Related papers: An improved Copson inequality

This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to Talagrand (\cite{TalL1L2}). As an application, the main result of this note is a…

Let $\boldsymbol{\alpha}\in \mathbb{R}^N$ and $Q\geq 1$. We consider the sum $\sum_{\boldsymbol{q}\in [-Q,Q]^N\cap\mathbb{Z}^N\backslash\{\boldsymbol{0}\}}\|\boldsymbol{\alpha}\cdot\boldsymbol{q}\|^{-1}$. Sharp upper bounds are known when…

We prove some extensions of Andrews inequality.

We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a…

In this paper, we improve on Fouvry and Radziwi{\l}{\l}'s results on unbalanced convolutions. In particular, we find that if $(\alpha_m)$ and $(\beta_n)$ are sequences supported on $m\sim M$ and $n\sim M$ where $\beta$ is equidistributed…

In this paper, we prove that the inequalities $\alpha [1/3 Q(a,b)+2/3 A(a,b)]+(1-\alpha)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <\beta [1/3 Q(a,b)+2/3 A(a,b)]+(1-\beta)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $\lambda [1/6 C(a,b)+5/6…

We consider a general class of sharp $L^p$ Hardy inequalities in $\R^N$ involving distance from a surface of general codimension $1\leq k\leq N$. We show that we can succesively improve them by adding to the right hand side a lower order…

We obtain an estimate for the cubic Weyl sum which improves the bound obtained from Weyl differencing for short ranges of summation. In particular, we show that for any $\varepsilon>0$ there exists some $\delta>0$ such that for any coprime…

Let $f$ be a nonnegative function supported on $(-1/4, 1/4)$. We show $$ \sup_{x \in \mathbb{R}}{\int_{\mathbb{R}}{f(t)f(x-t)dt}} \geq 1.28\left(\int_{-1/4}^{1/4}{f(x)dx} \right)^2,$$ where 1.28 improves on a series of earlier results. The…

In this note we improve the parameter $q$ that appears in Theorem 1 obtained by the author in [Math. Ineq. \& appl., Vol 19 (3) (2016), 1013-1030].

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear…

Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the…

Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a_1/q_1, ..., a_n/q_n with smaller denominators. We show that in the special cases of n=3 and n=4 and…

We find a new proof for the celebrated theorem of Keith and Zhong that a $(1,p)$-Poincar\'e inequality self-improves to a $(1,p-\epsilon)$-Poincar\'e inequality. The paper consists of a novel characterization of Poincar\'e inequalities and…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this…

In this paper, the improvement about the generalized Kolmogorov-type three series theorem, in the case of NQD random variables, is obtained by different method. Furthermore, the generalized Kolmogorov-type three series theorem is…

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n…

Let $M_{n,r}=(\sum_{i=1}^{n}q_ix_i^r)^{\frac {1}{r}}, r \neq 0$ and $M_{n,0}=\lim_{r \rightarrow 0}M_{n,r}$ be the weighted power means of $n$ non-negative numbers $x_i$ with $q_i > 0$ satisfying $\sum^n_{i=1}q_i=1$. For a real number…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow…