Relative Angular Derivatives

By Jonathan E. Shapiro

This paper appears in the Journal of Operator Theory, 46 (2001), 265-280.


We generalize the notion of the angular derivative of a holomorphic self-map, b, of the unit disk by replacing the usual difference quotient with a difference quotient relative to an inner function u. We relate the properties of this generalized difference quotient to properties of the Aleksandrov measures associated with the functions b and u. Six conditions are shown to be equivalent to each other, and these are used to define the notion of a relative angular derivative. We see that this generalized derivative can be used to reproduce some known results about ordinary angular derivatives, and the generalization is shown to obey a form of the product rule.

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