Angular derivatives for holomorphic self-maps of the disk
Let \(\varphi\) be a holomorphic function in the unit disk \(\mathbb{D}\) with \(\varphi(\mathbb{D})\subset \mathbb{D}\). Local properties of such functions near points where the angular derivative exists are studied. In particular, the following results are obtained. Theorem 2.4. Let \(\varphi\) have finite angular derivatives at \(\zeta_k\in \partial \mathbb{D}\), \(\omega_k:=\varphi(\zeta_k)\in \partial \mathbb{D}\), \(\alpha_k= |\varphi'(\alpha_k)|\), and let \[ q_k:=\frac1{\sqrt{\alpha_k\alpha_{k+1}}} \Bigl| \frac{\omega_k-\omega_{k+1}}{\zeta_k-\zeta_{k+1}} \Bigr|\quad\text{for}\quad k=1,2,3 \] with \(k=1\) instead of \(k=4\). Then, we have \[ q_1^2+q_2^2+q_3^2\leq 1+2q_1q_2q_3. \] Theorem 5.1. Let \(\zeta \in \partial \mathbb{D}\) and let the angular derivative \(\varphi'(\zeta)\neq \infty\) exist. If \(H\) is any horocycle at \(\zeta\), then \(\varphi(z)\to \varphi(\zeta)\) as \(z\to \zeta\), \(z\in H\). Theorem 5.1 generalized results by \textit{G. T. Cargo} [Can. J. Math. 14, 334--348 (1962; Zbl 0103.29802)] and \textit{P. R. Ahern} and \textit{D. N. Clark} [Mich. Math. J. 21, 115--127 (1974; Zbl 0277.30027)] proved for Blaschke products.
scientific article; zbMATH DE number 6821108
scientific article