On a multiplier operator induced by the Schwarzian derivative of univalent functions

In this paper we study a multiplier operator which is induced by the Schwarzian derivative of univalent functions with a quasiconformal extension to the extended complex plane. As applications, we show that the Brennan conjecture is satisfied for a large class of quasidisks. We also establish a new characterization of asymptotically conformal curves and of the Weil-Petersson curves in terms of the multiplier operator.