On the Chebyshev's Inequality for Weighted Means

Chebyshev-Grüss-type inequalities via discrete oscillations

The classical form of Grüss' inequality, first published by G. Grüss in 1935, gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. In the subsequent years, many variants of this inequality appeared in the literature. The aim of this paper is to introduce a different approach, presenting a new Chebyshev-Grüss-type inequality and applying it to different well-known linear, not necessarily positive, operators. Some conjectures are presented as well. We also compare the new inequalities with some older results. This new approach…

On the weighted Bojanov-Chebyshev Problem and the sum of translates method of Fenton

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erdélyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.

Inegalităţi de tip Chebyshev-Grüss pentru operatorii Bernstein-Euler-Jacobi

The classical form of Grüss' inequality was first published by G. Grüss in 1935 and gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. After that many variants of this inequality appeared in the literature. The aim of this paper is to consider some Chebyshev-Grüss-type inequalities and apply them to Bernstein-Euler-Jacobi (BEJ) operators of first and second kind. First and second moments of the operators are used to explain the situation.

加权切比雪夫多项式和正交多项式的下界

Lower Bounds for Weighted Chebyshev and Orthogonal Polynomials

本文推导了实直线紧致子集上加权切比雪夫和正交多项式Widom因子的最优渐近与非渐近下界，并推广了Bernstein和Widom的经典结果至更广权重类。

Mean convergence of Fourier-Akhiezer-Chebyshev series

We prove mean convergence of the Fourier series in Akhiezer-Chebyshev polynomials in $L^p$, $p>1$, using a weighted inequality for the Hilbert transform in an arc of the unit circle.

Some inequalities for Chebyshev polynomials

Askey and Gasper (1976) proved a trigonometric inequality which improves another trigonometric inequality found by M. S. Robertson (1945). Here these inequalities are reformulated in terms of the Chebyshev polynomial of the first kind $T_n$ and then put into a one-parametric family of inequalities. The extreme value of the parameter is found for which these inequalities hold true. As a step towards the proof of this result we establish the following complement to the finite increment theorem specialized to $T_n^{\prime}$: $$ T_n^{\prime}(1)-T_n^{\prime}(x)\geq (1-x)\,T_n^{\prime\prime}(x)\,,\q…

关于加权三角 Bojanov-Chebyshev 极值问题

On the weighted trigonometric Bojanov-Chebyshev extremal problem

本文研究了三角多项式的加权 Bojanov-Chebyshev 极值问题，通过 Fenton 平移求和法求解，并将结果推广至广义三角多项式。

Pointwise Remez inequality

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded $1$ on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik-Widom bounds for the extremal value, thereby providing a complete asym…

Some new refined Hardy type integral inequalities

In this paper, by using Jensen's inequality and Chebyshev integral inequality, some generalizations and new refined Hardy type integral inequalities are obtained. In addition, the corresponding reverse relation are also proved.

Finite Sections of Weighted Hardy's Inequality

We study finite sections of weighted Hardy's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant.

Finite Sections of Weighted Carleman's Inequality

We study finite sections of weighted Carleman's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant.

On the weighted Ostrowski type integral inequality for double integrals

In this paper, we establish new an inequality of weighted Ostrowski-type for double integrals involving functions of two independent variables by using fairly elementary analysis.

关于Montgomery-Vaughan加权希尔伯特不等式中的最优常数

On the optimal constant in the Montgomery-Vaughan weighted Hilbert inequality

该研究证明Montgomery-Vaughan加权希尔伯特不等式的最优常数严格大于π，通过显式极限反例完成证明，否定了Montgomery等人提出的相关问题。

A weighted inequality for potential type operators

We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on $ \mathbb R $ endowed with a measure which needs not to be doubling.

The weighted Fourier inequality, polarity, and reverse Hölder inequality

We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex set and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse Hölder class, is used to formulate the sufficient condition.

A Weighted Ostrowski Type inequality for L$_{1}\left[ a,b\right] $ and applications

The aim of this paper is to obtain some generalized weighted Ostrowski inequalities for differentiable mappings. Some well known inequalities can be derived as special cases of the inequalities obtained here. In addition, perturbed mid-point inequality and perturbed trapezoid inequality are also obtained. The inequalities obtained here have direct applications in Numerical Integration, Probability Theory, Information Theory and Integral Operator Theory. Some of these applications are discussed.

对应于消失权重的切比雪夫多项式

Chebyshev polynomials corresponding to a vanishing weight

本文研究单位圆上对应消失权重$(z-1)^s$的加权切比雪夫多项式，将Lachance等结论推广至非整数$s$，并联系双纽线等类别，关键推广了Erdős–Lax不等式。

On a weighted variable spaces $L_{p(x), ω}$ for $0< p(x)< 1$ and weighted Hardy inequality

In this paper a weighted variable exponent Lebesgue spaces $L_{p(x), ω}$ for $0< p(x)< 1$ is investigated. We show that this spaces is a quasi-Banach spaces. Note that embedding theorem between weight variable Lebesgue spaces is proved. In particular, we show that $L_{p(x), ω}(Ω)$ for $0< p(x)< 1$ isn't locally convex. Also, in this paper a some two-weight estimates for Hardy operator are proved.

A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities

In this note we prove a multilinear version of the reverse Hölder inequality in the theory of Muckenhoupt $A_p$ weights. We give two applications of this inequality to the study of multilinear weighted norm inequalities. First, we prove a structure theorem for multilinear $A_{\vec{p}}$ weights; second, we give a new sufficient condition for multilinear, two-weight norm inequalities for the maximal operator.

多参数分数次积分的加权$L^p$范数不等式

Weighted $L^p$-norm inequality of multi-parameter fractional integration

本文研究核在每个坐标子空间上具有奇异性的强分数次积分算子，证明了当多参数theta-bump特征有限时，相应的双权$L^p$范数不等式成立。

一个加权解耦不等式及其在极大Bochner-Riesz问题中的应用

A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

该研究证明了p=2n/(n-1)时的加权L^pℓ^p解耦估计，将其应用于极大Bochner-Riesz问题，得到三维空间中超出实插值指数的结果，并改进了平面情形的相关结论。

Weighted Ostrowski Type inequality Involving Integral Means

The ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The aim of this paper is to establish a new inequality using weight function which generalizes the inequalities of Dragomir, Wang and Cerone .The current article obtains bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval. A variety of earlier results are recaptured as particular instances of the current development. Applications for cumulative distribution function are also discussed.

On the necessity of bump conditions for the two-weighted maximal inequality

We study the necessity of bump conditions for the boundedness of the Hardy-Littlewood maximal operator $M$ from $L^p(v)$ into $L^p(w)$, where $1<p<\infty$. The conditions in question are obtained by replacing the average of $σ=v^{-\frac{1}{p-1}}$ in the Muckenhoupt $A_p$-condition by an average with respect to certain Banach function space, and are known to be sufficient for the two-weighted maximal inequality. We show that these conditions are in general not necessary for the boundedness of $M$ from $L^p(v)$ into $L^p(w)$.

关于希尔伯特不等式的Montgomery--Vaughan加权推广

On the Montgomery--Vaughan weighted generalization of Hilbert's inequality

本文研究希尔伯特不等式的Montgomery--Vaughan加权推广的最优常数问题，通过含参数不等式族得到相关常数的上下界，指出现有方法无法达到猜想的常数π，该问题仍待解决。

$l_2$ 中的加权 Hardy 不等式

Weighted Hardy Inequality in $l_2$

研究加权 Hardy 不等式最小常数 $d_n$ 的渐近行为，确定其精确收敛速率并构造“几乎极值”序列。

Small polynomials with integer coefficients

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of po…

关于多变量加权Hermite-Hadamard不等式及其在加权多元均值中的应用

On the weighted hermite-hadamard inequality in multiple variables, application for weighted multivariate means

该研究针对单变量Hermite-Hadamard不等式的拓展需求，提出多变量加权形式的不等式变体，并将其应用于构造推广现有双变量均值的加权多元均值。

One more proof of the Erdős--Turán inequality, and an error estimate in Wigner's law

We prove the Erdos-Turan equidistribution inequality, using a construction due to Chebyshev, Markov, and Stieltjes. The method is applicable in a more general setting. As an example, we state another inequality that can be proved using this method. The inequality can be used to get an error estimate in Wigner's law which is sharp near the edge.

关于函数从$W^1_p$空间在边界小部分消失的加权Hardy型不等式

On the Weighted Hardy Type Inequality for Functions from $W^1_p$ Vanishing on Small Parts of the Boundary

本文证明了函数从$W^1_p$空间在边界小部分消失的加权Hardy型不等式，扩展了经典加权Hardy型不等式。

On the weighted inequality between the Gagliardo and Sobolev seminorms

We prove weighted inequalities between the Gagliardo and Sobolev seminorms and also between the Marcinkiewicz quasi-norm and the Sobolev seminorm. With $A_1$ weights we improve earlier results of Bourgain, Brezis, and Mironescu.

加权Sobolev空间上具有精确增长条件的尖锐高阶Adams不等式

Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces

本文在加权Sobolev空间上建立了具有精确增长条件的尖锐高阶Adams不等式，证明其临界常数与指数的最优性，并将其应用于一类常微分方程，得到了弱解概念和正则性理论。

Weighted Lépingle inequality

We prove an estimate for weighted $p$-th moments of the pathwise $r$-variation of a martingale in terms of the $A_{p}$ characteristic of the weight. The novelty of the proof is that we avoid real interpolation techniques.

约翰意义下的加权图与全局波尔查诺不等式

Weighted graphs in the sense of John and a global Poincaré inequality

本文研究了加权图的条件，以保证全局波尔查诺不等式的成立，该条件是约翰1961年提出的条件的离散类比。

单调极值函数与加权 Hilbert 不等式

Monotone extremal functions and the weighted Hilbert's inequality

本文寻找满足单调性条件的 signum 函数的指数型最优单侧控制函数，并以此给出加权 Hilbert-Montgomery-Vaughan 不等式的简单 Fourier 分析证明。

带应用的精确加权Cohen--Dahmen--Daubechies--DeVore不等式：（加权）临界Sobolev空间、Gagliardo--Nirenberg不等式与Muckenhoupt权

Sharp Weighted Cohen--Dahmen--Daubechies--DeVore Inequality with Applications to (Weighted) Critical Sobolev Spaces, Gagliardo--Nirenberg Inequalities, and Muckenhoupt Weights

本文建立了Cohen等人2003年提出的不等式的精确加权变体，通过发展好立方体方法追踪对A_p权常数的精确依赖，得到加权Sobolev空间刻画、插值及Gagliardo-Nirenberg不等式等应用，还给出了Muckenhoupt权的新刻画并解决了一个公开问题。

Refined Young inequality and its application to divergences

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also study some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys-Tsallis divergence and the Jensen-Shannon-Tsallis divergence.