Uniform non-critical approximation on Arakelyan sets (including horizontal strips)

Determine whether uniform approximation by non-critical entire functions is achievable on all closed subsets E of the complex plane for which the complement in the Riemann sphere CP^1 is connected and locally connected (Arakelyan sets). In particular, ascertain whether every function f in the class A(E) that is holomorphic on the interior of E with nowhere-vanishing derivative admits, for every ε > 0, an entire function F with nowhere-vanishing derivative such that sup_{z ∈ E} |F(z) − f(z)| < ε; resolve this specifically for the closed horizontal strip E = {z ∈ C : α ≤ Im z ≤ β}.

Background

The paper establishes Carleman approximation by non-critical holomorphic functions on semi-admissible subsets of open Riemann surfaces, under natural connectedness and local connectedness assumptions, and provides an alternative method yielding uniform non-critical approximation on certain sets not covered by the semi-admissible framework. The methods rely on Runge-type approximation for non-critical functions and, in the plane, on approximating derivatives and integrating, which requires additional geometric bounds (e.g., path-length constraints).

For general sets of Carleman approximation, interpolation across regions of nonempty interior necessitates solving a \bar{\partial}-equation, which jeopardizes control over critical points; standard solutions with Hörmander-type estimates can also degrade the approximation on the set. In the classical (criticality-unconstrained) setting, Arakelyan's theorem characterizes closed sets E ⊂ C with CP^1 \ E connected and locally connected as sets of uniform approximation by entire functions. Whether the same uniform approximation can be achieved with the additional constraint that the approximant be non-critical remains unresolved—even for simple examples such as horizontal strips.