Inversion of the l'Hôpital rule for functions holomorphic in a ball (Q1022216)

In previous papers the author has proved two theorems concerning possibility to invert the l'Hôpital rule in the case of analytic functions of one complex variable, namely, Theorem A. Let \(f, g\) be analytic functions in the unit disc \(\triangle\). Let \(W_{\eta}\) be a Stolz angle from \(\triangle\) of the value \(2\eta\) (\(\eta\in (0, \pi/2)\) with the vortex at \(z = 1\). Let there exists \(\lim_{W_{\eta}\ni z \rightarrow 1}\, \frac{f(z)}{g(z)} = A\), where \(A\) is a complex number. {\parindent5mm \begin{itemize}\item[1.] If the values of the function \(\frac{g^{\prime}(z)}{g(z)}(1 - z)\) are separated from zero at \(W_{\eta}\) for \(z \rightarrow 1\), then \(\lim_{W_{\eta-\varepsilon}\ni z \rightarrow 1}\, \frac{f^{\prime}(z)}{g^{\prime}(z)} = A\) for any \(\varepsilon\in (0, \eta)\). \item[2.] If \(\lim_{W_{\eta}\ni z \rightarrow 1}\, \frac{g^{\prime}(z)}{g(z)}(1 - z) = 0\), then for any \(\varepsilon\in (0, \eta)\), \(\lim_{W_{\eta-\varepsilon}\ni z \rightarrow 1}\, \frac{f^{\prime}(z)}{g(z)}(1 - z) = 0\). \end{itemize}} Theorem B. Let \(f, G\) be analytic functions in the unit disc \(\triangle\) and \(W_{\eta}\) be as in Theorem A. If the following conditions are satisfied {\parindent6mm \begin{itemize}\item[(1)] \(\lim_{W_{\eta}\ni z \rightarrow 1}\, {f(z)}{G^{\prime}(z)} = A\in {\mathbb C}\), \item[(2)] \(\lim_{W_{\eta}\ni z \rightarrow 1}\, \frac{(\log\, G^{\prime}(z))^{\prime}}{(\log\, G(z))^{\prime}} = B\in {\mathbb C}\), \item[(3)] the values of the function \(\frac{G^{\prime}(z)}{G(z)}(1 - z)\) are separated from zero in \(W_{\eta}\), \newline then for any \(\varepsilon\in (0, \eta)\) there exists \(\lim_{W_{\eta-\varepsilon}\ni z \rightarrow 1}\, {f^{\prime}(z)}{G(z)} = - A B\). \end{itemize}} If the conditions (1) and (2) are satisfied and \(\lim_{W_{\eta-\varepsilon}\ni z \rightarrow 1}\, \frac{G^{\prime}(z)}{G(z)}(1 - z) = 0\), then for any \(\varepsilon\in (0, \eta)\) there exists \(\lim_{W_{\eta-\varepsilon}\ni z \rightarrow 1}\, {f^{\prime}(z)}{G^{\prime}(z)}(1 - z) = 0\). In the present paper, analogous theorems are proved for functions holomorphic in the unit ball in \({\mathbb C}^n\).