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In this paper we study a multiplier operator which is induced by theSchwarzian derivative of univalent functions with a qc extension to theextended complex plane. As applications, we prove that the Brennan's conjectureis true for a large class of quasi-disks. We also establish a newcharacterization of the asymptotically conformal curves and the WP curves interms of the multiplier operator.

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