Optimal evaluation of a Toader-type mean by power mean

In this paper, we present the best possible parameters p, q such that the double inequality Mₚ(a,b)< T[A(a,b), Q(a,b)]< Mq(a,b) holds for all $a, b>0$ with a≠ b , and we get sharp bounds for the complete elliptic integral E(t)=∫ ₀π/2(1-t²sin²θ)1/2\,dθ of the second kind on the interval (0, √2/2) , where T(a,b)=2/π ∫ ₀π/2√a²cos²θ+b²sin²θ\,dθ , $A(a,b)=(a+b)/2$ , Q(a,b)=√(a²+b²)/2 , Mᵣ(a,b)=[(aʳ+bʳ)/2]1/r ( r≠0 ), and M₀(a,b)=√ab are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.