bibkey: krasniqi2026bernsteinboundary authors: Valmir Krasniqi year: 2026 title: “On the Alzer-Berg problem: an optimal Bernstein boundary and a uniqueness conjecture” doi: 10.48550/arXiv.2609.28707 url: https://arxiv.org/abs/2609.28707v1 claim: “For the literal family H(a,c;y)=(1+1/y)^(ay-c), the preprint supplies the regularized density and moment recurrence, strict necessary parameter bounds, an attained Bernstein boundary with nonempty positive contacts, and Conjecture 23 on uniqueness and quadratic contact.” strata_touched: [] license: citation-only triage: anchor

Krasniqi’s exact Bernstein boundary

Valmir Krasniqi, On the Alzer-Berg problem: an optimal Bernstein boundary and a uniqueness conjecture, arXiv:2609.28707v1, submitted 23 September 2026. The DOI above is the arXiv-issued identifier; the abstract record marks its DataCite registration as pending. The versioned URL fixes the cited statements.

For $a>0$, $0\le c<1$, the source defines $H_{a,c}(y)=(1+1/y{)}^{ay-c}$, with $b=-c$. Proposition 2, equations (2.11)–(2.14), gives strict necessary bounds for Bernstein pairs. Proposition 3 assumes $a+c<3$ and supplies the twice-subtracted density, its Laplace representation, and the equivalence between the Bernstein property and global density nonnegativity. Equations (3.17), (3.19) give the exact coefficient recurrence and derivatives at zero. Proposition 6 gives ${\alpha }_{\ast }<7/3$. Theorem 17 identifies the attained boundary $A(b)$ and supplies nonnegativity and nonempty positive contacts for $-1<b<0$.

Conjecture 23 asks whether every such interior boundary density has a unique positive zero of order two. The cited source supplies no proof of that conjecture. Its necessary bounds and analytic identities are intermediate suppliers for the boundary-contact argument.

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