Higher order Ostrowski type inequalities over Euclidean domains

The classical Ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative. This result is extended to functions on general domains using the L^∞ norm of its nth partial derivatives. For radial functions on balls the inequality is sharp.     

A Markov-Type Inequality for Multivariate Polynomials on a Convex Body

A Markov-type inequality for the k-homogeneous part of a multivariate polynomial on a convex centrally symmetric body is given and an extremal polynomial is found. This generalizes and extends some estimates for univariate and multivariate polynomials obtained by Markov, Bernstein, Visser, Reimer, and Rack.     

Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces

We establish Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces. These inequalities can be applied to some important operators in Fourier analysis, such as the Bochner-Riesz multiplier over the critical index, the generalized Bochner-Riesz mean and the generalized Able-Poisson operator. This work was supported by Key Academic Discipline of Zhejiang Province of China and National Natural Science Foundation of China (Grant Nos. 10571014, 10631080, 10671019)     

An Extension of an Inequality of Duffin and Schaeffer

Using some simple geometric observation, in 1941 Duffin and Schaeffer derived the following property of the Tchebycheff polynomial T_n(x) : |T_n(x + iy)| |T_n(1 + iy)| (-1 x 1, - < y < ). They combined this property of T_n with classical arguments from complex analysis such as the Gauss–Lucas theorem and Rouches theorem to obtain their famous refinement of the inequality of the brothers Markov. The aim of this work is to extend the above property of T_n to the class of ultraspherical polynomials P_n⁽⁾, 0. We then apply this result to obtain a generalization of the inequality of Duffin and Schaeffer.     

The full Markov-Newman inequality for Müntz polynomials on positive intervals

For a function defined on an interval let

The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let be an integer. Let be distinct real numbers. Let . Then

where the supremum is taken for all (the span is the linear span over ).

     

Markov-Type Inequalities for Products of Müntz Polynomials

Let Λ(λ_j)^∞_j=0 be a sequence of distinct real numbers. The span of {x^λ₀, x^λ₁, …, x^λ_n} over is denoted by M_n(Λ)span{x^λ₀, x^λ₁, …, x^λ_n}. Elements of M_n(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λ_j)^∞_j=0andΓ(γ_j)^∞_j=0be increasing sequences of nonnegative real numbers. Let

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Then

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18(n+m+1)(λ_n+γ_m).In particular ,

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Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b](0, ∞).     

Mean inequalities in certain Banach function spaces

Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.     

Extremal properties of the derivatives of the Newman polynomials

Let be a set of distinct positive numbers. The span of

over will be denoted by

Our main result of this note is the following.

Theorem. Suppose . Let be a non-negative integer. Then there are constants 0$"> and 0$"> depending only on , , and such that

where the lower bound holds for all and for all , while the upper bound holds when and and when , , and .

     

Hardy inequalities for fractional integrals on general domains

We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.     

The version of Newman's Inequality for lacunary polynomials

The principal result of this paper is the establishment of the essentially sharp Markov-type inequality

for every with distinct real exponents greater than and for every . A remarkable corollary of the above is the Nikolskii-type inequality

for every with distinct real exponents greater than and for every . Some related results are also discussed.

     

An extremal property of Jacobi polynomials in two-sided Chernoff-type inequalities for higher order derivatives

For a weight function generating the classical Jacobi polynomials, the sharp double estimate of the distance from the subspace of all polynomials of an arbitrary fixed order is established.

     

The number of certain integral polynomials and nonrecursive sets of integers, Part 1

Given 2$">, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the ``cube' with real coordinates from into . This directly translates to a nice statement in logic (more specifically recursion theory) with a corresponding phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable.

     

Density of multivariate homogeneous polynomials on star like domains

The famous Weierstrass theorem asserts that every continuous function on a compact set in ${\mathbb{R}}^d$ can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.     

Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations

This note generalizes the well known Lyapunov-type inequalities for second-order linear differential equations to certain 2M-th order linear differential equations with five types of boundary conditions. The usage of the best constant of some Sobolev-type inequalities clarify the process for obtaining such inequality and sharpen the result of Çakmak [2].     

Lyapunov-type integral inequalities for certain higher order differential equations

In this paper, first we will give a short survey of the most basic results on Lyapunov inequality, and next we obtain this-type integral inequalities for certain higher order differential equations. Our results are sharper than some results of Yang (2003) [20].     

Markov-type Inequalities for Surface Gradients of Multivariate Polynomials

Let K ^d be a compact set with a smooth boundary and consider a polynomial p of total degree n such that p_C(K)1. Then we show that D_Tp(x)=o(n²) for any x Bd K and T a tangential direction at x. Moreover, the o(n²) term is given in terms of the modulus of smoothness of Bd K.     

Numerical radius inequalities for operator matrices

We prove a numerical radius inequality for operator matrices, which improves an earlier inequality due to Hou and Du. As an application of this numerical radius inequality, we derive a new bound for the zeros of polynomials.     

A note on Hardy-type inequalities

We use a theorem of Cartlidge and the technique of Redheffer's ``recurrent inequalities" to give some results on inequalities related to Hardy's inequality.

     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.