Vector Fields on the Space of Functions Univalent Inside the Unit Disk via Faber Polynomials

TL;DR

This paper develops a method to describe Kirillov vector fields on univalent functions within the unit disk using Faber polynomials, leveraging their generating functions for a novel analytical approach.

Contribution

It introduces a new representation of Kirillov vector fields via Faber polynomials, connecting function theory with polynomial generating functions.

Findings

Explicit formula for Kirillov vector fields in terms of Faber polynomials

Utilization of generating functions to analyze univalent functions

Enhanced understanding of the structure of univalent function spaces

Abstract

We obtain the Kirillov vector fields on the set of functions $f$ univalent inside the unit disk, in terms of the Faber polynomials of $1/f(1/z)$. Our construction relies on the generating function for Faber polynomials.