calculus - Polygamma function series: $sum_{k=1}^{infty }left(Psi^{(1)}(k)right)^2$

Applying the Copson's inequality, I found:

$$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where
$\Psi^{(1)}(k)$ is the polygamma function.
Is it known any sharper bound for the sum $S$?
Thanks.

Answer

The upper bound can be improved using asymptofic series :