Upper bound for the Lempert function of smooth domains

Abstract

Abstract. An upper estimate for the Lempert function of any C 1+ε -smooth bounded domain in C n is found in terms of the boundary distance.

Boundary behavior of the Kobayashi distance in pseudoconvex Reinhardt domains

“…In the same paper the authors showed the opposite estimate for C 1+ε -smooth domains with z 1 , z 2 near ζ 0 ∈ ∂D. This estimate in the bounded case follows from the inequality for the Lempert function of bounded C 1+ε -smooth domains obtained by Nikolov, Pflug and Thomas [6]. It was also proved that the above estimate fails in the C 1 -smooth case.…”

Estimates of the Bergman distance on Dini-smooth bounded planar domains

“…Remark. (a) The Dini-smoothness is essential as an example of a C 1 -smooth bounded simply connected planar domain shows (see [8,Example 2]). (b) One of the missing properties of b D in comparison with c D and l D is monotonicity under inclusion of (planar) domains.…”

Estimates of invariant distances on “convex” domains

“…(this proof uses only the existence of an appropriate supporting (real) hyperplane and the formula for the Poincaré distance of the upper halfplane). On the other hand, by [13,Theorem 1], for any C 1+ε -smooth bounded domain there exists a constant c > 0 such that…”

“…The smoothness is essential as an example of a C 1 -smooth bounded C-convex planar domain shows (see [13,Example 2]). Moreover, using [16,p.…”