A note on additivity of polygamma functions

TL;DR

This paper investigates the additivity properties of the absolute values of polygamma functions, establishing their sub-additivity and super-additivity on specific intervals based on a unique root related to the functions.

Contribution

It proves the sub-additivity and super-additivity of the absolute polygamma functions on certain intervals, revealing new structural properties of these special functions.

Findings

Functions are sub-additive on $( ext{ln} heta_i, ext{infty})$

Functions are super-additive on $(- ext{infty}, ext{ln} heta_i)$

Identifies the unique root $ heta_i$ related to the additivity properties

Abstract

In the note, the functions $\abs{\psi^{(i)}(e^x)}$ for $i\in\mathbb{N}$ are proved to be sub-additive on $(\ln\theta_i,\infty)$ and super-additive on $(-\infty,\ln\theta_i)$, where $\theta_i\in(0,1)$ is the unique root of equation $2\abs{\psi^{(i)}(\theta)}=\abs{\psi^{(i)}(\theta^2)}$.