Composition operators with univalent symbol in Schatten classes

We study composition operators, induced by a sub-domain of the unit disc whose boundary intersects the unit circle at 1 and which has, in a neighborhood of 1, a polar equation 1−r=γ(|θ|)1−r=γ(|θ|) (see Fig. 1). We obtain an explicit characterization for the membership in Schatten p-classes, in terms of γ.