A double inequality for bounding Toader mean by the centroidal mean

In the paper, the authors find the best numbers $α$ and $β$ such that $\overline{C}\bigl(αa+(1-α)b,αb+(1-α)a\bigr)<T(a,b) <\overline{C}\bigl(βa+(1-β)b,βb+(1-β)a\bigr)$ for all $a,b>0$ with $a\ne b$, where $\overline{C}(a,b)={2\bigl(a^2+ab+b^2\bigr)}{3(a+b)}$ and $T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2{\cos^2θ}+b^2{\sin^2θ}}\,dθ$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$.

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