LAUSR

In this paper, we present the best possible parameters α ( r ) $\alpha (r)$ , β ( r ) $\beta (r)$ such that the double inequality [ α ( r ) M r ( a , b ) + ( 1 − α ( r ) ) N r ( a , b ) ] 1 / r < TD [ M ( a , b ) , N ( a , b ) ] < [ β ( r ) M r ( a , b ) + ( 1 − β ( r ) ) N r ( a , b ) ] 1 / r , $$\begin{aligned} {}[\alpha (r)M^{r}(a,b)+(1-\alpha (r))N^{r}(a,b)] ^{1/r} < &\operatorname{TD}\bigl[M(a,b),N(a,b)\bigr] \\ < &\bigl[\beta (r)M^{r}(a,b)+\bigl(1-\beta (r)\bigr)N^{r}(a,b) \bigr]^{1/r}, \end{aligned}$$ holds for all r ≤ 1 $r\leq 1$ and a , b > 0 $a,b>0$ with a ≠ b $a\neq b$ , where TD ( a , b ) : = ∫ 0 π / 2 a 2 cos 2 θ + b 2 sin 2 θ d θ $$ \operatorname{TD}(a,b):= \int ^{\pi /2}_{0}\sqrt{a^{2}\cos ^{2}\theta +b^{2}\sin ^{2} \theta }\,d\theta $$ is the Toader mean, and M , N are means. As applications, we attain the optimal bounds for the Toader mean in terms of arithmetic, contraharmonic, centroidal and quadratic means, and then we provide some new bounds for the complete elliptic integral of the second kind.