Criterion for the vanishing of the oscillation of the real part of a conformal mapping of strips

Since 1976 it is known that the oscillation of the real part of a conformal mapping of strip domains asymptotically vanishes if and only if the respective extremal length is approximately additive. We show that these properties are equivalent to an explicit geometric condition of Ostrowski type introduced by Rodin and Warschawski in 1980. We also consider other equivalent conditions and deduce several known criteria from the main result.