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4 changed condition for preservation of boundary of unit disk.

Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $\lim_{|q|\rightarrow 1} \xi(q)=q$. |q|=1$then$|\xi(q)|=1$. Is there a method from Teichmüller theory that allows us to explicitly write down the coefficients$b_{n}$of$s(\xi(q)) = \sum_{n=0}^{\infty} a_{n}\xi(q)^{n} = \sum_{n=0}^{\infty} b_{n}q^{n}$given some explicit$\xi$? 3 added 21 characters in body Say I have a lacunary$q$series$s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$, and I have a quasiconformal transformation$\xi$which preserves the boundary of the unit disk in$\partial\mathbb{D}^{1}$\mathbb{C}$ such that $\lim_{|q|\rightarrow 1} \xi(q)=q$. Is there a method from Teichmüller theory that allows us to explicitly write down the coefficients $b_{n}$ of $s(\xi(q)) = \sum_{n=0}^{\infty} a_{n}\xi(q)^{n} = \sum_{n=0}^{\infty} b_{n}q^{n}$ given some explicit $\xi$?

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