For every integer n≥ 0 and every nonzero polynomial p of degree n whose zeros all lie on the unit circle T, define Q_n(z)=1+z^n. For all 0≤ s≤ t≤∞, one has displaystyle fraclVert p rVert_slVert Q_n rVert_s≤fraclVert p rVert_tlVert Q_n rVert_t, where, for 0<r<∞, lVert f rVert_r=(int_T|f|^r,dm)^1/r with m normalized Haar measure on T, and lVert f rVert_0 and lVert f rVert_∞ denote the geometric mean and supremum norm, respectively.

Baernstein’s quasi-norm monotonicity conjecture

For every integer $n\ge 0$ and every nonzero polynomial $p$ of degree $n$ whose zeros all lie on the unit circle $\mathbb{T}$ , define $Q_n(z)=1+z^n$ . For all $0\le s\le t\le\infty$ , one has $\displaystyle \frac{\lVert p\rVert_s}{\lVert Q_n\rVert_s}\le\frac{\lVert p\rVert_t}{\lVert Q_n\rVert_t}$ , where, for $0<r<\infty$ , $\lVert f\rVert_r=\left(\int_{\mathbb{T}}|f|^r\,\mathrm{d}m\right)^{1/r}$ with $m$ normalized Haar measure on $\mathbb{T}$ , and $\lVert f\rVert_0$ and $\lVert f\rVert_\infty$ denote the geometric mean and supremum norm, respectively.

References

Primary source

arXiv

Additional references

Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero — arXiv — Teng Zhang

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed solved

A 2026 preprint claims to prove the conjecture in full, but the proof has not been independently verified.

Baernstein’s conjecture asserts that normalized quasi-norms of polynomials with unimodular roots increase with the exponent: for $0\le s\le t\le\infty$ , $\|p\|_s/\|1+z^n\|_s\le\|p\|_t/\|1+z^n\|_t$ . A. Baernstein II posed it in 2008.

Known results

Baernstein II, 2008: posed the monotonicity conjecture, whose consequences include the entropy conjecture.
2018 account: recorded the assertion as Conjecture 2.2 and stated that it remained unproved.

October 2026 claimed proof

On October 1, 2026, Teng Zhang’s arXiv preprint was reported as establishing the inequality for every $0\le s\le t\le\infty$ , with coefficient, product, and entropy consequences. This is a claimed complete proof, not an independently verified result.

Current status (as of October 2026): The conjecture is claimed proved by Teng Zhang’s preprint, but the general result remains unverified.

Sources

No solutions have been posted yet.