On the unique extremality of quasiconformal mappings with dilatation bounds

Concerning the problem of extremality of quasiconformal mappings with dilatation bounds. we discuss the unique extremality of the problem and prove the if part of a conjecture on the unique extremality ([G1], [R1]). To this end, we need to investigate a new extremal problem in the infinitesimal setting. In particular, we give a complete description of the unique infinitesimal extremality of partially zero Beltrami differentials.