Let $N$ be a positive integer, and consider the equation
\begin{equation} \frac{1}{2} \sum_{n=1}^{N} \psi^{(1)} \left( \frac{x+1-n}{2} \right) = \frac{N}{x}, \end{equation} in the real unkown $x > N - 1$, where $\psi^{(1)}(x)$ is the trigamma function.
I have conjectured that this equation has no solution. Does someone have an idea of a possible proof?
Any help is welcome.
NB From the series representation of the trigamma function we know that the map $(0,\infty) \ni x \mapsto \psi^{(1)}(x)$ is strictly decreasing. So our statement is proved if we can prove the inequality \begin{equation} \psi^{(1)}(x) > \frac{1}{x} \quad (x > 0), \end{equation} but I do not know how to prove it for now.
Our system has actually no solution as conjectured, since the inequality stated in the note above holds true. More precisely, one can prove that
\begin{equation} \psi^{(1)}(x) > \frac{1}{x} + \frac{1}{2x^2} \quad (x > 0). \end{equation} For a proof, see e.g. Elbert and Laforgia On Some Properties of the Gamma Function, Section 2; Gordon, A Stochastic Approach to the Gamma Function, Theorem 4; Alzer, On Some Inequalities for the Gamma Function, Theorem 9.