Let $\psi (x)$ denote the digamma function, that is, the logarithmic derivative of the classical Euler gamma function $\Gamma (x)$ . In the paper, the authors discover the monotonic properties of the functions \( \frac{\psi ^{(n)}(x)}{x\psi ^{(n+1)}(x)} \quad \text{and}\quad \psi ^{(n)}(x) \psi ^{(n)}\biggl(\frac{1}{x}\biggr) \) for $n\ge 0$ on $(0,\infty )$ . With the aid of these monotonic properties, the authors confirm the positivity of the function \( \psi (x)+x\psi '(x)-\psi (x)\psi \biggl(\frac{1}{x}\biggr) \) on $(0,\infty )$ . The authors also pose five open problems, generalizing the latter two of the three functions mentioned above and their related conclusions.
