A Herglotz–Nevanlinna function from the optimal discrete 𝑝-Hardy weight

It was recently proved by Fischer, Keller, and Pogorzelski in [Integr. Equ. Oper. Theory, 95 (2003), p. 17] that the classical discrete p p -Hardy inequality admits an improvement, and the optimal p p -Hardy weight ω p ω ₚ was determined therein. We prove that ω p ω ₚ directly corresponds to a Herglotz–Nevanlinna function, establish an integral representation for this function, and consequently confirm a slight modification of a conjecture on its absolute monotonicity from the aforementioned article.