A note on additivity of polygamma functions

Abstract

In the note, the functions $\abs{ψ^{(i)}(e^x)}$ 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 $2\abs{ψ^{(i)}(θ)}=\abs{ψ^{(i)}(θ^2)}$.

3 pages

References in corpus (4)

• Complete monotonicity of some functions involving polygamma functions
• Two new proofs of the complete monotonicity of a function involving the psi function
• Limit formulas for ratios of derivatives of the gamma and digamma functions at their singularities
• Limit formulas for ratios of polygamma functions at their singularities

Cited by in corpus (1)

• Some inequalities for the trigamma function in terms of the digamma function