On an inequality of different metrics for algebraic polynomials

Turán's inequality for ultraspherical polynomials revisited

We present a short proof that the normalized Turán determinant in the ultraspherical case is convex or concave depending on whether parameter $λ$ is positive or negative.

Nikolskii inequality for lacunary spherical polynomials

We prove that for $d\ge 2$, the asymptotic order of the usual Nikolskii inequality on $\mathbb{S}^d$ (also known as the reverse Hölder's inequality) can be significantly improved in many cases, for lacunary spherical polynomials of the form $f=\sum_{j=0}^m f_{n_j}$ with $f_{n_j}$ being a spherical harmonic of degree $n_j$ and $n_{j+1}-n_j\ge 3$. As is well known, for $d=1$, the Nikolskii inequality for trigonometric polynomials on the unit circle does not have such a phenomenon.

On Bernstein's inequality for polynomials

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 inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $L^p-norm$.

On constrained Markov-Nikolskii type inequality for $k-$absolutely monotone polynomials

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. Markov. In $1926,$ Bernstein found the exact constant in the Markov inequality for monotone polynomials. T. Erdelyi showed that the order of the constants in constrained Markov-Nikolskii inequality for $k-$ absolutely monotone polynomials is the same as in the classical one in case $0<p\le q\le\infty.$ In this paper, we find the exact order for all values of $0…

极小内积与多项式中的Bombieri型不等式

The apolar inner product and a Bombieri type inequality for polynomials

本文提出了一种基于极小内积的多项式Bombieri型不等式，适用于非齐次多项式。

一个代数的 Brascamp-Lieb 不等式

An Algebraic Brascamp-Lieb Inequality

本文利用 Zhang 和 Zorin-Kranich 的多线性 Kakeya 型不等式，对多项式与有理映射证明整体非线性 Brascamp-Lieb 不等式，并通过仿射不变权得到仅依赖映射次数的一致常数。

On a Cyclic Inequality Related to Chebyshev Polynomials

We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study a related homogeneous cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq \sum_{i=1}^n x_i \sum_{i=1}^n x_i^a x_{i+1}^b,$$ where $a,b,x_1,\ldots, x_n$ (with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the inequality holds when $a=b=1$ and $n\leq 8$ for all nonnegative numbers $x_1,\ldots, x_n$.

关于高维球面上偶 Legendre 多项式的一个逐点不等式

On a pointwise inequality for even Legendre polynomials in high dimensional spheres

本文针对高维球面建立含谱间隙效应的相邻偶 Legendre 多项式逐点不等式，从而改进并支撑 Boltzmann 方程 Fisher 信息研究中的关键结果。

An inequality for Jacobi polynomials of form $P_n^{(α_n,β_n)}(x)$

We prove an inequality for Jacobi polynomials that \begin{align} Δ_n(x):=P_n^{(α_n,β_n)}(x)P_n^{(α_{n+1},β_{n+1})}(x)- P_{n-1}^{(α_n,β_n)}(x)P_{n+1}^{(α_{n+1},β_{n+1})}(x)\le 0,\ \forall x\ge 1, \end{align} where $α_n=an$ and $β_n=bn$ for some $a,b\ge 0$. The above inequality has a similar taste as the Tuŕan type inequalities, but with $α_n$ and $β_n$ that depends linearly on $n$.

Sturm Theorem and a Refinement of Vietoris Inequality for Cosine Polynomials

In a recent work, the authors established a refinement of the well-known 1958 result of Vietoris on nonnegative cosine polynomials. In four places of the proof, use was made of the classical Sturm Theorem on determining the number of real roots of an algebraic polynomials in a given interval. Although absolutely rigorous, the Sturm procedure involves lengthy technical computations carried out with the help of the software MAPLE 13. This article provides such details which were omitted in the article presenting the recent work.

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…

涉及倒数和与和为1的正数乘积的不等式的最佳常数

The Best Constant For Inequality Involving Sum Of The Reciprocals And Product Of Positive Numbers With Unit Sum

本文通过新的优化论证确定了涉及倒数和与和为1的正数乘积的代数不等式的最佳参数值，并在三数情形下推广了欧拉不等式，应用于三角形外接圆与内切圆半径之比。

A Multiplier Version of the Bernstein Inequality on the Complex Sphere

We prove a multiplier version of the Bernstein inequality on the complex sphere. Included in this is a new result relating a bivariate sum involving Jacobi polynomials and Gegenbauer polynomials, which relates the sum of reproducing kernels on spaces of polynomials irreducibly invariant under the unitary group, with the reproducing kernel of the sum of these spaces, which is irreducibly invariant under the action of the orthogonal group.

Reverse Bernstein Inequality on the Circle

The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed degree. In fact, we derived a more general result for the lower bounds on higher derivatives. These bounds are better then those obtained by applying bound for the first derivative successively several …

The Large Sieve Inequality for Quadratic Polynomial Amplitudes

We provide here a modest improvement upon a large sieve inequality for quadratic polynomial amplitudes orginally due to Liangyi Zhao.

Cosine polynomials with restrictions on their algebraic representation

We prove that for any even algebraic polynomial $p$ one can find a cosine polynomial with an arbitrary small $l_1$-norm of coefficients such that the first coefficients of its representation as an algebraic polynomial in $\cos x$ coincide with those of $p$.

Convex order for convolution polynomials of Borel measures

We give necessary and sufficient conditions for Borel measures to satisfy the inequality introduced by Komisarski, Rajba (2018). This inequality is a generalization of the convex order inequality for binomial distributions, which was proved by Mrowiec, Rajba, Wąsowicz (2017), as a probabilistic version of the inequality for convex functions, that was conjectured as an old open problem by I.~Raşa. We present also further generalizations using convex order inequalities between convolution polynomials of finite Borel measures. We generalize recent results obtained by B.~Gavrea (2018) in the discr…

On $L^p$ Markov type inequality for some cuspidal domains in $\mathbb{R}^2$

The purpose of this paper is to study a Markov type inequality for algebraic polynomials in $L^p$ norm on two-dimensional cuspidal domains.

An inequality on Chebyshev polynomials

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials, and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative.

雅可比多项式的一个不等式：有限增量定理的补充

An inequality for Jacobi polynomials: a complement to Finite Increment Theorem

对于满足特定条件的雅可比多项式，证明了一个形如 P(1)-P(x) ≥ P'(x)(1-x) 的不等式，作为有限增量定理的补充。

Inequalities for Lorentz polynomials

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 not vanishing in the open unit disk. The result may be compared with Erdos's classical Markov-type inequality (1940) for polynomials of degree at most n having only real zeros outside the interval (-1,1).

Sharp Remez inequality

Let an algebraic polynomial $P_n(ζ)$ of degree $n$ be such that $|P_n(ζ)|\le 1$ for $ζ\in E\subset\mathbb{T}$ and $|E|\ge 2π-s$. We prove the sharp Remez inequality $$ \sup_{ζ\in\mathbb{T}}|P_n(ζ)|\le \mathfrak{T}_{n}\left(\sec \frac{s} 4\right),$$ where $\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$ P_n(e^{iz})=e^{i(nz/2+c_1)}\mathfrak{T}_n\left(\sec\frac s 4\cos \frac {z-c_0} 2\right), \quad c_0,c_1\in\mathbb{R}. $$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomia…

The sharp Remez-type inequality for even trigonometric polynomials on the period

We prove that $$\max_{t \in [-π,π]}{|Q(t)|} \leq T_{2n}(\sec(s/4)) = \frac 12 ((\sec(s/4) + \tan(s/4))^{2n} + (\sec(s/4) - \tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-π,π]: |Q(t)| \leq 1\}) \geq 2π-s\,, \qquad s \in (0,2π)\,,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A \subset {\Bbb R}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{2n}(\cos t) = \cos(2nt)$ for $t \in {\Bbb R}$. This inequality is sharp. We also prove that $$\max_{t \in [-π,π]}{|…

关于超球多项式的Turán不等式

On Turán inequality for ultraspherical polynomials

本文证明了归一化超球多项式满足带权重因子 $|x|^θ$ 的更强 Turán 不等式，并将类似推广结果扩展到由三项递推定义的对称正交多项式。

A sharpening of a problem on Bernstein polynomials and convex function and related results

We present a short proof of a conjecture proposed by I. Raşa (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Raşa (2017). The methods of our proof allow us to obtain some extended versions of this inequality as well as other inequalities given by I. Raşa. As a tool we use stochastic convex ordering relations. We propose also some generalizations of the binomial convex concentration inequality. We use it to insert some additional expressions between left and right sides of the Raşa inequalities.

Dicrete Analogues of the Laguerre Inequality

It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!} \ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two consecutive zeros of $f(x)$ is at least $\sqrt{4-\frac{6}{m+2}}.$ These inequalities are a surprisingly similar discrete analogue of higher degree generalizations of the Laguerre and Turan inequalities. Being applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds uniform in all parameters involved, on the polynomials and …

Sharp Hardy's inequality for Jacobi and symmetrized Jacobi trigonometric expansions

Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type parameters $α,β\in(-1,\infty)^d$, whereas in the function systems for $α,β\in[-1/2,\infty)^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy's inequality are obtained with the same restrictions on the parameters $α$ and $β$.

Positivity of Turán determinants for orthogonal polynomials

The orthogonal polynomials $p_n$ satisfy Turán'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 general criteria for orthogonal polynomials to satisfy Turán's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the $q$--ultraspherical polynomials.

Polynomial Hamiltonian Systems with Movable Algebraic Singularities

The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic continuation along a rectifiable curve, are at most algebraic branch points.

一个Stein-Tomas型估计与一个解耦不等式

A Stein-Tomas type estimate and a decoupling inequality

本文利用多项式划分方法证明了Stein-Tomas型不等式和弱解耦不等式，并揭示了它们与Waring问题的紧密联系。

On some Algebraic Properties for q-Meixner Multiple Orthogonal Polynomials of the First Kind

We study a new family of q-Meixner multiple orthogonal polynomials of the first kind. The discrete orthogonality conditions are considered over a non-uniform lattice with respect to different q-analogues of Pascal distributions. We address some algebraic properties, namely raising and lowering operators as well as Rodrigues-type. Based on the explicit expressions for the raising and lowering operator a high-order linear q-difference equation with polynomial coefficients for the q-Meixner multiple orthogonal polynomials of the first kind is obtained. Finally, we obtain the nearest neighbor recu…

Positivity of Turán determinants for orthogonal polynomials II

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Turán'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 criterion for orthogonal polynomials to satisfy Turán's inequality. This extends essentially the results of \cite{szw}. In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.

Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials

We present a new, very short proof of a conjecture by I. Raşa, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. Wąsowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Raşa inequality. We give also the corresponding inequalities for other distributions. Our methods all…

Sum of Square Proof for Brascamp-Lieb Type Inequality

Brascamp-Lieb inequality is an important mathematical tool in analysis, geometry and information theory. There are various ways to prove Brascamp-Lieb inequality such as heat flow method, Brownian motion and subadditivity of the entropy. While Brascamp-Lieb inequality is originally stated in Euclidean Space, discussed Brascamp-Lieb inequality for discrete Abelian group and discussed Brascamp-Lieb inequality for Markov semigroups. Many mathematical inequalities can be formulated as algebraic inequalities which asserts some given polynomial is nonnegative. In 1927, Artin proved that any non- neg…

The Kato-Ponce Inequality with Polynomial Weights

We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights $\langle x\rangle^a = (1+|x|^2)^{a/2}$ for $a\ge 0$. We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential $J^s = (1- \De)^{{s}/{2}}$ holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree $s$. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponc…

Global bounds on stable polynomials

A classical inequality of Szász bounds polynomials with no zeros in the upper half plane entirely in terms of their first few coefficients. Borcea-Brändén generalized this result to several variables as a piece of their characterization of linear maps on polynomials preserving stability. In this paper, we improve Szász's original inequality, use determinantal representations to prove Szász type inequalities in two variables, and then prove that one can use the two variable inequality to prove an inequality for several variables.

Inverse polynomial images which consists of two Jordan arcs - An algebraic solution

Inverse polynomial images of $[-1,1]$, which consists of two Jordan arcs, are characterised by an explicit polynomial equation for the four endpoints of the arcs.

An algebraic interpretation of the continuous big q-Hermite polynomials

The continuous big $q$-Hermite polynomials are shown to realize a basis for a representation space of an extended $q$-oscillator algebra. An expansion formula is algebraically derived using this model.

A sharp Bernstein-type inequality and application to the Carleson embedding theorem with matrix weights

We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.

Approximating classes of functions defined by operators of differentiation or operators of generalised translation by means of algebraic polynomials

In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of classes of functions defined by the order of best approximation by algebraic polynomials.

Generalizations of Goncalves' inequality

If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots $α_1$, ..., $α_N$, then Gonçalves' inequality states that $\|F\|_2^2$ is bounded below by $\abs{a_N}^2 (\prod_{n=1}^N \max\{1, \abs{α_n}^2\} + \prod_{n=1}^N \min\{1, \abs{α_n}^2\})$. We establish generalizations of this inequality for other $L_p$ norms, and derive additional lower bounds on the $L_p$ norms of a polynomial in terms of its coefficients.

二维实值多项式的无维马尔可夫-伯恩斯坦型不等式

Dimension-free Markov--Bernstein-type inequality for $\mathbb R^2$-valued polynomials

该研究证明了$\boldsymbol{R}^n$到$\boldsymbol{R}^2$的次数不超过$d$的多项式映射的无维马尔可夫-伯恩斯坦型不等式，所得高斯不等式解答了Nourdin等人关于马利亚万行列式期望下界的问题。

A family of quadratic polynomial differential systems with algebraic solutions of arbitrary high degree

We show that the algebraic curve $a_0(x)(y-r(x))+p_2(x)a'(x)=0,$ where $r(x)$ and $p_2(x)$ are polynomial of degree 1 and 2 respectively and $a_0(x)$ is a polynomial solution of the convenient Fucsh's equation, is an invariant curve of the quadratic planar differential system. We study the particular case when $a_0(x)$ is an orthogonal polynomials. We prove that that in this case the quadratic differential system is Liouvillian integrable.

Umbral-Algebraic Methods and Asymptotic Properties of Special Polynomials

A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial families, with the use of elementary algebraic tools.

An algebraic interpretation of the $q$-Meixner polynomials

An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary $q$-pseudorotation operators. These operators are built from $q$-exponentials of the $\mathcal{U}_q(\mathfrak{su}(1,1))$ generators. The orthogonality, recurrence relation, difference equation, and other properties of the $q$-Mexiner polynomials are systematically obtained in the proposed framework.

An algebraic interpretation of the multivariate $q$-Krawtchouk polynomials

The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations and recurrence relations. The extension to an arbitrary number of variables is presented