Optimal bounds for a Gaussian Arithmetic-Geometric type mean by quadratic and contraharmonic means

Abstract

In this paper, we present the best possible parameters ${\alpha }_i,{\beta }_i(i=1,2,3)$ and ${\alpha }_4,{\beta }_4\in (1/2,1)$ such that the double inequalities \begin{align*} α_1Q(a,b)+(1-α_1)C(a,b)&<AG_{Q,C}(a,b)<β_1Q(a,b)+(1-β_1)C(a,b),\\ \qquad\ Q^{α_2}(a,b)C^{1-α_2}(a,b)&<AG_{Q,C}(a,b)<Q^{β_2}(a,b)C^{1-β_2}(a,b),\\ \frac{Q(a,b)C(a,b)}{α_3Q(a,b)+(1-α_3)C(a,b)}&<AG_{Q,C}(a,b)<\frac{Q(a,b)C(a,b)}{β_3Q(a,b)+(1-β_3)C(a,b)},\\ C\left(\sqrt{α_4a^2+(1-α_4)b^2},\sqrt{(1-α_4)a^2+α_4b^2}\right)&<AG_{Q,C}(a,b)<C\left(\sqrt{β_4a^2+(1-β_4)b^2},\sqrt{(1-β_4)a^2+β_4b^2}\right) \end{align*} hold for all $a,b>0$ with $a\ne b$, where $Q(a,b)$, $C(a,b)$ and $AG(a,b)$ are the quadratic, contraharmonic and Arithmetic-Geometric means, and $A{G}_{Q,C}(a,b)=AG[Q(a,b),C(a,b)]$. As consequences, we present new bounds for the complete elliptic integral of the first kind. Keywords: Arithmetic-Geometric mean, Complete elliptic integral, Quadratic mean, Contraharmonic mean

arXiv comment13 pages