Faber Polynomials with common zero

TL;DR

This paper characterizes two classes of meromorphic univalent functions in the class Σ based on the root properties of their Faber polynomials, providing explicit formulas for these polynomials.

Contribution

It introduces two specific classes of functions with unique Faber polynomial root properties and derives explicit forms for their Faber polynomials.

Findings

Identified two classes of functions with specific Faber polynomial root behaviors.

Derived explicit formulas for Faber polynomials in these classes.

Established conditions on the roots of Faber polynomials for these functions.

Abstract

We describe the two sets of meromorphic univalent functions in the class $\Sigma$, for which the sequence of Faber polynomials $\{F_j\}_{j=1}^\infty $ have the roots with following properties respectively: $\sum_{j=1}^{n}|F_j(z_0)|=0<|F_{n+1}(z_0)|, $ $n\in\mathbb N, $ and $|F_1(z_0)|> 0=\sum_{j=2}^{\infty}|F_j(z_0)|$. We found an explicit form of Faber polynomials for such functions.