Uniform Quotient Mappings of the Plane

It is shown that if f is a mapping of the plane onto itself that is uniformly continuous with modulus of continuity Ω(r) which is o( √ r) as r → 0 and f is also co-uniformly continuous then f = P ◦ h where h is a homeomorphism of the plane and P is a complex polynomial. The same conclusion holds also under other assumptions on the moduli of uniform and co-uniform continuity. However, we also present an example showing that this does not hold for all uniform quotient mappings: There is a mapping of the plane onto itself whose moduli of uniform and co-uniform continuity are both of power type but it maps an interval to zero. We also discuss uniform quotient mappings of the plane onto the line. Subject classification: 54E15, 57N05.