Optimal combinations bounds of root-square and arithmetic means for Toader mean

We find the greatest value α ₁ and α ₂, and the least values β ₁ and β ₂, such that the double inequalities α ₁ S(a,b)?+?(1???α ₁) A(a,b)?T(a,b)?β ₁ S(a,b)?+?(1???β ₁) A(a,b) and \(S^{\alpha_{2}}(a,b)A^{1-\alpha_{2}}(a,b)< T(a,b)< S^{\beta_{2}}(a,b)A^{1-\beta_{2}}(a,b)\) hold for all a,b?>?0 with a?≠?b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b)?=?(a ²?+?b ²)/2]^1/2, A(a,b)?=?(a?+?b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.