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Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$ which are holomorphic on $X$ and such that $\varphi_n(x)\to x$, for every $x \in X$. Every convex domain has this property.

If $n=1$ is this property equivalent to $X$ being a Jordan domain?

If $n>1$ and $X$ is pseudoconvex what are some sufficient conditions on $X$ (other than convexity) so that it has this property?

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  • $\begingroup$ Can you explain what $\phi(x)$ is? $\endgroup$
    – Alexandre Eremenko
    Commented Aug 5, 2020 at 13:42
  • $\begingroup$ @AlexandreEremenko it was just an error, sorry $\endgroup$
    – erz
    Commented Aug 5, 2020 at 13:44
  • $\begingroup$ I guess it holds also for images of biholomorphic maps from the closed bounded starshaped domains into \mathbb{C}^n. $\endgroup$
    – Luka Thaler
    Commented Aug 6, 2020 at 14:17

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