On locally quasiconformal mappings

A homeomorphism w = f(z) of a domain D is called a locally quasiconformal mapping, if for each subdomain D. of D with D...D, the restriction of f(z) on D. is a quasiconformal mapping. We give some conditions for a measurable function µ(z) on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc.