Summary: For a domain \(\Omega \subset {\mathbb C}^n\) let \(H(\Omega )\) be the holomorphic functions on \(\Omega \) and for any \(k\in \mathbb N\) let \(A^k(\Omega )=H(\Omega )\cap C^k(\overline {\Omega })\). Denote by \({\mathcal A}_D^k(\Omega )\) the set of functions \(f\: \Omega \to [0,\infty )\) with the property that there exists a sequence of functions \(f_j\in A^k(\Omega )\) such that \(\{| f_j| \}\) is a nonincreasing sequence and such that \( f(z)=\lim _{j\to \infty }| f_j(z)| \). By \({\mathcal A}_I^k(\Omega )\) denote the set of functions \(f\: \Omega \to (0,\infty )\) with the property that there exists a sequence of functions \(f_j\in A^k(\Omega )\) such that \(\{| f_j| \}\) is a nondecreasing sequence and such that \( f(z)=\lim _{j\to \infty }| f_j(z)| \). Let \(k\in \mathbb N\) and let \(\Omega _1\) and \(\Omega _2\) be bounded \(A^k\)-domains of holomorphy in \(\mathbb C^{m_1}\) and \(\mathbb C^{m_2}\), respectively. Let \(g_1\in {\mathcal A}_D^k(\Omega _1)\), \(g_2\in {\mathcal A}_I^k(\Omega _1)\) and \(h\in {\mathcal A}_D^k(\Omega _2)\cap {\mathcal A}_I^k(\Omega _2)\). We prove that the domains \(\Omega =\{(z,w)\in \Omega _1\times \Omega _2: g_1(z)