The best bounds for Toader mean in terms of the centroidal and arithmetic means

In the paper, the authors discover the best constants $α_{1}$, $α_{2}$, $β_{1}$, and $β_{2}$ for the double inequalities $α_{1}\bar{C}(a,b)+(1-α_{1}) A(a,b)< T(a,b) <β_{1} \bar{C}(a,b)+(1-β_{1})A(a,b)$ and $\frac{α_{2}}{A(a,b)}+\frac{1-α_{2}}{\bar{C}(a,b)}<\frac1{T(a,b)} <\frac{β_{2}}{A(a,b)}+\frac{1-β_{2}}{\bar{C}(a,b)}$ to be valid for all $a,b>0$ with $a\ne b$, where $\bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\quad A(a,b)=\frac{a+b}2,$ and $$ T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2{\cos^2θ}+b^2{\sin^2θ}}\,\tdθ$$ are respectively the centroidal, arithmetic, and Toader means of two positive numbers $a$ and $b$. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.

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