Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral. (20th July 2016) - Wang, Hua; Qian, Wei-Mao; Chu, Yu-Ming
Abstract : We present the best possible parameters α 1, β 1, α 2, β 2 ∈ R and α 3, β 3 ∈ ( 1 / 2, 1 ) such that the double inequalities Q α 1 ( a, b ) A 1 - α 1 ( a, b ) < A G [ A ( a, b ), Q ( a, b ) ] < Q β 1 ( a, b ) A 1 - β 1 ( a, b ), α 2 Q ( a, b ) + ( 1 - α 2 ) A ( a, b ) < A G [ A ( a, b ), Q ( a, b ) ] < β 2 Q ( a, b ) + ( 1 - β 2 ) A ( a, b ), Q [ α 3 a + ( 1 - α 3 ) b, α 3 b + ( 1 - α 3 ) a ] < A G [ A ( a, b ), Q ( a, b ) ] < Q [ β 3 a + ( 1 - β 3 ) b, β 3 b + ( 1 - β 3 ) a ] hold for all a, b > 0 with a ≠ b, where A ( a, b ), Q ( a, b ), and A G ( 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.