By Jonathan E. Shapiro

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

**Abstract:**

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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