Boundary behavior of the linear part of a holomorphic mapping (Q1089128)

Let \(D\subset {\mathbb{C}}^ n\), \(n\geq 2\), be a domain with \(C^ 2\)- boundary, \(D=\{z: \rho (z)<0\}\) where \(\rho \in C^ 2(\bar D)\) and \(\text{grad} \rho /_{\partial D}\neq 0;\) \(L_{\zeta}\) be Levi's form of \(\partial D\) on the complex tangent place. The main result of the work is Theorem 1. Let f be a holomorphic function on D and let the non-tangent boundary value \((\frac{\partial \rho}{\partial z_ n}\frac{\partial f}{\partial z_ j}-\frac{\partial \rho}{\partial z_ j}\frac{\partial f}{\partial z_ n})^*=0\), \(j=1,...,n-1\), on some set \(E\subset \partial D\) of positive measure and \(L_{\zeta}\not\equiv 0\), \(\zeta\in E\). Then \(f\equiv\) const. The generalization of this theorem to holomorphic mapping is received too.