Related papers: An extremal problem for odd univalent polynomials

Let $A\subset \mathbf{N}$ be a finite set of $n=|A|$ positive integers, and consider the cosine sum $f_A(x)=\sum_{a\in A}\cos ax$. We prove that $$\min_x f_A(x)\leqslant -n^{ 1/7-o(1)},$$ thereby establishing polynomial bounds for the…

We show that the discrete Painlev\'e II equation with starting value $a_{-1}=-1$ has a unique solution for which $-1 < a_n < 1$ for every $n \geq 0$. This solution corresponds to the Verblunsky coefficients of a family of orthogonal…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve…

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^\alpha$, where $-1 < \alpha < 1$. We also give periodic versions of these results in which the majorants…

We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials,…

In this paper, we establish transcendental entire function $A(z)$ and polynomial $B(z)$ such that the differential equation $f''+A(z)f'+B(z)f=0$, has all non-trivial solution of infinite order. We use the notion of \emph{critical rays} of…

We derive bounds and asymptotics for the maximum Riesz polarization quantity $$M_n^p(A) := \max_{{\bold x}_1, {\bold x}_2, \ldots, {\bold x}_n \in A} {\min_{{\bold x} \in A}{\sum_{j=1}^n{\frac{1}{|{\bold x} - {\bold x}_j|^{p}}}}}$$ (which…

Consider the problem {ll} \Delta^2 u= \lambda e^{u} &\text{in} B u=\frac{\partial u}{\partial n}=0 &\text{on}\partial B, where $B$ is the unit ball in $\R^N$ and $\lambda$ is a parameter. Unlike the Gelfand problem the natural candidate…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary…

The extremal values of multivariate trigonometric polynomials are of interest in fields ranging from control theory to filter design, but finding the extremal values of such a polynomial is generally NP-Hard. In this paper, we develop…

We consider some known and some new properties of the family of polynomials introduced by Ted Suffridge in 1969. We begin by giving a brief overview of their extremal properties in classic and more recent work. We also give a compact form…

The present paper is a continuation of the author's previous works, in which necessary and sufficient local extrema at a stationary point of a polynomial or a power series (and thus of an analytic function) are given. It is known that for…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

We consider an extremal problem for polynomials, which is dual to the well-known Smale mean value problem. We give a rough estimate depending only on the degree.

We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schr\"odinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials.…

Let $\mathbb{F}_q$ be a finite field with $q=p^e$ elements, where $p$ is a prime and $e\geq 1$ is an integer. Let $\ell<n$ be two positive integers. Fix a monic polynomial $u(x)=x^n +u_{n-1}x^{n-1}+\cdots +u_{\ell+1}x^{\ell+1} \in…

We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$…

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, such polynomials have exponentially small bias. This also yields an exponential tail bound on the weight…

We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$…

Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient…