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For an orientation-preserving quasiconformal map, the almost-everywhere coefficient ${\mu }_f=f_{\bar{z}}/f_z$. It is a coordinate-dependent representative of a tensor; the modulus and the resulting maximal dilatation are coordinate-independent.

For a path family $\Gamma$ on a Riemann surface, take the supremum of $L_{\rho }(\Gamma {)}^2/A_{\rho }$ over measurable conformal metrics of finite positive area. This is unchanged under conformal equivalence, and a quasiconformal map of maximal dilatation $K$ changes it by a factor between $K^{-1}$ and $K$.

In local holomorphic coordinates, write the two positive singular values of $Df$ as $s_1\ge s_2$. Preservation of orientation gives $J_f=s_1{s}_2$, and quasiconformality gives $s_1/s_2\le K$.

For an admissible conformal metric $\rho |dw|$ on $Y$, define a density on $X$ by $\sigma (z)=\rho (f(z))s_1(z)$. Along each path,so $L_{\rho }(f(\Gamma ))\le L_{\sigma }(\Gamma )$. The change of variables formula and $s_1^2\le K{J}_f$ giveAlso $s_1^2\ge J_f$, so $A_{\sigma }\ge A_{\rho }>0$. Thus $\sigma$ has positive finite area and is admissible, andTaking the supremum over $\rho$ provesApplying the same argument to $f^{-1}$, which has the same bound on its maximal dilatation, also gives $K^{-1}\lambda (\Gamma ,X)\le \lambda (f(\Gamma ),Y)$. The inequalities remain valid for extended extremal lengths; no extremizing density need exist.

An orientation-preserving homeomorphism with locally $L^2$ weak derivatives and essentially bounded infinitesimal distortion. In holomorphic coordinates, its Beltrami coefficient satisfies $\Vert \mu {\Vert }_{\infty }<1$, and the maximal dilatation is $(1+\Vert \mu {\Vert }_{\infty })/(1-\Vert \mu {\Vert }_{\infty })$.

On a closed genus-at-least-two Riemann surface, a Teichmüller map uniquely minimizes maximal dilatation in its homotopy class among maps to the same target. The Reich–Strebel inequality forces equality of Beltrami coefficients for an extremal competitor; their conformal difference is homotopic to the identity and hence is the identity. In genus one, translations must be factored out.