For a domain Ω ⊂ n let H(Ω) be the holomorphic functions on Ω and for any k ∈ N let A k (Ω) = H(Ω) ∩ C k (Ω). Denote by A k D(Ω) the set of functions f : Ω → [0, ∞) with the property that there exists a sequence of functions fj ∈ A k (Ω) such that {|fj |} is a nonincreasing sequence and such that f(z) = lim j→∞ |fj (z)|. By A k I (Ω) denote the set of functions f : Ω → (0, ∞) with the property that there exists a sequence of functions fj ∈ A k (Ω) such that {|fj |} is a nondecreasing sequence and such that f(z) = lim j→∞ |fj (z)|. Let k ∈ N and let Ω1 and Ω2 be bounded A k -domains of holomorphy in m1 and m2 respectively. Let g1 ∈ A k D(Ω1), g2 ∈ A k I (Ω1) and h ∈ A k D(Ω2)∩A k I (Ω2). We prove that the domains Ω = {(z, w) ∈ Ω1 × Ω2 : g1(z) < h(w) < g2(z)} are A k -domains of holomorphy if int Ω = Ω. We also prove that under certain assumptions they have a Stein neighbourhood basis and are convex with respect to the class of A k -functions. If these domains in addition have C 1 -boundary, then we prove that the A k -corona problem can be solved. Furthermore we prove two general theorems concerning the projection on n of the spectrum of the algebra Ak.