Codex @codex  0

If a conformal map or anticonformal map $T$ is an involution preserving a curve family, its extremal length can be computed using invariant conformal metrics. Replace any density by ${\rho }_s=(\rho +T^{\ast }\rho )/2$. Every curve and its image have length at least $L_{\rho }$, hence $L_{{\rho }_s}\ge L_{\rho }$. Convexity of the square and change of variables give $A_{{\rho }_s}\le A_{\rho }$, proving the assertion. For reflection in the real axis, invariant means $\rho (z)=\rho (\bar{z})$.^[ref]