 Conformal modulus of a quadrilateral

Map a conformal quadrilateral to a rectangle of width $w$ and height $h$, with the chosen opposite boundary arcs mapped to its vertical sides. Their extremal length is $M=w/h$; the other pair has extremal length $1/M$. The constant conformal metric is extremal. For an arbitrary density, average horizontal joining lengths over height and apply the Cauchy-Schwarz inequality to obtain $L_{\rho }^{2}h\le w{A}_{\rho }$, proving the upper bound. Vertical joining segments prove the dual formula. Conformal invariance of extremal length transports both equalities back to the quadrilateral.