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additive constant Ahlfors-Sario annulus arbitrary assumption bounded set capacity functions p0y characterized coincides compact bordered surface compact set conclude condition conformal mappings Consequently consider const contained converges corresponding defined denote differentials Dirichlet integral disjoint disk of radius dividing cycle equivalent exhaustion exists extremal length extremal metric extremal property finite number functions F Green's function Grotzsch harmonic function harmonic measure ideal boundary component identity implies inequality isolated subboundary Jordan curve Lemma line segment logarithmic capacity maximum principle modulus function normal operator obtain open Riemann surface p*dp parameter plane region principal functions Proof of Theorem radial slit disk rectifiable curves regular region regular subregions relation Remark resp respect result Riemann sphere Sario satisfies semiweak sequence simply connected singularity subregions Q Suppose totally disconnected uniquely determined univalent univalent analytic functions vanishes weak zero