Related papers: Polynomial and rational inequalities on Jordan arc…

Borwein and Erd\'elyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs…

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green's functions with…

Jordan analytic curves which are invariant under rational functions are studied

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral…

We prove that every function that is meromorphic on the closure of an analytic Jordan domain and sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the…

Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational…

We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel.…

In this paper we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically…

We consider the series of the Bergman orthogonal polynomials associated with a bounded simply-connected domain in the complex plane, whose boundary is a Jordan curve. These are the polynomials that are orthonormal with respect to the area…

The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of…

We prove in this article the generalizations on the exponential Orlicz spaces Markov's - Bernstein's inequalities for algebraic polynomials and rational functions.

Given $n\geq1$ and $r\in[0, 1),$ we consider the set $\mathcal{R}_{n, r}$ of rational functions having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D},$ were $\mathbb{D}$ is the unit disc of the complex plane. We give an…

We introduce a new canonical height function for Jordan blocks of small eigenvalues for endomorphisms on smooth projective varieties over a number field. We prove that under an assumption on the eigenvalues of the endomorphism on the group…

We establish the strong asymptotics for Bergman orthogonal polynomials defined over Jordan domains with corners. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for domains bounded by…

In this paper, we investigate the problem of finding tight linear lower bounding functions for multivariate polynomials over boxes. These functions are obtained by the expansion of polynomials into Bernstein form and using the linear least…

This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/(sinh x) are proved.

In this paper we present a geometric proof of the following fact. Let $D$ be a Jordan domain in $\mathbb{C}$, and let $f$ be analytic on $cl(D)$. Then there is an injective analytic map $\phi:D\to\mathbb{C}$, and a polynomial $p$, such that…

We present a short proof of a conjecture proposed by I. Ra\c{s}a (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Ra\c{s}a (2017). The methods of our proof…

We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted.

We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when…