The additivity of polygamma functions

Abstract

In the note, the authors prove that the functions $\abs{\psi^{(i)}(e^x)}$ for $i\in\mathbb{N}$ are subadditive on $(\ln\theta_i,\infty)$ and superadditive on $(-\infty,\ln\theta_i)$, where $\theta_i\in(0,1)$ is the unique root of equation $2\abs{\psi^{(i)}(\theta)}=\abs{\psi^{(i)}(\theta^2)}$.