Generalized bounded boundary turning functions

For a Jordan domain with smooth boundary, the boundary rotation θ 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 θ < ∞ and the functions under such mappings are called functions of bounded boundary turning or rotation. We define a generalization of this concept and examine some of its properties. A few open problems are also stated for further explorations.