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LANDAU ’ S THEOREM FOR HOLOMORPHIC CURVES

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in Pn, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, we have f(0) ≤ K, where f denotes the norm of the derivative measured with respect to the Fubini-Study metric. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the FubiniStudy metric. Our estimate is based on the potential-theoretic method of Eremenko and Sodin.