Sharp Generalized Seiffert Mean Bounds for Toader Mean

For p ∈ [0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a, b) = p(a − b)/arctan[2p(a − b)/(a + b)], 0 < p ≤ 1, a ≠ b; (a + b)/2, p = 0, a ≠ b; a, a = b. In this paper, we find the greatest value α and least value β such that the double inequality Sα(a, b) < T(a, b) < Sβ(a, b) holds for all a, b > 0 with a ≠ b, and give new bounds for the complete elliptic integrals of the second kind. Here, denotes the Toader mean of two positive numbers a and b.