Seiffert mean of two positive numbers \(a\) and \(b\) is defined by \[ S_p(a, b) = p(a - b) / \arctan[2p(a - b)/(a + b)],\quad 0 < p \leq 1,\;a \neq b;\quad (a + b)/2,\;p = 0,\;a \neq b;\;a, a = b. \] In this paper, we find the greatest value \(\alpha\) and least value \(\beta\) such that the double inequality \(S_\alpha(a, b) < T(a, b) < S_\beta(a, b)\) holds for all \(a, b > 0\) with \(a \neq b\), and give new bounds for the complete elliptic integrals of the second kind. Here, \(T(a, b) = (2/\pi) \int^{\pi/2}_0 \sqrt{a^2\cos^2 \theta + b^2\sin^2\theta}\,d\theta\) denotes the Toader mean of two positive numbers \(a\) and \(b\).