GENERALIZATION OF THE JULIA–CARATHE´ODORY THEOREM TO THE CASE OF SEVERAL BOUNDARY FIXED POINTS

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