Faber polynomials with common zero

Abstract

We describe the set of meromorphic univalent functions in the class $$\Sigma $$ , for which the sequence of the Faber polynomials $$\{F_j\}_{j=1}^\infty $$ have the roots with following properties $$|F_n (z_0)|>0=\sum _{\begin{array}{c} j=1 j\not =n \end{array}}|F_j (z_0 )|$$ . For such functions we found an explicit form of the Faber polynomials as well as we discussed some properties.