Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let Hom + ( T ) be the class of all sense‐preserving homeomorphic self‐mappings of T = { z = x + i y ∈ C : | z | = 1 } . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for ( K , K ′ )‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δ ω )  =   g in unit disk D with associated boundary value conditions Δ ω | T = φ ∈ C ( T ) and ω | T = f ∗ ∈ Hom+ ( T ) . Second, we establish biLipschitz continuity for ( K , K ′ )‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when K ′ , | | φ | | ∞ : = sup z ∈ D | φ ( z ) | and | | g | | ∞ : = sup z ∈ D | g ( z ) | are small enough.