Boundary behavior in the Stolz angle of functions analytic in the disk
The authors study the behavior of analytic functions in the unit disc \(\Delta=\{z:|z|<1\}\) in a Stolz angle. There proved assertions of the type of the Hardy-Littlewood theorem and obtained estimates for the growth of functions in a Stoltz angle. In particular, they obtained the theorem: Suppose that \(f\) is analytic on \(\Delta\), \(c\in\mathbb{C}\) and \(\eta\in (0,\pi/2\rangle\) are fixed numbers, \(W_\eta\) is the domain \[ W_\eta= \biggl\{z\in\Delta: \bigl|\arg(1-z) \bigr|<\eta,\;|1-z|<\rho\biggr\},\;0<\rho< \cos\eta, \] (the Stolz angle with vertex at \(z=1)\). If the finite limit \(\lim_{W_\eta\ni z \to 1}[f(z)(1-z)^c]=A\) exists, then for any \(\varepsilon\in(0,\eta)\) the limit \(\lim_{W_{\eta-\varepsilon}\ni z\to 1}[f'(z)(1-z)^{c+1}] =Ac\) exists. If the function \(f(z)(1-z)^c\) is not bounded in \(W_\eta\) and \(\text{Re} c>0\), then the function \(f'(z)(1-z)^{c+1}\) is also not bounded in \(W_\eta\).
scientific article; zbMATH DE number 1928202