  We present a characterization of one-component inner functions in terms of
the location of their zeros and their associated singular measure. As
consequence we answer several questions posed by J. Cima and R. Mortini. In
particular we prove that for any inner function $\Theta$ whose singular set has
measure zero, one can find a Blaschke product $B$ such that $\Theta B$ is
one-component. We also obtain a characterization of one-component singular
inner functions which is used to produce examples of discrete and continuous
one-component singular inner functions.

Classical Analysis and ODEs;Complex Variables