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1. Introduction. Let Aq (z)'dz'denote the hyperbolic metric on a hype bolic region Q in the complex plane C. For convex regions we shall sharp lower bounds for A0 (z) in terms of the geometric quantity õQ (z distance from the point z to the boundary of Q. In § 3 we obtain a bound that applies to all hyperbolic convex regions. Then in § 4 we der a lower bound that is valid for any convex region with the property t 8q is uniformly bounded in Q. Each of these lower bounds leads to dist tion and covering theorems for a certain family of possibly multiple-v analytic functions defined in the unit disk. In particular, we obtain cl covering theorems for normalized convex univalent functions defin the unit disk, including the fact that Bloch-Landau constant is tt/such functions. In order to obtain these distortion and covering theor from the lower bounds for the hyperbolic metric, we require a gener tion of the principle of hyperbolic metric …