SHARP BOUNDS FOR THE TOADER MEAN OF ORDER 3 IN TERMS OF ARITHMETIC, QUADRATIC AND CONTRAHARMONIC MEANS

In the article, we present the best possible parameters alpha(1), beta(1), alpha(2), beta(2) is an element of R and alpha(3), beta(3) is an element of [1/2, 1] such that the double inequalities alpha C-1(a, b) + (1 - alpha(1))A(a, b) < T-3(a, b) < beta C-1(a, b) + (1 - beta(1))A(a, b), alpha C-2(a, b) + (1 - alpha(2))Q(a, b) < T-3(a, b) < beta C-2(a, b) + (1 - beta(2))Q(a, b), C(alpha(3); a, b) < T-3(a, b) < C(beta(3); a, b) hold for a, b > 0 with a not equal b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q(a, b) = root(a(2) + b(2))/2 is the quadratic mean, C (a, b) = (a(2) + b(2))/(a + b) is the contra-harmonic mean, C (p; a, b) = C[pa + (1 - p)b, pb + (1 - p)a] is the one-parameter contra-harmonic mean and T-3(a, b) = (2/pi integral(pi/2)(0) root a(3) cos(2) theta + b(3) sin(2) theta d theta)(2/3) is the Toader mean of order 3. (C) 2020 Mathematical Institute Slovak Academy of Sciences