Faber Polynomials with common zero

We describe the two sets of meromorphic univalent functions in the class $Σ$, 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.

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