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The boundary behavior of the Kobayashi metric. (English) Zbl 0760.32010
Let $$3/4\leq\lambda<1$$. Set $$m=1/(2-2\lambda)$$, and define $U=\{(w,z)\in\mathbb{C}^ 2: 1<| w|^ 2+| z|^ m<4\}.$ The author uses the euclidean distance from the boundary of $$U$$ to estimate the Kobayashi-Royden metric $$F^ U$$ in the direction $$\xi=(1,0)$$, as follows. There exist positive constants $$C_ 1$$ and $$C_ 2$$ such that $C_ 1\text{dist}(P,\partial U)^{-\lambda}\leq F^ U(P,\xi)\leq C_ 2\text{dist}(P,\partial U)^{-\lambda}$ for all $$P=(- 1-\delta,0)$$ with $$\delta>0$$ sufficiently small.

##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables
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##### References:
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