In the article, we prove that the double inequalities Mα(a,b)<Q(a,b)eG(a,b)/U(a,b)-1<Mβ(a,b),Mλ(a,b)<G(a,b)eQ(a,b)/V(a,b)-1<Mμ(a,b)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} M_{\alpha }(a,b)< & {} Q(a,b)e^{G(a,b)/U(a,b)-1}<M_{\beta }(a,b), \\ M_{\lambda }(a,b)< & {} G(a,b)e^{Q(a,b)/V(a,b)-1}<M_{\mu }(a,b) \end{aligned}$$\end{document}hold for all a,b>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$a, b>0$$\end{document} with a≠b\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$a\ne b$$\end{document} if and only if α≤2log2/(2+log2)=0.5147⋯\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha \le 2\log 2/(2+\log 2)=0.5147\cdots $$\end{document}, β≥2/3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\beta \ge 2/3$$\end{document}, λ≤2log2/(2-log2)=1.0607⋯\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\lambda \le 2\log 2/(2-\log 2)=1.0607\cdots $$\end{document} and μ≥4/3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mu \ge 4/3$$\end{document}, where Mp(a,b)=[(ap+bp)/2]1/p\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$M_{p}(a,b)=[(a^{p}+b^{p})/2]^{1/p}$$\end{document}(p≠0)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(p\ne 0)$$\end{document}, M0(a,b)=G(a,b)=ab\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$M_{0}(a,b)=G(a,b)=\sqrt{ab}$$\end{document}, Q(a,b)=(a2+b2)/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$$\end{document}, U(a,b)=(a-b)/[2˘arctan((a-b)/2ab)]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$U(a,b)=(a-b)/[\sqrt{2\breve{}}\arctan ((a-b)/\sqrt{2ab})]$$\end{document} and V(a,b)=(a-b)/[2sinh-1((a-b)/2ab)]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$V(a,b)=(a-b)/[\sqrt{2}\sinh ^{-1}((a-b)/\sqrt{2ab})]$$\end{document} are respectively the pth power, geometric, quadratic, first Yang and second Yang means, and sinh-1(x)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sinh ^{-1}(x)$$\end{document} is the inverse hyperbolic sine function.
