Faber Polynomials with common zero

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.

in Russian