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We obtain a necessary and sufficient condition for the operator ofintegration to be bounded on $H^\infty$ in a simply connected domain. The mainingredient of the proof is a new result on uniform approximation of Blochfunctions. This gives a full characterization of symbols of certain Volterraoperators that act on bounded analytic functions in the disc if the symbol isassumed to be univalent. Without this assumption the answer is not known, andas the example at the end of the paper shows, the natural answer is definitelyfalse.

Uniform approximation of Bloch functions and the boundedness of the integration operator on $H^\infty$