THE LOWER BOUNDS FOR THE HYPERBOLIC METRIC ON BLOCH REGIONS

Abstract

Let X be a hyperbolic region in the complex plane C such that the hyperbolic metrix λX (w)|dw| exists. Let R(X) = sup{δX (w) : w ∈ X} where δX (w) is the euclidean distance from w to ∂X. Here ∂X is the boundary of X. A hyperbolic region X is called a Bloch region if R(X) < ∞. In this paper, we obtain lower bounds for the hyperbolic metric on Bloch regions in terms of the distance to the boundary.

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