An extremal property of Fekete polynomials

An Extremal Property of Fekete Polynomials

Zeros of Fekete Polynomials

An Extremal Problem for Polynomials

Let f(z) = z" Hbe a polynomial such that the level set E = {z : |/(z)| < 1} is connected. Then max{|/'(z)| : z e E} < 2<1/")-1n2 , and this estimate is the best possible. The following question was asked in [1, Problem 4.8]. Let f be a polynomial of degree n, (1) f{z)~z», z^oo. Put Ef = {z : \f(z)\ < 1} , and assume that Ef is connected. Is it true that (2) max{\f'(z)\:z£Ef}. Pommerenke ...

Fekete Polynomials and Shapes of Julia Sets

We prove that a nonempty, proper subset S of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if S is bounded and Ĉ \ int(S) is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation i...

An Extremal Property of Turán Graphs

Let Fn,tr(n) denote the family of all graphs on n vertices and tr(n) edges, where tr(n) is the number of edges in the Turán’s graph Tr(n) – the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G with at most λ colors, and let f(n, tr(n), λ) = max{PG(λ) : G ∈ Fn,tr(n...