A note on the relationship between quasi-symmetric mappings and φ-uniform domains

The aim of this note is to construct a ψ-uniform domain G   in the complex plane CC such that the identity mapping id: (G,jG)→(G,kG)(G,jG)→(G,kG) is not an η-quasi-symmetric mapping for any homeomorphism η  : [0,∞)→[0,∞)[0,∞)→[0,∞). This result shows that the answer to the related open problem, posed by Hästö, Klén, Sahoo and Vuorinen, is negative.