Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

Abstract

We present the best possible parameters α1,β1,α2,β2∈R and α3,β3∈(1/2,1) such that the double inequalities Qα1(a,b)A1-α1(a,b)0 with a≠b, where A(a,b), Q(a,b), and AG(a,b) are the arithmetic, quadratic, and Gauss arithmetic-geometric means of a and b, respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.