OPTIMAL BOUNDS FOR TOADER MEAN IN TERMS OF ARITHMETIC AND CONTRAHARMONIC MEANS

We find the greatest value alpha(1) and alpha(2), and the least values beta(1) and beta(2), such that the double inequalities alpha C-1(a, b)+(1-alpha(1))A(a, b) < T(a, b) < beta C-1(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(a, b) hold for all a, b > 0 with a not equal b. As applications, we get new bounds for the complete elliptic integral of the second kind. Here, C(a, b) = (a(2) + b(2))/(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 denote the contraharmonic, arithmetic, and Toader means of two positive numbers a and b, respectively.