On the question of stability of the class of holomorphic functions in a closed domain (Q1375030)

A continuous function \(f\: D\to\mathbb C^m\), with \(D\) a domain in \(\mathbb C^n\), is said to be a function of the class \(\mathcal B_{n,m}(\varepsilon)\) if there exists a measurable mapping \(Q\: D\to\mathbb C^{nm\times nm}\) such that \( |Q|_{\infty }= \operatorname{ess\;sup}_{z\in D} |Q(z)|\leq \varepsilon \) and \(f\) satisfies the complex Beltrami equation \(f_{\bar{z}}(z)=Q(z)f_z\). The problem of stability of the class \(\mathcal B_{n,m}(0)\) of homomorphic functions in a closed domain can be formulated as follows: does there exists a function \(\mu\: [0, 1)\to [0,+\infty)\) such that 1) \(\lim\limits_{\varepsilon\to 0} \mu(\varepsilon) = \mu (0) =0\) and 2) for each mapping \(f\: \{ z\in \mathbb C^n \mid||z|<1\} \to \{ z\in \mathbb C^m \mid||z|<1\}\) of the class \(\mathcal B_{n,m}(\varepsilon)\), there exists a mapping \(h\: \{ z\in \mathbb C^n \mid||z|<1\} \to \mathbb C^m \) of the class \(\mathcal B_{n,m}(0)\) such that the inequality \(|f(z)-h(z)|\leq\mu (\varepsilon )\) holds for all \( z\in \mathbb C^n\), \(|z|<1\). The author shows that the class \(\mathcal B_{n,m}(0)\) of homomorphic functions is not stable in a closed domain for all \(n,m\geq 1\). The problem was motivated by the theory of quasiconformal mappings (see, \textit{A. P. Kopylov} [Stability in the \(C\)-norm of classes of mappings (Russian). Novosibirsk: Nauka (1990; Zbl 0772.30023)] for details and similar results). The proof is based on a result of \textit{A. Beurling} and \textit{L. Ahlfors} [Acta Math. 96, 125-142 (1956; Zbl 0072.29602)] on the boundary correspondence under quasiconformal mappings.