Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

A quasisymmetric homeomorphism defines an element of the universal Teichmüller space and a symmetric one belongs to its little subspace. We show example h real line \({\mathbb {R}}\) onto itself such that \(h^{-1}\) is not symmetric. This implies set all self-homeomorphisms does constitute group under composition. also consider same problem for strongly self-homeomorphism which defined by certain concept harmonic analysis. These results reveal difference sets from those unit circle.