Sum of univalent functions

Let $f,g$ be a pair of univalent functions on a proper subdomain $\Omega$ of $\mathbb C$. Does their sum $f+g$ necessarily omit any complex value? Similarly, can all holomorphic function be written as sums of univalent functions?

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No to both in as simple domain as the unit disk. 1) $(1-z)^{-2}-(1+z)^{-2}$ 2) Univalent functions cannot grow too fast near the boundary. –  fedja Jun 16 '12 at 0:36
Or $z$ and $1/z$ on ${\bf C} - \{0\}$. –  Nik Weaver Jun 16 '12 at 0:42