Let \(X\) and \(Y\) be two hyperbolic Riemann surfaces covered by the unit disc \(\Delta=\{z;|z|<1\}\). Let \(Q(f_0)\) be the class of all quasiconformal mappings \(f\) of \(X\) onto \(Y\) which are homotopic to a given function \(f_0\)(mod \(\partial X)\). Denote by \(K[f]\) the maximal dilatation of \(f\). Let \(K_0=\inf\{K[f];f\in Q(f_0)\}\). For \(K\geq K_0\), let \(Q_K(f_0)=\{f;f\in Q(f_0),K(f)\leq K\}\). For a weight function \(\rho (w)\) on \(Y\), the Dirichlet-Douglas functional is defined as \[ \mathcal D_\rho [f]=\iint_X \rho (f(z))(|f_z|^2 +|f_{\overline z}|^2) dx dy . \] In this paper, the author considers the extremals of several functionals defined on \(Q(f_0)\) and on \(Q_K(f_0)\). Some known results are clarified and extended. In particulary, a proof of the nonexistence of harmonic mappings in the case where \(X\) and \(Y\) are unit discs for \(f\in Q(f_0)\), and a complete characterization of a \(K\)-minimal mapping of \(\mathcal D_\rho\) for \(f\in Q_K(f_0)\) are given.