Explicit Faber Polynomials on Circular Sectors

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.

Faber Polynomials for Rational Mapping Functions

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...

Packing Polynomials on Multidimensional Integer Sectors

Denoting the real numbers and the nonnegative integers, respectively, by R and N, let S be a subset of Nn for n = 1, 2, . . ., and f be a mapping from Rn into R. We call f a packing function on S if the restriction f |S is a bijection onto N. For all positive integers r1, . . . , rn−1, we consider the integer sector I(r1, . . . , rn−1) = {(x1, . . . , xn) ∈ Nn | xi+1 6 rixi for i = 1, . . . , n...