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Are there any holomorphic functions on a connected domain in $\mathbb C$ that can not be written as a sum of two univalent (holomorphic and injective) functions? What about as a sum of finitely many univalent functions? Or even infinitely many?

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    @LVK You know, $az+bz^n$ is univalent for |a|>n|b|, and we can always represent any number as a sum of oscillating series with arbirtrarily slowly decaying terms and split any number into finitely many arbitrarily small parts. So the infinite case is not really that interesting...2012-10-06

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There is a growth obstruction for finite sum representation. Indeed, a theorem of Prawitz (1927) says that every univalent function on the unit disk belongs to the Hardy space $H^p$ for all $p<1/2$. Consequently, $f(z)=(1-z)^{-q}$ is not a finite sum of univalent functions when $q>2$.

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    Wow! What a nice result.2012-10-01