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Baernstein Quasi-Norm Monotonicity Conjecture

The Baernstein quasi-norm monotonicity conjecture compares normalized circle means of a polynomial with those of equally spaced roots on the unit circle (Agler and McCarthy 2021). For a nonzero polynomial P of polynomial degree n>=1 with all its roots on the unit circle, set Q_n(z)=1+z^n and define

||P||_r=(1/(2pi)int_0^(2pi)|P(e^(itheta))|^rdtheta)^(1/r)

for 0<r<infty. These means are polynomial norms for r>=1 and quasi-norms for 0<r<1. The endpoint ||P||_0 is the Mahler measure, while ||P||_infty=max_(|z|=1)|P(z)|. The conjecture states that

(||P||_s)/(||Q_n||_s)<=(||P||_t)/(||Q_n||_t)

for 0<=s<=t<=infty.

Zhang (2026) proved the conjecture and states that the original proof was completed without AI assistance. ChatGPT was subsequently used for presentation and proofreading, and Codex generated an accompanying Lean formalization that the author reports was checked by the Lean kernel (Zhang 2026).

See also

Mahler Measure, Polynomial Norm, Quasi-Norm, Unit Circle

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References

Agler, J. and McCarthy, J. E. "The Krzyż Conjecture and an Entropy Conjecture." J. Anal. Math. 144, 207-226, 2021. https://doi.org/10.1007/s11854-021-0178-z.Zhang, T. "Baernstein's Quasi-Norm Monotonicity Conjecture for Polynomials with Unimodular Zeros." 1 Oct 2026. https://arxiv.org/abs/2610.02009.

Cite this as:

Weisstein, Eric W. "Baernstein Quasi-Norm Monotonicity Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BaernsteinQuasi-NormMonotonicityConjecture.html

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