LetA(X)be the Banach space of integrable, holomorphic, quadratic differentials ϕ on a Riemann surfaceX. We characterize the points ofA(X)at which the norm is weak uniformly convex in terms of the infinitesimal form of Teichmuller's metric onQSmodSand we give a quantified version of this characterization. Sullivan's coiling property applies along any Beltrami line [t|ϕ|/ϕ|] for which ϕ is a point of weak uniform convexity and the amount of coiling is quantified by the quantified version of weak convexity. For a closed setJin, we letA(J)be the Banach space of integrable functions inwhich are holomorphic in the complement ofJ. We generalize Bers' approximation theorem by showing that rational functions with simple poles inJare dense inA(J). Density is with respect to theL1-norm over the whole complex plane, includingJ