Growth and covering theorems associated with the Roper-Suffridge extension operator

The Roper-Suffridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univalent function $f\in \operatorname{Hol}(\mathbb{D},\mathbb{C})$ to a (locally) univalent map $F\in \operatorname{Hol}(B_{n},\mathbb{C}^{n})$. 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 $f\in S$, the image of $F$ contains a ball of radius $1/2$.