Starlike and convex maps in Banach spaces

Ted Joe Suffridge

Vol. 46 (1973), No. 2, 575–589

DOI: 10.2140/pjm.1973.46.575

Abstract

Let X and Y be complex Banach spaces and let B = {x ∈ X : ∥x∥ < 1}. This paper concerns holomorphic maps f : B → Y which have local holomorphic inverses. That is, for each x ∈ B, there is a neighborhood N ⊂ Y of f(x) and a holomorphic function g : N → B such that g(f(x)) = x and f(g(y)) = y for all y ∈ N. Necessary and sufficient conditions are found which guarantee that such a map be one-to-one and map the unit ball B onto a domain which is convex or starlike with respect to 0.

Mathematical Subject Classification 2000

Primary: 46G20

Milestones

Received: 27 March 1972

Published: 1 June 1973