In this paper, we present the best possible parameters
$\alpha(r)$
and
$\beta(r)$
such that the double inequality
$$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$
holds for all
$r\leq 1$
and
$a, b>0$
with
$a\neq b$
, and we provide new bounds for the complete elliptic integral
$\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{...
