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Rotational properties of homeomorphisms with integrable distortion

We establish a modulus inequality, with weak assumptions on the Sobolev regularity, for homeomorphisms with integrable distortion. As an application, we find upper bounds for the pointwise rotation of planar homeomorphisms with 

 
 p
 p
 

-integrable distortion. When the mapping is entire we bound the local pointwise rotation and when the mapping is restricted to a bounded convex domain 

 
 
 Ω
 ⊂
 
 C
 
 
 \Omega \subset \mathbb {C}
 

 we concentrate on the rotation along the boundary. Furthermore, we show that these bounds are sharp in a very strong sense. Our examples will also prove that the modulus of continuity result, due to Koskela and Takkinen, for the homeomorphisms with 

 
 p
 p
 

-integrable distortion is sharp in this strong sense.