Zygmund-type Inequalities Research Articles

On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

Abstract In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

Generalization of Zygmund type inequalities for the sth derivative of polynomials

If p(z) is a polynomial of degree n and p(z) ≠ 0 in |z| < 1, it was proved by Hans and Lal [Anal. Math. 40, 105-115(2014)] that for any β ∈ C with |β| ≤ 1,1 ≤ s ≤ n, In this paper, we prove an inequality which gives an improved and generalized extension of the above inequality into Lγ norm.

(p, q)-Regular operators between Banach lattices

We study the class of (p, q)-regular operators between quasi-Banach lattices. In particular, a representation of this class as the dual of a certain tensor norm for Banach lattices is given. We also provide some factorization results for (p, q)-regular operators yielding new Marcinkiewicz–Zygmund type inequalities for Banach function spaces. An extension theorem for $$(q, \infty )$$ -regular operators defined on a subspace of $$L_q$$ is also given.

New Integral Mean Estimates for the Polar Derivative of a Polynomial

Let p(z) be a polynomial of degree n and for α ∈ C, let Dαp(z):= np(z) + (α − z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain certain integral inequalities concerning the polar derivative of a polynomial, which besides yielding some interesting results, also includes some well-known theorems as special cases. Moreover we refine some Zygmund type inequalities for the polar derivative of a polynomial and present compact generalizations of some prior results.

Marcinkiewicz–Zygmund type results in multivariate domains

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in $${\mathbb{R}^d}$$ . These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.

Upper and lower bounds of large deviations for some dependent sequences

Let $${\{X_{n}, n\geq1\}}$$ be a sequence of random variables with $${S_n=\sum_{i=1}^nX_i}$$ and $${M_n=\max \{X_1,X_2,\ldots, X_n\}}$$ . Under some suitable conditions, we establish the upper bound of large deviations for $${S_n}$$ and $${M_n}$$ based on some dependent sequences including acceptable random variables, widely acceptable random variables and a class of random variables that satisfies the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality. In addition, the lower bound of large deviations for some dependent sequences is also obtained. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables.

Some Zygmund type inequalities for the sth derivative of polynomials

For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that $$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$ for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the Lp mean extension of the above and other related results for the sth derivative of polynomials.

Polynomials with polar derivatives

In this paper, we present an integral inequality for the polar derivative of polynomials. Our theorem includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

Integral inequalities concerning polynomials with polar derivatives

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P 0 (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

Moment inequalities for m-negatively associated random variables and their applications

The moment inequalities for m-NA random variables, especially the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality are established and the Khintchine–Kolmogorov convergence theorem and the three series theorem for m-NA random variables are also obtained. As one application of the moment inequalities, we study the large deviation for least squares estimator in nonlinear regression models under some general conditions. As another application, we investigate the strong consistency for least squares estimator in multiple linear regression models based on m-NA random variables.

Moment bounds for dependent sequences in smooth Banach spaces

We prove a Marcinkiewicz–Zygmund type inequality for random variables taking values in a smooth Banach space. Next, we obtain some sharp concentration inequalities for the empirical measure of {T,T2,⋯,Tn}, on a class of smooth functions, when T belongs to a class of nonuniformly expanding maps of the unit interval.

Equivalent Moser type inequalities in [formula omitted] and the zero mass case

We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi–Tanaka inequality which we prove to be attained. Then, we consider smooth compactly supported functions with respect to the Dirichlet norm ‖∇⋅‖2, and we prove an optimal Lorentz–Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H1(R2) such as the Adachi–Tanaka inequality and a version of Ruf's inequality.

On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models

In this paper, some probability inequalities and moment inequalities for widely orthant-dependent (WOD, in short) random variables are presented, especially the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality. By using these inequalities, we further study the complete convergence for weighted sums of arrays of row-wise WOD random variables and give some special cases, which extend some corresponding ones for dependent sequences. As applications, we present some sufficient conditions to prove the complete consistency for the estimator of nonparametric regression model based on WOD errors by using the complete convergence that we established. At last, the choice of the fixed design points and the weight functions for the nearest neighbor estimates is proposed. Our results generalize some known results for independent random variables and some dependent random variables.

Weighted estimates of truncated potential kernels in the variable exponent setting

In connection with the applications to the study of classical operators of harmonic analysis in variable exponents Morrey spaces, we prove estimates of weighted norms of potential kernels truncated to balls B(x, r). We admit a class of radial type weights. The conditions on the validity of estimates are given in terms of Zygmund type inequalities on weight or in terms of their Matuszewska–Orlich indices

Some results concerning growth of polynomials

In this paper, we establish inequalities for polynomials with restricted zeros, which in particular yields interesting generalizations of some Zygmund type inequalities for polynomial.

Some integral mean estimates for polynomials

In this paper we establish Lq inequalities for polynomials, which in particular yields interesting generalizations of some Zygmund-type inequalities.

Mean convergence of orthogonal Fourier series and interpolating polynomials

For a family of weight functions that include the general Jacobi weight functions as special cases, exact condition for the convergence of the Fourier orthogonal series in the weighted L p space is given. The result is then used to establish a Marcinkiewicz--Zygmund type inequality and to study weighted mean convergence of various interpolating polynomials based on the zeros of the corresponding orthogonal polynomials.

Zygmund type inequalities for double singular Cauchy-Stieltjes integral

For the double singular Cauchy-Stiltjes integral over spanning set of a bicylindric domain, a Zygmund type inequality connecting partial and mixed moduli of continuity of the singular integral and its density, is obtained. On this basis, some spaces are constructed, invariant with respect to the double singular integral. Mathematics subject classification (1991): 30A10.