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We adapt Sarason's proof of the Julia-Caratheodory theorem to the class ofSchur-Agler mappings of the unit ball, obtaining a strengthened form of thistheorem. In particular those quantities which appear in the classical theoremand depend only on the component of the mapping in the complex normal directionhave K-limits (not just restricted K-limits) at the boundary.

An improved Julia-Caratheodory theorem for Schur-Agler mappings of the unit ball