We study the ring Q-homeomorphisms with respect to p-modulus, p ˃ 2, in the complex plane and establish lower bounds for the area of an image of a disc. The extremal problem concerning a minimization of the area functional is solved.
We consider the class of ring Q-homeomorphisms with respect to p-modulus in R n with p > n, and obtain a lower bound for the volume of images of a ball under such mappings. In particular, the following theorem is proved in the paper: Let D be a bounded domain in R n , n 2 and let f : D → R n be a ring Q-homeomorphism with respect to p-modulus at a point x 0 ∈ D with p > n, and the function Q satisfies the condition q x0 (t) q 0 t −α , q 0 ∈ (0, ∞) , α ∈ [0, ∞) for a.e. t ∈ (0, d 0), d 0 = dist(x 0 , ∂D). Then for all r ∈ (0, d 0) the estimate m(f B(x 0 , r)) Ω n (p − n α + p − n
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