Related papers: Lower bound for cyclic sums of Diananda type

Given an integer $m\geq2$, the Hardy--Littlewood inequality (for real scalars) says that for all $2m\leq p\leq\infty$, there exists a constant $C_{m,p}% ^{\mathbb{R}}\geq1$ such that, for all continuous $m$--linear forms…

For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function $\sgn(x)$ on the two symmetric intervals $[-1,-a]\cup[a,1]$ by rational functions with the only possible poles of degree $2k-1$ at the origin and of…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…

Let $A^2(D)$ be the Bergman space over the open unit disk $D$ in the complex plane. Korenblum conjectured that there is an absolute constant $c \in (0,1)$ such that whenever $|f(z)|\le |g(z)|$ in the annulus $c<|z|<1$ then $||f(z)|| \le…

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = \sigma + it$) is used to obtain estimates for $$…

We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical…

Two sharp lower bounds for the length of a longest cycle $C$ of a graph $G$ are presented in terms of the lengths of a longest path and a longest cycle of $G-C$, denoted by $\overline{p}$ and $\overline{c}$, respectively, combined with…

Let $ x = [0;a_1,a_2,...]$ be the decomposition of the irrational number $x \in [0,1]$ into regular continued fraction. Then for the derivative of the Minkowski function $?(x)$ we prove that $?'(x) = +\infty$ provided $ \limsup_{t\to…

A theorem on the existence of exactly $N$ limit cycles around a critical point for the Lienard system $\ddot{x}+f(x) \dot{x}+g(x) =0$ is proved. An alogrithm on the determination of a desired number of limit cycles for this system has been…

Let $\theta >2$ be real and non-integral with integer part $n = \lfloor \theta \rfloor$ and let $ \phi (x)$ be a generalised polynomial with leading term $x^\theta.$ We establish a mean value estimate for the exponential sum…

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot…

We show that for every $0 < \epsilon \leq 1$ and integer $k\geq 1$, there exists an integer $n = n(\epsilon,k)$ so that for all primes $p$, and integers $0 \leq a \leq p-1$, there exist integers $1 \leq x_1 < ... < x_n \leq p^\epsilon$ such…

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $C(k,\ell)$ is a digraph consisting of two internally vertex disjoint directed paths of lengths $k$ and $\ell$ with the same initial vertex and terminal vertex.…

We revisit the Callias index formula for Dirac-type operators $L$ in odd space dimension $n$, and prove that \begin{align} \text{ind} \, (L) =\bigg(\frac{i}{8\pi}\bigg)^{\frac{n-1}{2}}\frac{1}{2(\frac{n-1}{2})!} \lim_{\Lambda…

The logarithm of the number of binary n-variable bent functions is asymptotically less than $11(2^n)/32$ as n tends to infinity. Keywords: boolean function, Walsh--Hadamard transform, plateaued function, bent function, upper bound

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be a Banach algebra with its standard norm || ||_n and the pointwise product; so, there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || f ||_{n} || g…

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and…

For positive integers $n$, $m$, and $h$, we let $\rho \hat{\;}(\mathbb{Z}_n, m, h)$ denote the minimum size of the $h$-fold restricted sumset among all $m$-subsets of the cyclic group of order $n$. The value of $\rho \hat{\;}(\mathbb{Z}_n,…

Given an irrational alpha and an x in the unit interval, the set of balanced times, for which the same number of (k*alpha+x) (modulo one) are less than or equal to one half as are larger than one half, is in general infinite, but sparse in…