The Best Bounds for Toader Mean in Terms of the Centroidal and Arithmetic Means

In the paper, the authors discover the best constants alpha(1), alpha(2), beta(1), and beta(2) for the double inequalities alpha(1)(C) over bar (a, b) + (1 - alpha(1))A(a, b) < T(a, b) < beta(1)(C) over bar (a, b) + (1 - beta(1))A(a, b) and alpha(2)/A(a, b) + 1 - alpha(2)/C(a, b) < 1/T(a, b) < beta(2)/A(a, b) + 1 - beta(2)/(C) over bar (a, b) to be valid for all a, b > 0 with a not equal b, where (C) over bar (a, b) = 2(a(2) + ab + b(2))/3(a + b), 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 are respectively the centroidal, arithmetic, and Toader means of two positive numbers a and b. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.