Optimal bounds for Gaussian arithmetic-geometric mean with applications to complete elliptic integral
Summary: We present the best possible parameters \(\alpha_1, \beta_1, \alpha_2, \beta_2 \in \mathbb{R}\) and \(\alpha_3, \beta_3 \in(1 / 2,1)\) such that the double inequalities \(Q^{\alpha_1}(a, b) A^{1 - \alpha_1}(a, b) < \mathrm{AG} [A(a, b), Q(a, b)] < Q^{\beta_1}(a, b) A^{1 - \beta_1}(a, b)\), \(\alpha_2 Q(a, b) +(1 - \alpha_2) A(a, b) < \mathrm{AG} [A(a, b), Q(a, b)] < \beta_2 Q(a, b) +(1 - \beta_2) A(a, b)\), \(Q [\alpha_3 a +(1 - \alpha_3) b, \alpha_3 b +(1 - \alpha_3) a] < \mathrm{AG} [A(a, b), Q(a, b)] < Q [\beta_3 a +(1 - \beta_3) b, \beta_3 b +(1 - \beta_3) a]\) hold for all \(a, b > 0\) with \(a \neq b\), where \(A(a, b)\), \(Q(a, b)\), and \(\mathrm{AG}(a, b)\) 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.
scientific article; zbMATH DE number 6614000
scientific article