Inequalities for angular derivatives and boundary interpolation
Let \(\mathbb D\) be the open unit disk in the complex plane, and let \(F\) be a Schur function in \(\mathbb D\) (that is, \(F\) is a holomorphic self-mapping of \(\mathbb D\)). An interior point \(\xi\in \mathbb D\) is called a fixed point of \(F\) if \(F(\xi)=\xi\). A boudary point \(\xi\in \partial \mathbb D\) is called a fixed point of \(F\) if \(F(\xi):=\angle\lim_{z\to\xi}F(z)=\xi\), where \(\angle\lim_{z\to\xi}F(z)\) stands for the angular limit of \(F\) at \(\xi\). Furthermore, if \(F\) has a finite angular derivative at a boundary fixed point \(\xi\in \partial \mathbb D\), that is, the angular derivative at \(\xi\) \(F'(\xi):=\angle\lim_{z\to \xi} \dfrac{F(z)-F(\xi)}{z-\xi}\) exists finitely, we say that \(\xi\) is a regular point of \(F\). More generally, if for \(\xi\in \partial \mathbb D\), the angular limit \(F(\xi):=\angle\lim_{z\to\xi}F(z)\) exists with \(F(\xi)\in\partial \mathbb D\) [and \(F'(\xi)\) exists, resp.], we say that \(\xi\) is a contact [regular contact] point of \(F\). The classical Julia-Wolff-Carathéodory theorem in conjunction with the Denjoy-Wolff theorem in Complex analysis asserts that if \(F\) is fixed point-free in \(\mathbb D\), then the so-called Denjoy-Wolff point \(\xi\) of \(F\) is on \(\partial \mathbb D\), and the angular derivative \(F'(\xi)\) at \(\xi\) exists finitely and is a positive number (\(\leq 1\)). \textit{C. C. Cowen} and \textit{C. Pommerenke} [J. Lond. Math. Soc., II. Ser. 26, 271--289 (1982; Zbl 0476.30001)] proved that if \(F\) has several boundary regular fixed (or mutual contact) points \(\xi_1\), \(\xi_2\), \(\cdots\), \(\xi_N\), then the angular derivatives at these points are subject to certain inequalities. (Here \(\xi_1\), \(\xi_2\), \(\ldots\), \(\xi_N\in \partial \mathbb D\) is said to be a family of mutual contact points of \(F\) if \(F(\xi_1)=F(\xi_2)=\cdots=F(\xi_N)=\omega\) for some \(\omega\in \partial \mathbb D\).) In this paper, the authors develop a unified approach to establish inequalities for angular derivatives of \(F\), and apply them to study the boundary Nevanlinna-Pick interpolation problem with a (not necessarily finite) collection of either mutual contact points or boundary fixed points of \(F\). In particular, they show that the Cowen-Pommerenke inequalities are sufficient for the solvability of the interpolation problems. Furthermore, their methods leads to simple explicit formulas for solutions of interpolation problems.
scientific article; zbMATH DE number 6147913
scientific article