Related papers: An improved Copson inequality
The aim of this work is to improve Wilker inequalities near the origin and {\pi}/2.
A well known result in the theory of uniform distribution modulo one (which goes back to Fej\'er and Csillag) states that the fractional parts $\{n^\alpha\}$ of the sequence $(n^\alpha)_{n\ge1}$ are uniformly distributed in the unit…
Based on a new idea of factorization, we prove an improved discrete Rellich inequality and discuss its optimality. We also give a conjecture on improved higher order discrete Hardy-like inequalities and formulate an open problem for the…
In this paper, we sharpen and generalize Carlson's double inequality for the arc cosine function.
$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…
One aspect of Poisson approximation is that the support of the random variable of interest is often finite while the support of the Poisson distribution is not. In this paper we will remedy this by examining truncated negative binomial (of…
In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the…
In this paper we give a conditional improvement to the Elekes-Szab\'{o} problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of…
Let $x=[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of $x\in[0,1)$. We prove that the Hausdorff dimension of \begin{equation*}E_{even}=\{x\in[0,1)\colon a_{2n}(x)\to\infty\ (n\to\infty)\}.\end{equation*} is 1/2. In general,…
In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s},…
We present a recent analysis of epsilon'/epsilon in the 1/N_c expansion. We show that the 1/N_c corrections to the matrix element of Q_6 are large and positive, indicating a Delta I = 1/2 enhancement similar to the one of Q_1 and Q_2 which…
In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack…
We study the simultaneous estimation of multiple phases as a discretised model for the imaging of a phase object. We identify quantum probe states that provide an enhancement compared to the best quantum scheme for the estimation of each…
Given $n\in N$ and $x,\gamma\in R$, let \begin{equation*} ||\gamma-nx||^\prime=\min\{|\gamma-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime…
We study the systematic inclusion of QED corrections in the evolution of parton distributions. O(alpha) corrections modify the evolution equation for parton distributions. They introduce additional parton distributions, like the photon…
Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in…
We show that sequences of the form $\alpha n^{\theta} \pmod{1}$ with $\alpha > 0$ and $0 < \theta < \tfrac{43}{117} = \tfrac{1}{3} + 0.0341 \ldots$ have Poissonian pair correlation. This improves upon the previous result by Lutsko,…
A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha)$ in the continued fraction expansion of $\alpha$ converges to a number $K = 2.6854520\ldots$ (Khinchin's…
Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le \rho$, where $\rho>1$. Cassels proved that, under an additional restriction on $\rho$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le…
In this paper, we present a solution to the inequality $$ \bigg( \int_0^{\infty} \bigg( \int_x^{\infty} \bigg( \int_0^t h \bigg)^q w(t)\,dt \bigg)^{r / q} u(x)\,ds \bigg)^{1/r}\leq C \, \bigg( \int_0^{\infty} h^p v \bigg)^{1 / p}, \quad h…