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Certain geometric function theory results are obtained for holomorphic mappings on the unit ball. Specifically, the mappings studied are one-to-one onto domains that are starlike with respect to the origin. For such a mapping f (z), sharp estimates are derived for| f (z)| in terms of| z|. Also, a generalization of the Koebe covering theorem is proved. As a corollary of the work, a new proof is given that, in C n for n≥ 2, a ball and a polydisc are not biholomorphically equivalent.