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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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Appendix B. Length-Area-Modulus Inequalities

This appendix will first study the conformal geometry of a rectangle. (Compare Lemma 17.1.) Let image = [0, Δx] × [0, Δy] be a closed rectangle in the plane of complex numbers z = x + iy, and let R = (0, Δx) × (0, Δy) be its interior. By a conformal metric on R we mean a metric of the form

image

where z image ρ(z) > 0 is any strictly positive real-valued function which is defined and continuous throughout the open rectangle. ...

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