Optimal evaluation of a Toader-type mean by power mean

In this paper, we present the best possible parameters p, q is an element of R such that the double inequality M-p(a, b) < T[A(a, b), Q(a, b)] < M-q(a, b) holds for all a, b > 0 with a not equal b, and we get sharp bounds for the complete elliptic integral epsilon(t) = integral(pi/2)(0) (1 - t(2) sin(2) theta)(1/2) d theta of the second kind on the interval (0,root 2/2), where T(a, b) = 2 pi integral(pi/2)(0) root a(2) cos(2) theta + b(2) sin(2) theta d theta, A(a, b) = (a + b)/2, Q(a, b) = root(a(2) + b(2))/2, M-r(a, b) = [(a(r) + b(r))/2](1/r) (r not equal 0), and M-0(a, b) = root ab are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.