The Bernstein inequality for slice regular polynomials

多项式的某些 Bernstein 型 $L_p$ 不等式

Certain Bernstein-type $L_p$ inequalities for polynomials

本文将已知的多项式 Bernstein 型不等式推广到 $L_p$ 范数，放宽了参数 $α$ 的条件，并证明了零点受限多项式的类似不等式。

Bernstein-Remez inequality for Nash functions: A complex analytic approach

Consider an open, bounded set $Ω\subset \mathbb{C}$, a positive integer $k$ and a compact $\mathcal{K}\subset Ω$ of cardinality strictly greater than $k$. We prove that, for any function $f$ which is holomorphic in $\overline Ω$, and whose graph satisfies $S(z,f(z))=0$ for some polynomial $S\in\mathbb{C}[z,w]$ of degree at most $k$ (hence $f$ is an algebraic function), the quantity $\max_{\overlineΩ}|f|/\max_{\mathcal{K}}|f|$ is bounded by a constant that only depends on $k$, $Ω$, $\mathcal{K}$ but not on $f$ (estimates of this kind are called Bernstein-Remez inequalities). This result has bee…

八元数上切片正则函数的改进与精细化 Bohr 型不等式

Improved and Refined Bohr-Type Inequalities for Slice Regular Functions over Octonions

本文研究八元数上切片正则函数的 Bohr 型不等式，给出其推广、改进与精细化版本，并证明所有结果都是精确的。

On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable

The goal of this paper is to introduce and study some geometric properties of slice regular functions of quaternion variable like univalence, subordination, starlikeness, convexity and spirallikeness in the unit ball. We prove a number of results, among which an Area-type Theorem, Rogosinski inequality, and a Bieberbach-de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from $\mathbb R^4$ to itself. As a tool for subordination we define a suitable notion of composition of slice regular funct…

Convergence speed of weighted Bergman kernels towards extremal functions

We construct inner products by the Bernstein-Markov inequality on spaces of holomorphic sections of high powers of a line bundle. The corresponding weighted Bergman kernel functions converge to an extremal function. We obtain a uniform convergence speed.

Bernstein-Szegő measures on the two dimensional torus

We present a new viewpoint (namely, reproducing kernels) and new proofs for several recent results of J. Geronimo and H. Woerdeman on orthogonal polynomials on the two dimenional torus (and related subjects). In addition, we show how their results give a new proof of Ando's inequality via an equivalent version proven by Cole and Wermer. A simple necessary and sufficient condition for two variable polynomial stability is also given.

关于保持多项式间不等式的算子

On operators preserving inequalities between polynomials

该综述探讨了Bernstein型多项式不等式极值性质的算子方面，指出将多项式映射到多项式且具零保持性的线性算子自然保持Bernstein不等式。

Polynomial estimates over exponential curves in $\mathbb C^2$

For any complex $α$ with non-zero imaginary part we show that Bernstein-Walsh type inequality holds on the piece of the curve $\{(e^z,e^{αz}) : z \in \mathbb C\}$. Our result extends a theorem of Coman-Poletsky \cite{CP10} where they considered real-valued $α$.

Random Bernstein-Markov factors

For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials, and show that the expected (average) and almost sure value of $\Vert P_n' \Vert/\Vert P_n\Vert$ is often different from the classical deterministic upper bound $n$. In particular, for circles of radii less than one, the ratio $\Vert P_n' \Vert/\Vert P_n\Vert$ is almost surely bounded as $n$ tends to infinity, and its expected value is uniformly bounded for all…