Riemann mapping

Question

Let in the complex plane be a bounded Jordan region T (that is a bounded and simply connected set with the boundary a Jordan curve), containing the origin, with its Riemann mapping onto the open unit disk, having in its Taylor expansion all the coefficients as real numbers.

Question : There exists a sequences of strictly decreasing (in the sense of set inclusion) and converging to T, Jordan domains T_{n} (including strictly T), so that their corresponding Riemann mappings, f_{n}, converge uniformly in the closure of T, to the Riemann mapping f and in addition, all f_{n} having their Taylor expansions with real coefficients ?

Remark. In the case when we do not require that the coefficients in the Taylor expansions of the Riemann mappings be all real numbers, the answer to the above questions is YES, see e.g. Theorem 4, p. 32, in the J.L. Walsh's book-"Interpolation and Approximation by Rational Functions in the Complex Domain."

Answer 1

A holomorphic function $f$ has a Taylor series with real coefficients if and only if it is symmetric with respect to the real line, i.e. satisfies $f(\bar z)=\overline{f(z)}$. Let's call such functions real maps.

If the Riemann map $f$ is a real map, then $T=f^{-1}(\mathbb D)$ is symmetric with respect to the real line. The converse is also true by uniqueness of a Riemann map with $f(0)=0$ and $f'(0)>0$.

The question thus reduces to finding a sequence of symmetric Jordan domains $T_n$ decreasing to the closure of $T$, for then the Riemann maps $f_n$ (which are then real maps) converge to $f$ locally uniformly by Carathéodory kernel theorem.

You can take for example $T_n := \mathbb C \setminus g(\overline{\mathbb{D}}(0,1-2^{-n}))$ where $g$ is the Riemann map from the unit disk to $\widehat{\mathbb{C}}\setminus \overline T$, with a simple pole of positive residue at the origin.