On the Hurwitz zeta function with an application to the beta-exponential distribution

TL;DR

This paper establishes a monotonicity property of the Hurwitz zeta function, deriving inequalities for polygamma functions and linking these to the cumulants of the beta-exponential distribution through a probabilistic interpretation.

Contribution

It introduces a new monotonicity property of the Hurwitz zeta function and connects it to the cumulants of the beta-exponential distribution, providing novel inequalities.

Findings

Monotonicity property of the Hurwitz zeta function

Chain of inequalities for polygamma functions

Probabilistic interpretation via beta-exponential distribution

Abstract

We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the beta-exponential distribution.