TO THE THEORY OF MAPPINGS WITH FINITE AREA DISTORTION D. Kovtonyuk and V. Ryazanov November 17, 2004 (helsinki.tex) Abstract In all dimensions k =1, ..., n − 1, we show that mappings f in R n with finite distortion of hyperarea satisfy certain modulus inequalities in terms of inner and outer dilatation of the mappings. 1 Introduction Quasiconformal and quasiregular mappings have been recently generalized to sev- eral directions, see e.g. [AIKM], [GI], [HK], [IKO 1 ], [IKO 2 ], [IM], [IR], [IS], [KKM 1 ], [KKM 2 ], [KO], [MRSY 1 ], [MRSY 2 ], [MV 1 ], [MV 2 ], [RSY 1 ] - [RSY 3 ]. In all those generalizations the modulus techniques play a key role. The following concept was proposed in [MRSY 1 ]. Let D be a domain in R n ,n ≥ 2, and let Q : D → [1, ∞] be a measurable function. A homeomorphism f : D → R n = R n  {∞} is called a Q−homeomorphism if M (f Γ) ≤  D Q(x) · ρ n (x) dm(x) (1.1) for every family Γ of paths in D and every admissible function ρ for Γ. Recall that, given a family of paths Γ in R n , a Borel function ρ : R n → [0, ∞] is called admissible for Γ, abbr. ρ ∈ adm Γ, if  γ ρ ds ≥ 1 (1.2) for each γ ∈ Γ. The (conformal) modulus of Γ is the quantity M (Γ) = inf ρ∈adm Γ  D ρ n (x) dm(x) (1.3) with the measure and the integral by Lebesgue. In the work [MRSY 2 ], the concept has been extended to mappings with branch- ing. Note that the modulus inequality (1.1) in the definition of a Q−homeo- morphism has first appeared for n = 2 in connection with the so-called BMO−qua- siconformal mappings, see [RSY 1 ] - [RSY 3 ], cf. also V (6.6) in [LV] in the the- ory of quasiconformal mappings. In this paper, we consider the modulus of 1