TY - JOUR

T1 - Inequalities for the carathéodory and poincaré metrics in open unit balls

AU - Earle, Clifford J.

AU - Harris, Lawrence A.

PY - 2011/4

Y1 - 2011/4

N2 - We generalize a known inequality relating the Euclidean and hyperbolic metrics in Poincaré's unit ball model of hyperbolic space. Our generalization applies to Schwarz-Pick metrics in the open unit ball of any complex Banach space. We study the case of equality, both in the general case and the case when the open unit ball is homogeneous. For open unit balls of Hilbert spaces, we explicitly determine the case of equality, and we prove a distortion theorem that quantifies the failure of equality when the inequality is strict. An analogous distortion theorem for real hyperbolic space follows readily.

AB - We generalize a known inequality relating the Euclidean and hyperbolic metrics in Poincaré's unit ball model of hyperbolic space. Our generalization applies to Schwarz-Pick metrics in the open unit ball of any complex Banach space. We study the case of equality, both in the general case and the case when the open unit ball is homogeneous. For open unit balls of Hilbert spaces, we explicitly determine the case of equality, and we prove a distortion theorem that quantifies the failure of equality when the inequality is strict. An analogous distortion theorem for real hyperbolic space follows readily.

KW - Bounded symmetric domains

KW - Complex geodesics

KW - Hyperbolic space

KW - JB-Triples

KW - Schwarz-Pick systems

UR - http://www.scopus.com/inward/record.url?scp=79952284130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952284130&partnerID=8YFLogxK

U2 - 10.4310/PAMQ.2011.v7.n2.a1

DO - 10.4310/PAMQ.2011.v7.n2.a1

M3 - Article

AN - SCOPUS:79952284130

SN - 1558-8599

VL - 7

SP - 253

EP - 273

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

IS - 2

ER -