BiBloch mappings and composition operators from Bloch type spaces to BMOA

The problem of constructing functions f1f1, f2f2 analytic in the unit disc DD of the complex plane satisfying|f1′(z)|+|f2′(z)|≍ψ(11−|z|),z∈D, is solved for a wide class of weights ψ that includes normal weights. Several applications are given. Particularly we can generalize known characterizations on Bloch-BMO pullbacks.