Bloch’s theorem for mappings of bounded and finite distortion

We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural form. Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author was visiting at the University of Michigan. He wishes to thank the department for hospitality.