A Schwarz Lemma for Convex Domains in Arbitrary Banach Spaces

In this note the following new version of the Schwarz lemma is proved: If f is a holomorphic function mapping a bounded convex domain D1 of a complex Banach space into a convex domain D2 of another complex Banach space and f a.sb, then the image by f of the set of points in D1 lying at a distance greater than r from the frontier of D1 is at a positive distance from the frontier of D2 . 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.