Injectivity criteria of linear combinations of harmonic quasiregular mappings

Abstract

In this paper, we consider linear combinations of harmonic K-quasiregular mappings f_j = h_j + g_i (j = 1, 2) of the class Har(k; φ_j), where k ∈ [0,1), ||ω_fj||_∞ = ||g'_j/h'_j||_∞ ≤ k < 1, k = (1 - K)/(1 + K), and φ_j = h_j + e^iθg_j is a univalent analytic function. We provide sufficient conditions for the linear combinations of mappings in these classes to be univalent and for the image domains to be linearly connected. Furthermore, we consider under which conditions the linear combination f is bi-Lipschitz.