Sharp bounds for Gauss Lemniscate functions and Lemniscatic means
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^...
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AIMS Press
2021-05-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2021437?viewType=HTML |
| _version_ | 1828892239622832128 |
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| author | Wei-Mao Qian Miao-Kun Wang |
| author_facet | Wei-Mao Qian Miao-Kun Wang |
| author_sort | Wei-Mao Qian |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-13T13:30:49Z |
| format | Article |
| id | doaj.art-821db16019aa4c94867cd429cf3d1606 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-13T13:30:49Z |
| publishDate | 2021-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-821db16019aa4c94867cd429cf3d16062022-12-21T23:44:10ZengAIMS PressAIMS Mathematics2473-69882021-05-01677479749310.3934/math.2021437Sharp bounds for Gauss Lemniscate functions and Lemniscatic meansWei-Mao Qian0Miao-Kun Wang11. School of Continuing Education, Huzhou Vocational & Technical College (Huzhou Radio & Television University), Huzhou, 313000, China2. Department of Mathematics, Huzhou University, Huzhou 313000, ChinaFor $ 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.https://www.aimspress.com/article/doi/10.3934/math.2021437?viewType=HTMLlemniscate functionlemniscatic meanarithmetic meangeometric meansharp bounds |
| spellingShingle | Wei-Mao Qian Miao-Kun Wang Sharp bounds for Gauss Lemniscate functions and Lemniscatic means AIMS Mathematics lemniscate function lemniscatic mean arithmetic mean geometric mean sharp bounds |
| title | Sharp bounds for Gauss Lemniscate functions and Lemniscatic means |
| title_full | Sharp bounds for Gauss Lemniscate functions and Lemniscatic means |
| title_fullStr | Sharp bounds for Gauss Lemniscate functions and Lemniscatic means |
| title_full_unstemmed | Sharp bounds for Gauss Lemniscate functions and Lemniscatic means |
| title_short | Sharp bounds for Gauss Lemniscate functions and Lemniscatic means |
| title_sort | sharp bounds for gauss lemniscate functions and lemniscatic means |
| topic | lemniscate function lemniscatic mean arithmetic mean geometric mean sharp bounds |
| url | https://www.aimspress.com/article/doi/10.3934/math.2021437?viewType=HTML |
| work_keys_str_mv | AT weimaoqian sharpboundsforgausslemniscatefunctionsandlemniscaticmeans AT miaokunwang sharpboundsforgausslemniscatefunctionsandlemniscaticmeans |