Bounded turning circles are weak-quasicircles

Bounded Turning Circles Are Weak-quasicircles

We show that a metric Jordan curve Γ is bounded turning if and only if there exists a weak-quasisymmetric homeomorphism φ : S1 → Γ.

Bilipschitz Homogeneous Curves in R2 Are Quasicircles

Generalized bounded boundary turning functions

For a Jordan domain with smooth boundary, the boundary rotation h is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. The domain is said to have bounded turning or rotation if h < 1 and the functions under such mappings are called functions of bounded boundary turning or rotation. We define a generalization of this concept and ...

Bi-lipschitz Homogeneous Curves in R Are Quasicircles

We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.

Upper bound for functions of bounded turning

For normalized analytic functions in the unit disk, we consider subclasses of bounded turning. The geometric representation is introduced, the radii of convexity (close to convex) are calculated and some subordination relations are suggested. Moreover, the upper bound of the pre-Schwarzian norm for these functions is computed. AMS subject classifications: 30C45