On the local asymptotics of Faber polynomials

The Faber Polynomials for Circular Sectors

The Faber polynomials for a region of the complex plane, which are of interest as a basis for polynomial approximation of analytic functions, are determined by a conformai mapping of the complement of that region to the complement of the unit disc. We derive this conformai mapping for a circular sector {;: \z\ < 1, |argz| < i/a}, where a > 1, and obtain a recurrence relation for the coefficient...

Lattice Paths and Faber Polynomials

The rth Faber polynomial of the Laurent series f(t) = t + f0 + f1/t + f2/t + · · · is the unique polynomial Fr(u) of degree r in u such that Fr(f) = tr + negative powers of t. We apply Faber polynomials, which were originally used to study univalent functions, to lattice path enumeration.

Derivatives of Faber Polynomials and Markov Inequalities

We study asymptotic behavior of the derivatives of Faber polynomials on a set with corners at the boundary. Our results have applications to the questions of sharpness of Markov inequalities for such sets. In particular, the found asymptotics are related to a general Markov-type inequality of Pommerenke and the associated conjecture of Erdős. We also prove a new bound for Faber polynomials on p...

Faber Polynomials for Rational Mapping Functions

Zeros and Local Extreme Points of Faber Polynomials Associated with Hypocycloidal Domains∗

Faber polynomials play an important role in different areas of constructive complex analysis. Here, the zeros and local extreme points of Faber polynomials for hypocycloidal domains are studied. For this task, we use tools from linear algebra, namely, the Perron-Frobenius theory of nonnegative matrices, the Gantmacher-Krein theory of oscillation matrices, and the Schmidt-Spitzer theory for the ...