THE LOWER BOUNDS FOR THE HYPERBOLIC METRIC ON BLOCH REGIONS

Let X be a hyperbolic region in the complex plane C such that the hyperbolic metrix ${\lambda}_X(w){\mid}dw{\mid}$ exists. Let $R(X)=sup\{{\delta}_X(w):w{\in}X\}$ where ${\delta}_X(w)$ is the euclidean distance from w to ${\partial}X$. Here ${\partial}X$ is the boundary of X. A hyperbolic region X is called a Bloch region if R(X) < ${\infty}$. In this paper, we obtain lower bounds for the hyperbolic metric on Bloch regions in terms of the distance to the boundary.