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

Abstract

In the article, we prove that the double inequalities1+(6p−7)r′p+(5p−6)r′πtanh−1⁡(r)2r<K(r)<1+(6q−7)r′q+(5q−6)r′πtanh−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∈(0,1) if and only if p≥π/2=1.570796⋯ and q≤89/69=1.289855⋯, where K(r)=∫0π/2(1−r2sin2⁡t)−1/2dt 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′=1−r2, and A(1,r)=(1+r)/2, G(1,r)=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.