On Koebe-Bloch theorem under some integral constraints

The paper establishes a Koebe–Bloch-type theorem for families of open discrete mappings satisfying an inverse Poletsky inequality with respect to p-modulus under integral constraints. Let F_{Φ,p}^{a,b,δ}(D) be the family of all open discrete mappings f : D → R^n (n ≥ 2) for which there exists a measurable function Q_f satisfying the inverse Poletsky inequality at every point and the global integrability condition ∫_{R^n} Φ(Q_f(y)) / (1 + |y|^2)^n dm(y) < ∞, with h(f(a), f(b)) ≥ δ. Assuming that Φ is an increasing convex function satisfying the divergence condition ∫δ^∞ dτ / (τ (Φ^{-1}(τ))^{1/(p-1)}) = ∞ and that the family is equicontinuous at two distinct points a, b ∈ D, the authors prove that for every compact set K ⊂ D and every ε > 0 there exists r_0 = r_0(ε, K) > 0 (independent of f) such that f(B(x_0, ε)) ⊃ B_h(f(x_0), r_0) for all f ∈ F{Φ,p}^{a,b,δ}(D) and all x_0 ∈ K. The result provides a uniform openness property under integral constraints on distortion and is published in [2].