Stronger inequality when no $x_i$ is at the "lower solution"

If all $x_i$s are given by the higher root of equation 2, one gets

$$2\left(D-1\right)=\Sigma_{i} r_i + \sqrt{r_{i}^2-4\alpha D^2}$$

$$\Sigma_{i}r_i+2=2D-\Sigma_{i }\sqrt{r_{i}^2-4\alpha D^2}$$

With the same argument as previously,

$$\frac{r_{s}}{\sqrt{\alpha}} \ge 2+ \Sigma_{i}r_i +\Sigma_{i}\sqrt{r_{i}^2-r_{s}^2},$$ (6)

with $r_s=\min_i r_i$.