Let B2,p:={z∈C2:|z1|2+|z2|p<1} (0 < p < 1). Then, B2,p (0 < p < 1) is a non-convex complex ellipsoid in C2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ∂B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2,p, where z0 is any smooth boundary point of B2,p.
Le HE
,
Zhenhan TU
. THE SCHWARZ LEMMA AT THE BOUNDARY OF THE NON-CONVEX COMPLEX ELLIPSOIDS[J]. Acta mathematica scientia, Series B, 2019
, 39(4)
: 915
-926
.
DOI: 10.1007/s10473-019-0401-5
[1] Burns D M, Krantz S G. Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J Amer Math Soc, 1994, 7(3):661-667
[2] Chelst D. A generalized Schwarz lemma at the boundary. Proc Amer Math Soc, 2001, 123(11):3275-3278
[3] Franzoni T, Vesentini E. Holomorphic Maps and Invariant Distances. Amsterdan:North-Holland, 1980
[4] Garnett J B. Bounded Analytic Functions. New York:Academic press, 1981
[5] Huang X J. A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains. Can J Math, 1995, 47:405-420
[6] Krantz S G. The Schwarz lemma at the boundary. Complex Var Elliptic Equ, 2011, 56(5):455-468
[7] Liu T S, Tang X M. Schwarz lemma at the boundary of the Egg Domain Bp1,p2 in Cn. Canad Math Bull, 2015, 58(2):381-392
[8] Liu T S, Tang X M. Schwarz lemma at the boundary of strongly pseudoconvex domain in Cn. Math Ann, 2016, 366:655-666
[9] Osserman R. A sharp Schwarz inequality on the boundary. Proc Amer Math Soc, 2000, 128(12):3513-3517
[10] Pflug P, Zwonek W. The kobayashi metric for non-convex complex ellipsoids. Complex Var Theory Appl, 1996, 29(1):59-71
[11] Wang X P, Ren G B. Boundary Schwarz Lemma for Holomorphic Self-mappings of Strongly Pseudoconvex Domains. Complex Anal Oper Theory, 2017, 11:345-358
[12] Wu H. Normal families of holomorphic mappings. Acta Math, 1967, 119:193-233
