bibkey: stampachwaclawek2026birman authors: František Štampach; Jakub Waclawek year: 2026 title: “Optimal discrete p-Hardy–Rellich–Birman inequalities” doi: null url: https://arxiv.org/abs/2605.25238v1 claim: “Conjecture 5.6(iii) asserts a convergent expansion in negative powers of n with entirely non-negative coefficients for the alternative Birman weight at every positive order and p > 1.” strata_touched: [] license: citation-only triage: anchor

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Source: https://arxiv.org/abs/2605.25238v1

Section 2, equations (2.2) and (2.3), pp. 3–4, define the difference operators, nonlinear discrete p-Laplacian and signed-power convention. Equation (2.15) and Remark 2.12, p. 8, specify the normalized convergent series and explain the proposed improvement by truncation. Equations (2.16) and (2.17), p. 9, define the alternative parameter sequence and its weight. Section 5, Conjecture 5.6, p. 28, states the three conjectural properties of that alternative weight.

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The alternative discrete Birman weight

Conjecture 5.6 starts with:

Let $\ell \in \mathbb{N}$ and $p>1$, and let ${\tilde{\rho }}^{(\ell ,p)}$ be defined by (2.17) and (2.16).

Part (iii), p. 28, states verbatim:

For all $n\ge \ell$, the terms ${\tilde{\rho }}_n^{(\ell ,p)}$ admit a power series expansion in negative powers of $n$ with entirely non-negative coefficients; cf. (2.15).

The real sequence used here is literally (2.16):

$${\stackrel{n(\ell ,p)=n1-1/pj=1\prod \ell -1(n-j)(n\in N0),g}{\mathfrak{g}}}_n^{(\ell ,p)}=0(n<0).$$

Equation (2.2) defines $(\nabla u{)}_n=u_n-u_{n-1}$ and $(\mathrm{div}u{)}_n=u_{n+1}-u_n$. Equation (2.3) defines $-{\Delta }_p^{(\ell )}u=(-1{)}^{\ell }{\mathrm{div}}^{\ell }({\nabla }^{\ell }u{)}^{⟨p-1⟩}$, where ${\nu }^{⟨a⟩}=\nu |\nu {|}^{a-1}$ and $0^{⟨a⟩}=0$. Equation (2.17) divides this expression for $u={\stackrel{by\; (g}{\mathfrak{g}}}_n{)}^{p-1}$.

In the convergent (2.15) normalization, let $c_0=(1/q{)}_{\ell }^p$ and $c_k=c_0{A}_k$ for $k\ge 1$. Since $p>1$ and $\ell \ge 1$, $c_0>0$. Non-negative normalized coefficients therefore imply the weaker assertion encoded by claim: there is one non-negative coefficient sequence giving $n^{\ell p}{\tilde{\rho }}_n$ for every $n\ge \ell$.

At $\ell =2$ and $p=11/10$, rational enclosures at $n=100,200,400$ force the secant slopes at $x=1/n$ to decrease. Non-negative power-series coefficients force those slopes to increase, so the alternative weight has no such expansion. This refutes only part (iii).

Remark 2.12 concerns the source’s original weight $\rho$, whereas Conjecture 5.6 concerns $\tilde{\rho }$. The positivity results for $\ell =1$ and $p=2$ remain applicable; the failure for the alternative weight does not refute the original-weight conjecture in Remark 2.12.

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