Abstract. Universal upper bounds for the Kobayashi and quasihyperbolic distances near Dini-smooth boundary points of domains in C n and R n , respectively, are obtained.
It is shown that any non-degenerate C-convex domain in C n is uniformly squeezing. It is also found the precise behavior of the squeezing function near a Dini-smooth boundary point of a plane domain.Denote by B n the unit ball in C n . Let M be an n-dimensional complex manifold, and p ∈ M. For any holomorphic embedding f :f 's exist, and s M (p) = 0 otherwise. If inf M s M > 0, then M is said to be uniformly squeezing.Many properties and applications of the squeezing function and the uniformly squeezing manifolds have been explored by various authors, see e.g. [2,3,4,5,6,7,8].By [8, Theorem 2.1], any convex bounded domain in C n is uniformly squeezing. Our first aim is to extend this result to a larger class of domains.A domain D in C n is called C-convex if any non-empty intersection of D with a complex line is a simply connected domain. Then C n \ D is a union of hyperplanes (see e.g. [1, Theorem 2.3.9]). This easily implies that if D is degenerate, i.e. containing complex lines, then D is linearly equivalent to C × D ′ , and hence s D = 0.On the other hand, we have the following.2010 Mathematics Subject Classification. 32F45.
Precise behavior of the Carathéodory, Kobayashi and Bergman metrics and distances near smooth boundary points of planar domains is found under different assumptions of regularity.The most precise known estimates on the boundary behavior of the Carathéodory, Kobayashi and Bergman metrics (γ D , κ D and β D ) of a bounded strictly pseudoconvex domain D in C n with C 3 -or C 4 -smooth boundary can be found in [2,3,6] (for the C 2,ε -smooth case see [8]).Setwhere χ ∂D (a) is the signed curvature of ∂D at a.2010 Mathematics Subject Classification. 32F45, 32A25. Key words and phrases. Carathéodory metric and distance, Kobayashi metric and distance, Bergman kernel, metric and distance.M. Trybula and L. Andreev are partially supported by the Bulgarian National Science Found under contract DFNI-I 02/14.
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