On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind

In the article, we prove that the double inequalities 1+(6p-7)r'/p+(5p - 6)r' pi tanh(-1)(r)/2r < kappa (r) < 1+(6q-7)r'/q+(5q-6)r'/q+(5q-6)r' pi tanh(-1)(r)/2r, qA(1,r) + (5q-6)G(1,r)/A(1,r + (6q-7)G(1,r) L (1,r) AGM (1,r) < pA(1,r) + (5p - 6)G(1,r)/A(1,r) + (6p - 7)G(1,r) L(1, r) hold for all r is an element of(0,1) if and only if p >= pi/2 = 1.570796 ... and q <= 89/69 = 1.289855 ... , where kappa(r) = integral(pi/2)(0)(1 - r(2) sin(2)t)(-1/2)dt is the complete elliptic integral of the first kind, tanh(-1) (r) = log[(1+r)/(1-r)]/2 is the inverse hyperbolic tangent function, r' = root 1-r(2), and A(1, r) = (1 + r)/2, G(1, r) = root r, L(1,r) = (r = 1 /log r and AGM(1, r) are the arithmetic, geometric, logarithmic and Gaussian arithmetic-geometric means of 1 and r, respectively. (C) 2018 Elsevier Inc. All rights reserved.