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

Abstract

Let $m$ denote the normalized Haar measure on the unit circle $\mathbb{T}$. For $0<r<\infty$, define $\Vert f{\Vert }_r:={\left({\int }_{\mathbb{T}}|f{|}^r\mathrm{d}m\right)}^{1/r}$, with $\Vert f{\Vert }_0$ and $\Vert f{\Vert }_{\infty }$ 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 $\mathbb{T}$, $$\frac{\Vert p{\Vert }_s}{\Vert Q_n{\Vert }_s}\le \frac{\Vert p{\Vert }_t}{\Vert Q_n{\Vert }_t},0\le s\le t\le \infty .$$ 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 ${\Vert {\prod }_{j=1}^N(1-z^{s_j})\Vert }_{\infty }\ge 2\sqrt{N}$ for all positive integers $s_1,\dots ,s_N$ and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.

arXiv comment23 pages. All comments are welcome!