Inequality $\sum\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum \frac{1}{x + n^2} $ 
$x\geq0$, then, we have

$$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2} $$


The problem is not easy, even $x=1$. Any help will be appreciated

I think this approach is also valid, and simpler.
A term-wise subtraction (LHS - RHS) gives:
$$
\frac{x}{(x+n^2)^2} - \frac{1}{2(x+n^2)} = \frac{2x - (x+n^2)}{2(x+n^2)^2} = \frac{x-n^2}{2(x+n^2)^2}
$$
for sufficiently large number of terms, this can be proved to be less than zero.
