Growth and covering theorems associated with the Roper-Suffridge extensionoperator

The Roper-Suffridge extension operator, originally introduced in the contex t of convex functions, provides a way of extending a (locally) univalent fu nction f is an element of Hol( D; C) to a (locally) univalent map F is an e lement of Hol(B-n; C-n). If f belongs to a class of univalent functions whi ch satisfy a growth theorem and a distortion theorem, we show that F satisf ies a growth theorem and consequently a covering theorem. We also obtain co vering 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 is an element of S, the image of F contains a ball of radius 1/2.