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Relative angular derivatives. (English) Zbl 1002.46021
Summary: We generalize the notion of the angular derivative of a holomorphic self-map $$b$$, of the unit disk, by replacing the usual difference quotient $${{b(z)-b(z_0)}\over{z-z_0}}$$ with a difference quotient relative to an inner function $$u$$, $${{1-b(z)}\over{1-u(z)}}$$. We relate properties of this generalized difference quotient to the 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.

##### MSC:
 4.6e+23 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)