Generalization of the Julia–Carathéodory Theorem to the Case of Several Boundary Fixed Points

Holomorphic self-maps of the unit disk with boundary fixed points are investigated. In 1982, Cowen and Pommerenke established an interesting generalization of the classical Julia–Carathéodory theorem, which allowed them to derive a sharp estimate for the derivative at the Denjoy–Wolff point on the class of functions with an arbitrary finite set of boundary fixed points. In this paper, we obtain a new generalization of the Julia–Carathéodory theorem, which contains the Cowen–Pommerenke result as a special case; moreover, it is an effective tool for solving various problems on classes of functions with fixed points.