Let $D$ and $\Omega$ be proper subdomains of the complex plane ${\Bbb C}$, and let $f\:D\to\Omega$ be a $C^2$-mapping. Also, let $\sigma$ and $\rho$ be smooth positive densities for metrics in $D$ and $\Omega$, respectively. We denote the complex partial derivatives of $f$ by $f_z$ and $f_{\overline{z}}$ and similarly for higher-order derivatives. Write $w=f(z)\in\Omega$. We say that $f$ is $\rho$-harmonic if $$f_{z\overline{z}}+f_z f_{\overline{z}}((\log\rho)_w\circ f)=0$$ in $D$. This is the Euler-Lagrange equation for the functional $$\int_{D}\rho(f(z))(|f_z|^2+|f_{\overline{z}}|^2)dxdy.$$ If, instead, we consider the functional $$\int_{D}\sigma(z)(|f_z|^2+|f_{\overline{z}}|^2)dxdy,$$ we obtain the equation $$2 f_{z\overline{z}}\sigma(z)+f_z\sigma_{\overline{z}}+f_{\overline{z}}\sigma_z =0,$$ whose solutions the authors call $\sigma$-harmonic. \par The authors obtain identities in the form of partial differential equations when $f$ is a sense-preserving diffeomorphism of $D$ onto $\Omega$, for $f$ or its inverse function $g$, under the assumption that one of the functions is $h$-harmonic for a suitable density $h$. For example, if $f$ is $\sigma$-harmonic, then $g$ satisfies a certain PDE involving $\sigma$, and furthermore $g$ is $\delta$-harmonic for a density $\delta(w)$ if, and only if, a certain PDE involving $f$, $\sigma$, and $\delta$ is satisfied. The proofs are by means of computations. A number of examples in annuli are given.