In this note the following new version of the Schwarz lemma is proved: If f is a holomorphic function mapping a bounded convex domain D-1 of a complex Banach space into a convex domain D-2 of another complex Banach space and f(a) = b, then the image by f of the set of points in D-1 lying at a distance greater than r from the frontier of D-1 is at a positive distance from the frontier of D-2. This distance depends only upon a, b, and r, and it can be estimated specifically in terms of the norms of the Banach spaces. Our result extends several earlier theorems. (C) 1996 Academic Press, Inc. [References: 8]
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