The Roper-Suffridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univalent function f implied by Hol(D, C) to a (locally) univalent map F implied by Hol(B_n, C~n). If f belongs to a class of univalent functions which satisfy a growth theorem and a distortion theorem, we show that F satisfies a growth theorem and consequently a covering theorem. We also obtain covering theorems of Bloch type: If f is convex, then the image of F (which, as shown by Roper and Suffridge, is convex) contains a ball of radius #pi#/4. If f implied by S, the image of F contains a ball of radius 1/2.
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