In the article, we present the best possible parameters α1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _1$$\end{document}, α2\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _2$$\end{document}, α3\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _3$$\end{document}, α4\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _4$$\end{document}, β1\documentclass[12pt]{minimal}
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\begin{document}$$\beta _1$$\end{document}, β2\documentclass[12pt]{minimal}
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\begin{document}$$\beta _2$$\end{document}, β3\documentclass[12pt]{minimal}
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\begin{document}$$\beta _3$$\end{document} and β4\documentclass[12pt]{minimal}
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\begin{document}$$\beta _4$$\end{document} on the interval (0, 1) such that the double inequalities Gα1(a,b)<LMGAa,b<Gβ1(a,b)\documentclass[12pt]{minimal}
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\begin{document}$${G_{{\alpha _1}}}(a,b)< L{M_{GA}}\left( {a,b} \right) < {G_{{\beta _1}}}(a,b)$$\end{document}, Gα2(a,b)<LMAGa,b<Gβ2(a,b)\documentclass[12pt]{minimal}
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\begin{document}$${G_{{\alpha _2}}}(a,b)< L{M_{AG}}\left( {a,b} \right) < {G_{{\beta _2}}}(a,b)$$\end{document}, Qα3(a,b)<LMAQa,b<Qβ3(a,b)\documentclass[12pt]{minimal}
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\begin{document}$${Q_{{\alpha _3}}}(a,b)< L{M_{AQ}}\left( {a,b} \right) < {Q_{{\beta _3}}}(a,b)$$\end{document} and Qα4(a,b)<LMQAa,b<Qβ4(a,b)\documentclass[12pt]{minimal}
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\begin{document}$${Q_{{\alpha _4}}}(a,b)< L{M_{QA}}\left( {a,b} \right) < {Q_{{\beta _4}}}(a,b)$$\end{document} hold for a,b>0\documentclass[12pt]{minimal}
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\begin{document}$$a,b > 0$$\end{document} with a≠b\documentclass[12pt]{minimal}
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\begin{document}$$a \ne b$$\end{document}, where Gp(a,b)\documentclass[12pt]{minimal}
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\begin{document}$${G_p}(a,b)$$\end{document} and Qp(a,b)\documentclass[12pt]{minimal}
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\begin{document}$${Q_p}(a,b)$$\end{document} are respectively the one-parameter geometric and quadratic means, LMGA(a,b)\documentclass[12pt]{minimal}
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\begin{document}$$L{M_{GA}}(a,b)$$\end{document}, LMAG(a,b)\documentclass[12pt]{minimal}
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\begin{document}$$L{M_{AG}}(a,b)$$\end{document}, LMAQ(a,b)\documentclass[12pt]{minimal}
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\begin{document}$$L{M_{AQ}}(a,b)$$\end{document} and LMQA(a,b)\documentclass[12pt]{minimal}
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\begin{document}$$L{M_{QA}}(a,b)$$\end{document} are four lemniscatic means of a and b. As applications, some new bounds for the arc lemniscate functions are given.
