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Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero

  • Teng Zhang
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Abstract

Let m denote the normalized Haar measure on the unit circle 𝕋. For 0<r<∞, define lVert frVertr:=(∫𝕋|f|r d m)1/r, with lVert frVert0 and lVert frVert∞ interpreted as the geometric mean and the supremum norm, respectively. Set Qn(z)=1+zn. We prove that, for every nonzero polynomial p of degree n whose zeros all lie on 𝕋, (lVert prVerts)/(lVert QnrVerts) ≤ (lVert prVertt)/(lVert QnrVertt), 0≤ s≤ t≤∞. This settles Baernstein's quasi-norm monotonicity conjecture. As corollaries, we obtain an Lr extension of Visser's coefficient inequality, the sharp O'Hara--Rodriguez inequality and its higher-power analogues, the Erdős--Szekeres product bound $ \left\lVert\prod_{j=1}^N(1-z^{s_j})\right\rVert_\infty\ge2\sqrt N $ for all positive integers s1,…,sN and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026baernsteins,
  title = {{Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero}},
  author = {Teng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02009},
  eprint = {2610.02009},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02009},
}
RIS
TY  - UNPB
TI  - Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero
AU  - Teng Zhang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02009
UR  - https://arxiv.org/abs/2610.02009
AB  - Let m denote the normalized Haar measure on the unit circle 𝕋. For 0<r<∞, define lVert frVert_r:=(∫_𝕋|f|^r d m)^1/r, with lVert frVert_0 and lVert frVert_∞ interpreted as the geometric mean and the supremum norm, respectively. Set Q_n(z)=1+z^n. We prove that, for every nonzero polynomial p of degree n whose zeros all lie on 𝕋, (lVert prVert_s)/(lVert Q_nrVert_s) ≤ (lVert prVert_t)/(lVert Q_nrVert_t), 0≤ s≤ t≤∞. This settles Baernstein's quasi-norm monotonicity conjecture. As corollaries, we obtain an L^r extension of Visser's coefficient inequality, the sharp O'Hara--Rodriguez inequality and its higher-power analogues, the Erdős--Szekeres product bound $ \left\lVert\prod_{j=1}^N(1-z^{s_j})\right\rVert_\infty\ge2\sqrt N $ for all positive integers s_1,…,s_N and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.
ER  -