Abstract
By the method of extremal length, three general theorems are proven which find, as corollaries, sharp coefficient bounds for functions univalent in the exterior of the unit disc, with a standard normalization but also assuming a finite number of initial coefficients are zero, which possess a K-quasiconformal extension to a ring subdomain of the unit disc. All extremal functions are exhibited. Similar estimates are found for functions similar to the above but which omit a fixed disc centered at the origin.
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