= Length-area exhaustion of the complex plane
{title2=$L(R_j)/S(R_j)\to0$}
For a nonconstant <meromorphic function> on the plane, let $S(R)$ be its normalized pulled-back spherical area on $|z|<R$, and $L(R)$ the spherical length of its boundary image counted with <multiplicity>. The <Cauchy-Schwarz inequality> gives $L(R)^2\le8\pi^2R S'(R)$. If $L/S$ stayed above a positive constant for all large $R$, integration of $(1/S)'\le-c/R$ would make $1/S$ negative. Thus there are arbitrarily large radii with $L/S$ arbitrarily small, even when the total area is finite.
Back to article page
