We show that, for a class of moduli functions ω(δ), 0 ≤ δ ≤ 2, the property, provided u is a quasiregular mapping. Our class of moduli functions includes ω(δ) = δ α (0 < α ≤ 1), so our result generalizes earlier results on Hölder continuity (see [1]) and Lipschitz continuity (see [2]).
We prove some sharp extremal distance results for functions in weighted Bergman spaces on the upper halfplane. We also prove new analogous results in the context of bounded strictly pseudoconvex domains with smooth boundary.
We prove that ω u (δ) ≤ Cω f (δ), where u : B n → R n is the harmonic extension of a continuous map f : S n−1 → R n , if u is a K-quasiregular map. Here C is a constant depending only on n, ω f and K and ω h denotes the modulus of continuity of h.
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