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摘要
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.
