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

TL;DR

This work connects the improved optimal discrete $p$-Hardy weight $oldsymbol{ω}_{p}$ to a Herglotz–Nevanlinna function by introducing $f_p(z)=-z^{p-1}oldsymbol{ω}_{p}(z)$ and proving it admits an integral representation with a positive density $ ho_p$ supported on $(-1,1)$. Consequently, the authors establish that $f_p$ is a Herglotz–Nevanlinna function for all $p>1$, obtain the density-based integral representation, and deduce that the scaled weight $x^{-p}oldsymbol{ω}_{p}(1/x)$ is absolutely monotone on $[0,1]$, with positive even moments $m_{2k}^{(p)}$ admitting a combinatorial formula. The results extend a conjecture from prior work to non-integer $p$ through the density $ ho_p$ and provide explicit moment expansions, illuminating the positivity structure behind the improved Hardy inequality on graphs. Overall, the paper builds a bridge between discrete Hardy improvements and complex-analytic function theory, enabling precise positivity and monotonicity statements for all $p>1$.

Abstract

It was recently proved by Fischer, Keller, and Pogorzelski in [Integr. Equ. Oper. Theory, 95(24), 2023] that the classical discrete $p$-Hardy inequality admits an improvement, and the optimal $p$-Hardy weight $ω_{p}$ was determined therein. We prove that $ω_{p}$ 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.

Theorems & Definitions (18)

• Theorem 1
• Corollary 2
• Remark 3
• Remark 4
• Theorem 5
• Theorem 6
• Lemma 7
• proof
• Lemma 8
• proof