Elementary Properties of Extremal Length
The extremal length satisfies a few simple monotonicity properties. First, it is clear that if, then . Moreover, the same conclusion holds if every curve contains a curve as a subcurve (that is, is the restriction of to a subinterval of its domain). Another sometimes useful inequality is
This is clear if or if, in which case the right hand side is interpreted as . So suppose that this is not the case and with no loss of generality assume that the curves in are all rectifiable. Let satisfy for . Set . Then and, which proves the inequality.
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