LetFbe the family of functions schlicht in the annulus R:= {w:ρ<|w|<1} (ρ<) mappingRonto the unit disk minus some continuum. This family of functions was introduced by Gaier [2], and a first distortion theorem was proved by Duren-Schiffer [1], Gaier Huckemann, [3], and independently by Gehring-af Hällström [4]. Here we characterize the familyFand prove a further distortion theorem forF.