A New Variant of the Schwarz{pick{ahlfors Lemma
نویسنده
چکیده
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 shrunk. The method of proof is also different in that it 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 effects—some beneficial 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 |z| < R1 and suppose that |f(z)| < R2 and f(0) = 0. Then |f(z)| ≤ R2 R1 |z| for |z| < R1. (1) It is also generally noted that strict inequality holds for every z 6= 0 unless f is of the special form f(z) = R2 R1 ez, for some real α. (2) As immediate corollaries, one has: Corollary 1 (Liouville’s Theorem). A bounded analytic function in the entire plane is constant. Proof. R2 is fixed, and R1 may be chosen arbitrarily large. Corollary 2. If R1 = R2, then |f (0)| ≤ 1. (3) A slightly less obvious, but still elementary corollary is The methods and results of this paper derive from a paper of Antonio Ros [R], and in particular, from Lemma 6 of that paper. Research at MSRI is supported in part by NSF grant DMS-9701755. 1
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A new variant of the Schwarz { Pick { Ahlfors Lemma 159 What Pick
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 ...
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تاریخ انتشار 1998