Optimal combinations bounds of root-square and arithmetic means for Toader mean

We find the greatest values alpha(1) and alpha(2), and the least values beta(1) and beta(2), such that the double inequalities alpha S-1(a, b) (1 - alpha(1))A(a, b) < T (a, b) < beta S-1(a, b) + (1 - beta(1))A(a, b) and S-alpha 2 (a, b)A(1-alpha 2) (a, b) < T (a, b) < S-beta 2 (a, b) A(1-beta 2) (a, b) hold for all a, b > 0 with a not equal b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a, b) = [(a(2) + b(2))/2](1/2), A(a, b) = (a + b)/2, and T (a, b) = 2/pi integral(pi/2)(0)root a(2)cos(2) theta + b(2)sin(2) theta d theta denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.