 Length-area exhaustion of the complex plane

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 $L(R{)}^2\le 8{\pi }^{2}R{S}^{\prime }(R)$. If $L/S$ stayed above a positive constant for all large $R$, integration of $(1/S{)}^{\prime }\le -c/R$ would make $1/S$ negative. Thus there are arbitrarily large radii with $L/S$ arbitrarily small, even when the total area is finite.