A double inequality for bounding Toader mean by the centroidal mean

In this paper, the authors find the best numbers α and β such that C¯αa+(1−α)b,αb+(1−α)a<T(a,b)<C¯βa+(1−β)b,βb+(1−β)a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{C}\left(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\right)<T(a,b) <\overline{C}\left(\beta a+(1-\beta)b,\beta b+(1-\beta)a\right) $$\end{document}for all a,b>0 with a≠b, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)}$\end{document} and T(a,b)=2π∫0π/2a2cos2𝜃+b2sin2𝜃d𝜃\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T(a,b)=\frac {2}{\pi }{\int }_{0}^{{\pi }/{2}}\sqrt {a^{2}{\cos ^{2}{\theta }}+b^{2}{\sin ^{2}{\theta }}}\, \mathrm {d} \theta $\end{document} denote respectively the centroidal mean and Toader mean of two positive numbers a and b.