A sharp lower bound for the complete elliptic integrals of the first kind

Let K(r) be the complete elliptic integrals of the first kind and arthr denote the inverse hyperbolic tangent function. We prove that the inequality 2/pi K(r) > [1 - lambda +lambda (arthr/r)(q)](1/q) holds for r is an element of (0, 1) with the best constants lambda = 3/4 and q = 1/10. This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic-geometric mean in terms of logarithmic and arithmetic means.