A new variant of the Schwarz { Pick { Ahlfors Lemma 159 What Pick
نویسنده
چکیده
We prove a \general shrinking lemma" that resembles the Schwarz{Pick{ Ahlfors Lemma and its generalizations, but diiers in applying to maps of a nite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that all distances are shrunk. The method of the proof is also diierent in that relates the shrinking of the Schwarz{Pick{Ahlfors{type lemmas to the comparison theorems of Riemannian geometry. We start by reviewing the history of Schwarz{type lemmas, with remarks about the eeects-some beneecial and some not-of successive generalizations. There are minor variations in the way the Schwarz lemma is usually stated. Here is one of the standard formulations. Lemma 1 (The Schwarz Lemma). Let f(z) be analytic on a disk j z j< R 1 and suppose that j f(z) j< R 2 and f(0) = 0. Then j f(z) jj R 1 R 2 j z j for j z j< R 1 : (0.1) It is also generally noted that strict inequality holds for every z 6 = 0 unless f is of the special form f(z) = R 1 R 2 e ii z (0.2) 1 The methods and results of this paper derive from a paper of Antonio RossR], and in particular, from Lemma 6 of that paper. A slightly expanded version will appear in the Notices of the AMS.
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A New Variant of the Schwarz{pick{ahlfors Lemma
We prove a “general shrinking lemma” that resembles the Schwarz– Pick–Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that all distances are shru...
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