Sharp bounds for Gauss Lemniscate functions and Lemniscatic means

Abstract

For $ a, b > 0 $ with $ a\neq b $, the Gauss lemniscate mean $ \mathcal{LM}(a, b) $ is defined by $ \begin{equation*} \mathcal{LM}(a,b) = \left\{\begin{array}{lll} \frac{\sqrt{a^2-b^2}}{\left[{ {\rm{arcsl}}}\left(\sqrt[4]{1-b^2/a^2}\right)\right]^2}, \ &a>b,\\ \frac{\sqrt{b^2-a^2}}{\left[{ {\rm{arcslh}}}\left(\sqrt[4]{b^2/a^2-1}\right)\right]^2},\ &a<b, \end{array}\right. \end{equation*} $ where $ { {\rm{arcsl}}}(x) = \int_{0}^{x}\frac{dt}{\sqrt{1-t^4}} \ (|x| < 1) $ and $ {\rm{arcslh}}(x) = \int_{0}^{x}\frac{dt}{\sqrt{1+t^4}}\ (x\in \mathbb{R}) $ is the arc lemniscate sine and hyperbolic arc lemniscate sine functions respectively. In this paper, we mainly establish sharp two-parameter bounds for four symmetric and homogeneous means derived from $ \mathcal{LM}(a, b) $, $ \mathcal{LM}_{\mathcal{GA}}(a, b) = \mathcal{LM}(\mathcal{G}(a, b), \mathcal{A}(a, b)) $, $ \mathcal{LM}_{\mathcal{AG}}(a, b) = \mathcal{LM}(\mathcal{A}(a, b), \mathcal{G}(a, b)) $, $ \mathcal{LM}_{\mathcal{AQ}}(a, b) = \mathcal{LM}(\mathcal{A}(a, b), \mathcal{G}(a, b)) $ and $ \mathcal{LM}_{\mathcal{QA}}(a, b) = \mathcal{LM}(\mathcal{A}(a, b), \mathcal{G}(a, b)) $. The obtained results lead to several asymptotical inequalities for Lemniscate functions. Here $ \mathcal{A}(a, b) = (a+b)/2 $, $ \mathcal{G}(a, b) = \sqrt{ab} $ and $ \mathcal{Q}(a, b) = \sqrt{(a^2+b^2)/2} $ are the classical arithmetic, geometric, and quadratic means.

Keywords

• lemniscate function
• lemniscatic mean
• arithmetic mean
• geometric mean
• sharp bounds