Hi, my question is :

Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. $ Does there exist a complex analytic diffeomorphism $F$ ( analytic in two complex variables ) whose domain is either $D^2\subset \mathbb{C^2}$ or the complex 2-dimensional unit open polydisk $\Omega ={{(z,w): |z|^2+|w|^2 < 1}}$ in $\mathbb{C^2}$ such that its restriction to $D\subset D^2$ or $D\subset \Omega $ is $f$ ? By restriction , I mean $ F(z,0) = f(z) $ in the case of $ D \subset D^2 \subset \mathbb {C}^2 $ .

The range of $F$ does not necessarily have to be $D^2$ or $\Omega$, but it would be even better if they are !

If this is a very well-known result, you can cite a reference.

Is the same result true in 1-dimension as well , i.e. replacing $D$ by $I\subset R$ and changing the complex-analytic/conformal diffeomorphism $F$ accordingly , i.e. asking that domain of $F$ is $I^2$ or $D$ with restriction $ f $ ?

Thank you .