Related papers: The Sharp Erd\H{o}s-Tur\'an Inequality

Here we consider the hypergraph Tur\'an problem in uniformly dense hypergraphs as was suggested by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density $\pi_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for…

For rational functions, we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erd\"os-Lax and Tur\'an to rational functions R. In return these…

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the…

In 1939 Tur\'{a}n raised the question about lower estimations of the maximum norm of the derivatives of a polynomial $p$ of maximum norm $1$ on the compact set $K$ of the complex plain under the normalization condition that the zeroes of…

Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove…

The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…

In 1993, Fishburn and Graham established the following qualitative extension of the classical Erd\H{o}s-Szekeres theorem. If $N$ is sufficiently large with respect to $n$, then any $N\times N$ real matrix contains an $n\times n$ submatrix…

There exists a positive function $\psi(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$\Omega${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${,…

Recently, the higher order Tur\'{a}n inequalities for the Boros-Moll sequences $\{d_\ell(m)\}_{\ell=0}^m$ were obtained by Guo. In this paper, we show a different approach to this result. Our proof is based on a criterion derived by Hou and…

In 1964, Erd\H{o}s proposed the problem of estimating the Tur\'an number of the $d$-dimensional hypercube $Q_d$. Since $Q_d$ is a bipartite graph with maximum degree $d$, it follows from results of F\"uredi and Alon, Krivelevich, Sudakov…

Let an algebraic polynomial $P_n(\zeta)$ of degree $n$ be such that $|P_n(\zeta)|\le 1$ for $\zeta\in E\subset\mathbb{T}$ and $|E|\ge 2\pi -s$. We prove the sharp Remez inequality $$ \sup_{\zeta\in\mathbb{T}}|P_n(\zeta)|\le…

We show how Turan's inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0$ for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities.…

In 1959, Erd\H{o}s and Szekeres posed a series of problems concerning the size of polynomials of the form $$ P_n(z) = \prod_{j=1}^n (1 - z^{s_j}), $$ where $s_1, \dots, s_n$ are positive integers. Of particular interest is the quantity $$…

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to $1$, then the sum of absolute values of its terms is less than or equal to $1$ for the subdisk…

Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erd\H{o}s, Ko, and Rado in 1961 established that any $k$-uniform intersecting family on $[n]$ has a maximum size of…

Originally, Tur\'{a}n's inequality states that if $(P_n(x))_{n\in\mathbb{N}_0}$ is the sequence of Legendre polynomials, then $\Delta_n(x):=P_n^2(x)-P_{n+1}(x)P_{n-1}(x)\geq0$ for all $n\in\mathbb{N}$ and $x\in[-1,1]$. Gasper specified the…

We prove an Erd\H{o}s--Tur\'an type inequality for compact Lie groups, from which we deduce an effective version of Deligne's equidistribution theorem.

The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for…

Bukh and Conlon used random polynomial graphs to give effective lower bounds on $\mathrm{ex}(n,\mathcal{T}^\ell)$, where $\mathcal{T}^\ell$ is the $\ell$th power of a balanced rooted tree $T$. We extend their result to give effective lower…

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Tur\'an's theorem, the…