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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Hardy inequality admits an improvement, and the optimal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Hardy weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript p"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was determined therein. We prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript p"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.