Faber polynomials with common roots

Abstract

We describe two sets of meromorphic univalent functions in the class $\Sigma$, for which the sequences 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