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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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    $\begingroup$ 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. $\endgroup$
    – fedja
    Commented Jun 16, 2012 at 0:36
  • $\begingroup$ Or $z$ and $1/z$ on ${\bf C} - \{0\}$. $\endgroup$
    – Nik Weaver
    Commented Jun 16, 2012 at 0:42

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