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

TL;DR

This paper investigates a multiplier operator derived from the Schwarzian derivative of univalent functions with quasiconformal extensions, applying it to verify the Brennan conjecture for quasidisks and characterizing special curves.

Contribution

It introduces a new multiplier operator linked to the Schwarzian derivative and uses it to address conjectures and characterize specific classes of curves in complex analysis.

Findings

Brennan conjecture verified for a large class of quasidisks

New characterization of asymptotically conformal and Weil-Petersson curves

Establishment of properties of the multiplier operator in complex analysis

Abstract

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.