Quasi-Nearly Subharmonic Functions and QC Mappings

Let G be a domain in ℝ n , $$\mathbb {R}^n,$$ f : G → ℝ n $$f: G \to \mathbb {R}^n$$ a harmonic map, and A $$\mathcal {A}$$ a class of self-homeomorphisms of G. We study in this chapter what can be said about the functions of the form f ∘ h , h ∈ A $$f \circ h, h \in \mathcal {A}$$ . For example, we show that if n = 2 and A $$\mathcal {A}$$ is the class of conformal maps, then the functions in this class are also harmonic. However, if A $$\mathcal {A}$$ is the class of harmonic maps, or quasiconformal harmonic maps, this statement is no longer true.