Béla Nagy

We establish the existence and uniqueness of rational conformal maps of minimal degree \(n+1\) for opening up \(n\) arcs. In earlier results, the degree was exponential in \(n\). We also discuss two related problems. (a) We establish existence of rational functions of minimal degree with prescribed critical values, and show that the number of (suitably normalized) rational functions is given in terms of the Hurwitz numbers. (b) We consider the problem of finding rational functions of minimal degree with prescribed critical points, where we establish existence of solutions by considering certain polynomial equations, and where the number of normalized solutions is bounded from above by a Catalan number. We illustrate our results with two examples.

For a fixed positive integer n consider continuous functions $$K_1,\dots $$K 1 , ⋯ , $$ K_n:[-1,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$K n : [ - 1 , 1 ] → R ∪ { - ∞ } that are concave and real valued on $$[-1,0)$$[ - 1 , 0 ) and on (0, 1], and satisfy $$K_j(0)=-\infty $$K j ( 0 ) = - ∞ . Moreover, let $$J:[0,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$J : [ 0 , 1 ] → R ∪ { - ∞ } be upper bounded and such that $$[0,1]\setminus J^{-1}(\{-\infty \})$$[ 0 , 1 ] \ J - 1 ( { - ∞ } ) has at least $$n+1$$n + 1 elements, but it is arbitrary otherwise. For $$x_0:=0<x_1<\dots < x_n \le x_{n+1}:=1$$x 0 : = 0 < x 1 < ⋯ < x n ≤ x n + 1 : = 1 , so called nodes, and for $$t\in [0,1]$$t ∈ [ 0 , 1 ] consider the sum of translates function $$F(x_1,\ldots ,x_n,t):=J(t)+\sum _{j=1}^n K_j(t-x_j)$$F ( x 1 , … , x n , t ) : = J ( t ) + ∑ j = 1 n K j ( t - x j ) , and the vector of interval maximum values $$m_j:=m_j(x_1,\ldots ,x_n):=\max _{t\in [x_j,x_{j+1}]}F(x_1,\ldots ,x_n,t)$$m j : = m j ( x 1 , … , x n ) : = max t ∈ [ x j , x j + 1 ] F ( x 1 , … , x n , t ) ( $$j=0,1,\ldots ,n$$j = 0 , 1 , … , n ). We describe the structure of the arising interval maxima as the nodes run over the n-dimensional simplex. Applications presented here range from abstract moving node Hermite–Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov’s problem to more abstract results of interpolation theoretic flavour.

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erd\'elyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.

MSC: 26A51, 41A50 DOI: 10.21538/0134-4889-2022-28-4-262-272 In previous papers we investigated so-called sum of translates functions $F(\mathbf x,t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb R}:=\mathbb R\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb R}$ is a fixed

The famous Jones–Ruscheweyh theorem states that n distinct points on the unit circle can be mapped to n arbitrary points on the unit circle by a Blaschke product of degree at most $$n-1$$n - 1 . In this paper, we provide a new proof using real algebraic techniques. First, the interpolation conditions are rewritten into complex equations. These complex equations are transformed into a system of polynomial equations with real coefficients. This step leads to a “geometric representation” of Blaschke products. Then another set of transformations is applied to reveal some structure of the equations. Finally, the following two fundamental tools are used: a Positivstellensatz by Prestel and Delzell describing positive polynomials on compact semialgebraic sets using Archimedean module of length N. The other tool is a representation of positive polynomials in a specific form due to Berr and Wörmann. This, combined with a careful calculation of leading terms of occurring polynomials finishes the proof.

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons.