Approximation by holomorphic functions with controlled growth (Q1896468)

A function \(\theta : (0,1) \to (0,1)\) is said to be a nonbounded growth if \(\theta\) is decreasing and \(\lim_{x \to 0} \theta (x) = \infty\). Let \(\Omega\) be a bounded pseudoconvex domain in \(\mathbb{C}^n\) with a \(C^\infty\) boundary, let \(\varphi\) be a continuous function in \(\Omega\) and \(\theta\) be a given nonbounded growth. Under these conditions the authors prove that there exists a function \(f\), holomorphic in \(\Omega\) and a bijection \(\beta : (0,1) \to (0,1)\) such that: (1) \(|f(z) |\leq \theta (\text{dist} (z\), complement of \(\Omega))\), \(\forall z \in \Omega\); (2) for almost all points \(w\) of strict pseudoconvexity of \(\partial \Omega\) \(\lim_{\varepsilon \to 0} [f(w - \varepsilon n_w) - \varphi (w - \varepsilon n_w)] = 0\), where \(n_w\) is the exterior normal in \(w\). This proposition extend a result obtained by \textit{M. Ortel} and \textit{W. Schneider} [Math. Scand. 56, 287-310 (1985; Zbl 0599.30003)] in the unit disc and those of A. Iordan and the second author in the unit ball of \(\mathbb{C}^n\).