Related papers: Certain Bernstein-type $L_p$ inequalities for poly…

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this…

Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of \[\bigg|P(Rz)-\alpha P(z)+\beta\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|\alpha|\biggr\}P(z)\bigg|, \ \text{for} \ z \in…

We prove new Bernstein and Markov type inequalities in $L^p$ spaces associated with the normal and the tangential derivatives on the boundary of a general compact $C^\alpha$-domain with $1\leq \alpha\leq 2$. These estimates are also applied…

Let $P(z)$ be a polynomial of degree $n$ having no zero in $|z|<k$ where $k\geq 1,$ then for every real or complex number $\alpha$ with $|\alpha|\geq 1$ it is known \begin{equation*} \underset{|z|=1}{\max}|D_\alpha P(z)|\leq…

If $P(z)$ be a polynomial of degree at most $n$ which does not vanish in $|z| < 1$, it was recently formulated by Shah and Liman \cite[\textit{Integral estimates for the family of $B$-operators, Operators and Matrices,} \textbf{5}(2011), 79…

We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'\|_{L_p(\omega)} \le C(p,\omega)\, {n}\,\|T_n\|_{L_p(\omega)},\qquad 0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain…

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,…

A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of…

Let \( P(z) \) be a polynomial of degree \( n \) and $\alpha \in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_\alpha P(z) \) and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. The polar derivative \( D_\alpha…

Let $ P(z) $ be a polynomial of degree $ n $ and for any real or complex number $\alpha,$ let $D_\alpha P(z)=nP(z)+(\alpha-z)P^{\prime}(z)$ denote the polar derivative with respect to $\alpha.$ In this paper, we obtain generalizations of…

We give the sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum_{k=0}^{n}a_kz^k$, $a_n\not=0,$ for the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ to be true for all $z\in\overline{\mathbb…

If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots $\alpha_1$, ..., $\alpha_N$, then Gon\c{c}alves' inequality states that $\|F\|_2^2$ is bounded below by $\abs{a_N}^2 (\prod_{n=1}^N \max\{1, \abs{\alpha_n}^2\}…

For $1< p <2$ we obtain sharp inequalities for the supremum of products of homogeneous polynomials on $L_p(\mu)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in the…

The paper studies Bernstein type inequalities for restrictions of holomorphic polynomials to graphs $\Gamma_f\subset\mathbb C^{n+m}$ of holomorphic maps $f:\mathbb C^n\rightarrow\mathbb C^m$. We establish general properties of exponents in…

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m} a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m} |a_{\alpha}|^{r}…

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

In this paper, we are concerned with Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of…

Due to the invalidation of the Gauss-Lucas type result for quaternionic polynomials, we first give in this paper an alternative proof of the Bernstein inequality in $L^{p} (1\leq p \leq+\infty)$ for slice regular polynomials by the Fej\'er…

Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let…

For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we study the inequality \[ \|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial…