Polynomial inequalities on certain algebraic hypersurfaces

It is known that if E is a C^{infty } determining set, then E is a Markov set if and only if it has Bernstein’s property. This article provides the equivalent of this result for compact subsets of some algebraic varieties.

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

Polynomial inequalities and Markov's inequality in weightedL p -spaces

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened. Bibiography: 13 titles.

In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

In this work, we discuss generalizations of the classical Bernstein and\nMarkov type inequalities for polynomials and we present some new inequalities\nfor the $k$th Fr\\'echet derivative of homogeneous polynomials on real and\ncomplex $L_{p}(\\mu)$ spaces. We also give applications to homogeneous\npolynomials and symmetric multilinear mappings in $L_{p}(\\mu)$ spaces. Finally,\nBernstein's inequality for homogeneous polynomials on both real and complex\nHilbert spaces has been discussed.\n

Markov–Bernstein Type Inequalities for Multivariate Polynomials on Sets with Cusps

We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space $${{\mathcal {M}}}^{p(\cdot ),\lambda (\cdot )}$$ with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials.

There are fine extensions of the univariate Bernstein-Szegő inequality for multivariate polynomials considered on a convex domain K. The current one estimates the gradient of the polynomial P at a point x ∈ K by constant times degree, ‖P‖C(K) and a geometrical factor. The best constant is within [2, 2√2]. In this note we disprove the conjecture (based on some particular cases) that the best constant is 2.

On Nikol’skii type inequality between the uniform norm and the integral [formula omitted]-norm with Laguerre weight of algebraic polynomials on the half-line

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions $$f:\mathbb {R}^n \rightarrow \mathbb {R}$$ f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalities $$\begin{aligned} \Vert g- \mathbb Eg\Vert _p \le C(p)\Vert \nabla g\Vert _p. \end{aligned}$$ ‖ g - E g ‖ p ≤ C ( p ) ‖ ∇ g ‖ p . Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $$C(p) \le C\sqrt{p}$$ C ( p ) ≤ C p ) or the Poincaré inequality (then $$C(p) \le Cp$$ C ( p ) ≤ C p ). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of $$f$$ f . When the underlying measure is Gaussian and $$f$$ f is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős–Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.

Markov and Bernstein type inequalities for polynomials

In this article, the synchronization issue of certain edge-colored networks is investigated. We start with a linear subspace determined by the colored edges and show that the invariant kernel of this linear subspace is precisely consists of all the synchronous states. Moreover, this invariant kernel is characterized by a cluster of algebraic equations. Especially, for network described by a polynomial vector field, due to the property of Noetherian rings, this invariant kernel can be determined by a finite number of algebraic equations. Furthermore, the equivalence between network synchronization and the asymptotic behavior of the aforementioned invariant kernel are proved. Based on the colored edges and the invariant kennel, we decompose the original network twice, arriving at a three-layer network: the first two layers, referred to as the external part, can only synchronize to an equilibrium point, while the third layer, known as the internal part, possesses the same colored edges. For this three-layer network, we construct two Lyapunov-type functions for the external part and the internal part, respectively, to establish the synchronization criteria. In particular, our criteria involve linear matrix inequalities and polynomial inequalities of smaller scale, which can be solved by the existing semi-definite programming tools. Finally, this article provides three examples to illustrate the effectiveness and advantages of the theoretical results presented.

Equivalence of the global and local Markov inequalities in the complex plane

In the present note, we will introduce the definition of generalized p convex function. We will investigate some properties of generalized p convex function. Moreover, we will develop Jensen’s type, Schur type, and Hermite Hadamard type inequalities for generalized p convex function.