Related papers: On reverse Markov-Nikol'skii inequalities for poly…

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For…

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…

Let $K\subset \mathbb{C}$ be a convex compact set, and let $\Pi_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Tur\'an type inverse Markov factor is defined by $M_n(K)=\inf_{P\in \Pi_n(K)}…

We prove a few interesting inequalities for Lorentz polynomials including Nikolskii-type inequalities. A highlight of the paper is a sharp Markov-type inequality for polynomials of degree at most n with real coefficients and with derivative…

We consider the classical problem of estimating norms of higher order derivatives of algebraic polynomial via the norms of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A.…

For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$…

For a convex domain K in the complex plane C, the well-known general Markov inequality asserting that a polynomial p of degree n ||p'|| < c(K) n^2 ||p|| holds. On the other hand for polynomials in general, ||p'|| can be arbitrarily small as…

P. Tur\'an was the first to derive lower estimations on the uniform norm of the derivatives of polynomials $p$ of uniform norm $1$ on the interval I:=[-1,1] and the disk D:=$\{z \in C~:~|z| \le 1\}$, under the normalization condition that…

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the…

Let $P_{<n}(z)$ be the Rudin-Shapiro polynomial of degree $n-1$. We show that $|P_{<n}(z)|\le \sqrt{6n-2}-1$ for all $n\ge0$ and $|z|=1$, confirming a longstanding conjecture. This bound is sharp in the case when $n=(2\cdot 4^k+1)/3$ and…

Polynomials with coefficients in $\{-1,1\}$ are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vall\'ee Poussin sums,…

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space $\Pi_n^d$ of spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$ as…

We prove Nikol'skii type inequalities which for polynomials on the $n$-dimensional torus $\mathbb{T}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0 <p <q < \infty$). Among other things we…

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is…

Let $P(z)=\sum_{n=0}^Na_nz^n$ be a Littlewood polynomial of degree $N$, meaning that $a_n\in\{\pm 1\}$. We say that $P$ is reciprocal if $P(z)=z^NP(1/z)$. Borwein, Erd\'elyi and Littmann posed the question of determining the minimum number…

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Tur\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general…

Let $D_\alpha P(z)=nP(z)+(\alpha-z)P^{\prime}(z)$ denote the polar derivative of a polynomial $P(z)$ of degree $n$ with respect to a point $\alpha\in\mathbb{C}.$ In this paper, we present a correct proof, independent of Laguerre's theorem,…

We prove that there is an absolute constant $c > 0$ such that for every $$a_0,a_1, \ldots,a_n \in [1,M]\,, \qquad 1 \leq M \leq \frac 14 \exp \left( \frac n9 \right)\,,$$ there are $$b_0,b_1,\ldots,b_n \in \{-1,0,1\}$$ such that the…

Schur studied limits of the arithmetic means $s_n$ of zeros for polynomials of degree $n$ with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that $\limsup_{n\to\infty}…

We study mappings differentiable almost everywhere, possessing the $N$-Luzin property, the $ N^{\,-1}$-property on the spheres with respect to the $(n-1)$-dimensional Hausdorff measure and such that the image of the set where its Jacobian…