Given a homeomorphism, \(w=H(e^{i\theta})\), \(0\leq \theta \leq 2\pi\), of the unit circumference \(\partial U\), we denote by Q(H) the class of quasiconformal homeomorphisms of U onto itself with boundary values H on \(\partial U\). The extremal dilatation for the class Q(H) is \textit{\(K_ H=\inf \{K[f]:\) \(f\in Q(H)\},\) where \[ K[f]=ess \sup [(| f_ z| +| f_{\bar z}|)/(| f_ z| -| f_{\bar z}|). \] An element \(f^*\) of Q(H) is an extremal mapping if \(K[f^*]=K_ H\). The main purpose of the present contribution is to elucidate how certain concrete estimates on \(K_ H\) from below, which under certain conditions are best possible, are obtained in terms of the Dirichlet integrals of harmonic mapping between U induced by H. (An essential careless mistake is found in the introduction.) }