The Inverse of a $$\sigma (z)$$ σ ( z ) -Harmonic Diffeomorphism

Abstract

A new kind of functional, analogous to the Douglas–Dirichlet functional, is defined as $$\begin{aligned} E'[f]=\displaystyle \iint _{\Omega }\sigma (z)(|f_{z}|^{2}+ |f_{\overline{z}}|^{2})\mathrm{d}x\mathrm{d}y \end{aligned}$$ for $$f\in {C^{2}}$$ on $$\Omega $$ with a conformal metric density $$\sigma (z)$$ . A critical point of this new functional is said to be a $$\sigma (z)$$ -harmonic mapping. We consider the harmonicity of the inverse function of a $$\sigma (z)$$ -harmonic diffeomorphism and obtain a necessary and sufficient condition, which improves on the corresponding result for Euclidean harmonic mappings. In addition, a property of the inverse function of $$\rho $$ -harmonic mappings is investigated and an example is given.