Inequalities for the Polar Derivative of a Polynomial

复Banach空间中带Fréchet导数与面积项的Bohr型不等式

Bohr-type inequalities with Fréchet derivative and Area terms in Complex Banach spaces

该研究将单位圆盘上的Bohr型不等式推广至复Banach空间全纯映射情形，利用Fréchet导数等工具得到显式精确Bohr半径，还证明了含导数、系数与面积项的精细不等式的对应半径为指定方程的唯一正解。

关于多项式极导数的广义 $L_p$ 不等式注记

A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial

本文研究了多项式极导数的广义 $L_p$ 不等式，针对所有零点位于给定圆盘内的多项式，建立了新的不等式并推广了已有结论。

关于多项式广义极导数的一些不等式

Certain Inequalities for the generalized polar derivative of a polynomial

本文研究了多项式广义极导数的不等式，扩展了已知结果并估计了其最大模。

Some Inequalities for the Polar Derivative of Some Classes of Polynomials

In this paper, we investigate an upper bound of the polar derivative of a polynomial of degree $n$ $$p(z)=(z-z_m)^{t_m} (z-z_{m-1})^{t_{m-1}}\cdots (z-z_0)^{t_0}(a_0+\sum\limits_{ν=μ} ^{n-(t_m+\cdots+t_0)} a_νz^ν)$$ where zeros $z_0,\ldots,z_m$ are in $\{z:|z|<1\}$ and the remaining $n-(t_m+\cdots+t_0 )$ zeros are outside $\{z:|z|<k\}$ where $k \geq 1.$ Furthermore, we give a lower bound of this polynomial where zeros $z_0,\ldots,z_m$ are outside $\{z:|z|\leq k\}$ and the remaining $n-(t_m+\cdots+t_0 )$ zeros are in $\{z:|z|<k\}$ where $k\leq 1.$

Weighted norm inequalities for derivatives on Bergman spaces

An equivalent norm in the weighted Bergman space $A^p_ω$, induced by an $ω$ in a certain large class of non-radial weights, is established in terms of higher order derivatives. Other Littlewood-Paley inequalities are also considered. On the way to the proofs, we characterize the $q$-Carleson measures for the weighted Bergman space $A^p_ω$ and the boundedness of a Hörmander-type maximal function. Results obtained are further applied to describe the resolvent set of the integral operators $T_g(f)(z)=\int_0^z g'(ζ)f(ζ)\,dζ$ acting on $A^p_ω$.

具有指定极点和受限零点的有理函数的Turan型不等式

Turan type inequalities for rational functions with pescribed poles and restricted zeros

该研究建立了原点带s重零点、极点指定的有理函数的Turan型不等式，并由此推导出多项式及其极导数的相关不等式，丰富了复分析不等式领域的结果。

由边界函数估计的alpha-调和函数的一些最优不等式

Some optimal inequalities for alpha-harmonic functions estimated by their boundary functions

本文利用边界函数的$L^p$范数和Lipschitz常数，分别对单位圆盘上的alpha-调和函数及其一阶偏导函数进行估计，获得了一系列最优或渐近最优的不等式。

多维的精细波尔不等式

Multidimensional analogues of the refined Bohr type inequalities

本文研究多维情形下的波尔不等式，通过改进经典不等式并引入径向导数，提出多维全纯函数的精确估计方法。

Zygmund's type inequality to the polar derivative of a polynomial

In this paper we improve a result recently proved by Irshad et al. [On the Inequalities Concerning to the Polar Derivative of a Polynomial with Restricted Zeroes, Thai Journal of Mathematics, 2014 (Article in Press)] and also extend Zygmund's inequality to the polar derivative of a polynomial.

On an inequality concerning the polar derivative of a polynomial

In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

Bernstein- and Markov-type inequalities

This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general sets.

Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball

The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.

Weighted norm inequalities for de Branges--Rovnyak spaces and their applications

Let $\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty(\mathbb{C}_+)$. We study the boundary behavior of the derivatives of functions in $\mathcal{H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(μ)} \le C\|f\|_{\mathcal{H}(b)}$, where $f \in \mathcal{H}(b)$ and $μ$ is a Carleson-type measure on $\mathbb{C}_+\cup\mathbb{R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for $\mathcal{H}(b)$ spaces. These results extend Cohn and Volberg--Treil embedding theo…

拟正则Bergman空间中的平均导数与反向不等式

Averaged derivatives and reverse inequalities in quasiregular Bergman spaces

本文研究$\boldsymbol{R}^n$单位球上拟正则映射的加权Bergman空间，通过平均导数刻画Bergman可积性，确定支配集的几何条件，扩展了Bergman理论并补充了拟正则Hardy空间的相关研究。

从算子理论视角看Schwarz-Pick型不等式

Schwarz-Pick type inequalities from an operator theoretical point of view

从算子理论视角，利用von Neumann不等式证明并推广了Schwarz-Pick型不等式，包括多点情形、算子版本及多圆盘与高阶导数情形，并借助判别簇增强了双圆盘上的Schwarz-Pick引理。

平均$n$叶函数的Bernstein型不等式

Bernstein-type inequalities for mean $n$-valent functions

该研究推导了单位圆盘内平均$n$叶函数导数的新积分估计，拓展补充了已有相关结果，并改进了Dolzhenko的若干有理逼近逆定理。