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I posted this on Math.Stack.Exchange with no luck, so I thought it would be perhaps better suited for this site.

We recall that a domain of holomorphy is a domain in $\mathbb{C}^n$ that is holomorphically convex.

It is quite standard in many textbooks, for example Shabat's Introduction to Complex Analysis - Part II, that if $D$ is a domain of holomorphy, then $f(D)$ is a domain of holomorphy if $f$ is a biholomorphic mapping.

I cannot find any proofs of necessity however. Can we weaken the assumptions on $f$ at all?

Moreover, what about $\mathscr{F}$-convex domains, for some class $\mathscr{F}$? That is, under what conditions is the image of an $\mathscr{F}$-convex set again $\mathscr{F}$-convex?

Natural choices for $\mathscr{F}$ include holomorphic functions, polynomials and rational functions.

If $\mathscr{F}$ is taken to be linear functions, then we recall that such functions that map $\mathscr{F}$-convex sets to $\mathscr{F}$-convex sets are the affine mappings.

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    $\begingroup$ Cartan and Thullen proved an open connected set $D\subset \mathbf{C}^n$ is a domain of holomorphy if and only if it is a Stein space. For such $D$, is your question to give interesting necessary conditions on a holomorphic $f:D \to \mathbf{C}^n$ so $f(D)$ is open and a domain of holomorphy? If $f$ has discrete fibers then its image is open (see 4.3 in Ch. 5 in Coherent Analytic Sheaves). The map $q:D\to f(D)$ must then be flat, so if its analytic fibers are finite with constant rank then $q$ is a finite flat surjection, so $f(D)$ is Stein by the "Theorem B" criterion for Steinness. $\endgroup$
    – nfdc23
    Commented Dec 4, 2017 at 5:10
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    $\begingroup$ @nfdc23 .q finite surjective is enough.See proposition 73.1 page 313 Kaup and Kaup Holomorphic functions in Several Complex Variables. $\endgroup$
    – Mohan Ramachandran
    Commented Dec 4, 2017 at 17:53

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