Question:
Given a compact subset $K \subset \mathbb{C}$, and (non-polynomial) functions $h_i:K\rightarrow \mathbb{C}$, with $h_i\in \mathcal{A}(K)$ (that is each $h_i$ is holomorphic on the interior of $K$), are there any known sufficient criteria for $\{h_i\}$ to be dense in $\mathcal{A}(K)$?
Related:
A more general version of this question was asked on MathOverflow 7 years ago, but with inconclusive answer.
Nachbin's theorem says that an involutive subalgebra which separates points and tangent vectors is dense in the space of smooth functions. But the $h_i$ being holomorphic precludes them from forming an involutive subalgebra.
Mergelyan's theorem provides a result for the denseness of polynomials in $\mathcal{A}(K)$ so long as $\mathbb{C}\backslash K$ is connected. But I would not like to assume the $h_i$ to be polynomials.
Various Stone-Weierstrass theorems are concerned with continuity instead of holomorphicity and also require involutive sub-algebras.
Thanks for any help!
Edit: Corrected a misformulation of Nachbin's formulation: I originally said it guarantees denseness in continuous functions as opposed to smooth functions.
This is a long comment that is definitely not an answer but indicates a direction to look into if of course of interest.
First note that this is a very vague question and unless more details about why is of interest, it's going to be hard to answer - probably looking at research around Mergelyan's theorem gives stuff that may be of interest for general $K$
On the other hand, universality properties in the critical strip of (analytic continuations of) Dirichlet series based on Dirichlet characters and their derivatives say, give infinitely many such classes of dense analytic functions on disks as essentially any nontrivial linear combination (for same character modulus so the Davenport Heilbronn function for example) or derivative of such (eg $\zeta'(s)$) has dense vertical translates in some specific critical strip disc and of course this can be moved to any disc in the plane, so one gets such a class for each function as above.
A good article for this direction is Bagchi A joint universality theorem for Dirichlet L-functions