Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

Abstract In this paper, we present the best possible parameters α ( r ) $\alpha(r)$ and β ( r ) $\beta(r)$ such that the double inequality [ α ( r ) A r ( a , b ) + ( 1 − α ( r ) ) Q r ( a , b ) ] 1 / r < T D [ A ( a , b ) , Q ( a , b ) ] < [ β ( r ) A r ( a , b ) + ( 1 − β ( r ) ) Q r ( a , b ) ] 1 / r $$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$ holds for all r ≤ 1 $r\leq 1$ and a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , and we provide new bounds for the complete elliptic integral E ( r ) = ∫ 0 π / 2 ( 1 − r 2 sin 2 θ ) 1 / 2 d θ $\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta$ ( r ∈ ( 0 , 2 / 2 ) ) $(r\in (0, \sqrt{2}/2))$ of the second kind, where T D ( a , b ) = 2 π ∫ 0 π / 2 a 2 cos 2 θ + b 2 sin 2 θ d θ $TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$ , A ( a , b ) = ( a + b ) / 2 $A(a,b)=(a+b)/2$ and Q ( a , b ) = ( a 2 + b 2 ) / 2 $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ are the Toader, arithmetic, and quadratic means of a and b, respectively