SHARP INEQUALITIES FOR THE TOADER MEAN OF ORDER-1 IN TERMS OF OTHER BIVARIATE MEANS

In the article, we present the best possible parameters alpha(1), alpha(2), alpha(3), alpha(4), beta(1), beta(2), beta(3), beta(4) is an element of R such that the double inequalities alpha(1)/H(a,b) + 1 - alpha(1)/G(a,b) < 1/T-1(a,b) < beta(1)/H(a,b) + 1 - beta(1)/G(a,b), alpha(2)/H(a,b) + 1 - alpha(2)/A(a,b) < 1/T-1(a,b) < beta(2)/H(a,b) + 1 - beta(2)/G(a,b), alpha(3)/H(a,b) + 1 - alpha(3)/L(a,b) < 1/T-1(a,b) < beta(3)/H(a,b) + 1 - beta(3)/L(a,b), alpha(4)/H(a,b) + 1 - alpha(4)/P(a,b) < 1/T-1(a,b) < beta(4)/H(a,b) + 1 - beta(4)/L(a,b), hold for all a,b > 0 with a not equal b, and provide several new bounds for the complete elliptic integral of the second kind, where T-1 (a,b) = (2/pi integral(pi/2)(0) root a(-1) cos(2) theta + b(-1) sin(2) theta d theta)(2) is the Toader mean of order 1, and H(a,b) = 2ab/(a + b), G(a,b) = root ab, L(a,b) = (a - b)/(log a log b), P(a,b) (a - b)/[2arcsin((a - b)/(a + b))] and A(a,b)= (a + b)/2 are the harmonic, geometric, logarithmic. Seiffert and athlunetic means, respectively.

Automated summary

In this article, a set of parameters and their inequalities are proposed to study new bounds for the complete elliptic integral of the second kind.