Let $m$ denote the normalized Haar measure on the unit circle $\mathbb{T}$. For $0<r<\infty$, define $\lVert f\rVert_{r}:=\left(\int_{\mathbb{T}}|f|^{r}\,\mathrm{d} m\right)^{1/r}$, with $\lVert f\rVert_{0}$ and $\lVert f\rVert_{\infty}$ 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 $\mathbb{T}$, $$ \frac{\lVert p\rVert_s}{\lVert Q_n\rVert_s} \le \frac{\lVert p\rVert_t}{\lVert Q_n\rVert_t}, \qquad 0\le s\le t\le\infty. $$ 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 Erd\H{o}s--Szekeres product bound $ \left\lVert\prod_{j=1}^N(1-z^{s_j})\right\rVert_\infty\ge2\sqrt N $ for all positive integers $s_1,\ldots,s_N$ and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{}<1\}$. For $\beta>1$, let $\mathcal{N}(\beta)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\}<\beta$ for $z \in \mathbb{D}$. The main purpose of this...
For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the q...
Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $\gamma_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely th...
Let $\mathcal{S}^ r$ denote the Schatten--von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be...
We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}...
A. Fragkos, Ben Krause, Michael T. Lacey· 0 citations
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