Analytic Characterization of $t$-Quasicircles and Conformal Mappings onto $t$-Quasidisks

Abstract

In this paper, we introduce a new class of mappings, termed $(\rho ,t)$-quasisymmetric mappings, which generalizes the classical concept of quasisymmetric mappings. Using this broader class of mappings, we provide an analytic characterization of $t$-quasicircles. This result can be viewed as a $t$-quasisymmetric analogue of a classical theorem by Tukia and Väisälä \cite{TV}. Furthermore, we study conformal mappings from the unit disk $\mathbb{D}$ onto $t$-quasidisks and show that their boundary values are "almost`` $(\rho ,t^2)$-quasisymmetric. This result extends the Quasicircle Theorem to the case of $t$-quasicircles.