On the boundary smoothness of conformal mappings between domains with nonsmooth boundaries

The boundary smoothness of conformal mappings between domains with nonsmooth boundaries is characterized by its modulus of continuity by assuming that the quality of an arbitrary Jordan curve is characterized by its modulus of oscillation. A connected bounded domain in the plane with Jordan boundary is found to be univalent conformal mapping from the domain to the unit disk. A univalent conformal mapping from the unit disk to a bounded domain with rectifiable Jordan boundary satisfies Warschawski inequality. A mapping with Jordan boundary to a bounded domain with Jordan boundary are defined by quantities depending on a small quantities. The conformal mapping is extended over the closure of the arc and the homomorphism from one union set to an intersection set.