On asymptotically sharp bi-Lipschitz inequalities of quasiconformal mappings satisfying inhomogeneous polyharmonic equations

Abstract

Suppose that f is a K-quasiconformal ((K, K ′ )-quasiconformal resp.) self-mapping of the unit disk D, which satisfies the following: (1) the inhomogeneous polyharmonic equation. . , n − 1} and T denotes the unit circle), and (3) f (0) = 0, where n ≥ 2 is an integer and K ≥ 1 (K ′ ≥ 0 resp.). The main aim of this paper is to prove that f is Lipschitz continuous, and, further, it is bi-Lipschitz continuous when ϕ j ∞ are small enough for j ∈ {1, . . . , n}. Moreover, the estimates are asymptotically sharp as K → 1 (K ′ → 0 resp.) and ϕ j ∞ → 0 for j ∈ {1, . . . , n}, and thus, such a mapping f behaves almost like a rotation for sufficiently small K (K ′ resp.) and ϕ j ∞ for j ∈ {1, . . . , n}.