Suppose $x > \frac{1}{2}$. Let $\psi^{(0)}$ and $\psi^{(1)}$ denote the digamma and trigamma functions, respectively. Does the following inequality hold for any such $x$?
$\psi^{(0)}(x) - \psi^{(0)}\left(x + \frac{1}{2} \right) + x \left( \psi^{(1)}(x) - \psi^{(1)}\left(x + \frac{1}{2}\right) \right) > 0$
Graphing it suggests that it does, but I don't know how to prove it. Thanks in advance!