Is the estimation $$ \frac{n}{2}-1 < \sum_{i=1}^n\frac{1}{1+x_i^2} $$ right, if $x\in B_1(0)\subset\mathbb{R}^n$? I guess $B_1(0)$ is the open unit ball.
Anyhow it is:
$$ \sum_{i=1}^n\frac{1}{1+x_i^2}<\frac{n}{2} $$ on $B_1(0)$, so
$$ \frac{n}{2}-\sum_{i=1}^{n}\frac{1}{1+x_i^2}>0. $$
Then $$ \frac{n}{2}-\sum_{i=1}^{n}\frac{1}{1+x_i^2}-1 $$ does not have to be negative, I think.