TY - JOUR

T1 - The Julia-Carathéodory theorem on the bidisk revisited

AU - McCarthy, John E.

AU - Pascoe, James E.

N1 - Publisher Copyright:
© Bolyai Institute, University of Szeged.

PY - 2017

Y1 - 2017

N2 - The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carathéodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.

AB - The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carathéodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.

KW - Boundary behavior of holomorphic functions

KW - Function theory on the bidisk

KW - Julia-Caratheodory Theorem

UR - http://www.scopus.com/inward/record.url?scp=85021042583&partnerID=8YFLogxK

U2 - 10.14232/actasm-016-311-x

DO - 10.14232/actasm-016-311-x

M3 - Article

AN - SCOPUS:85021042583

SN - 0001-6969

VL - 83

SP - 165

EP - 175

JO - Acta Scientiarum Mathematicarum

JF - Acta Scientiarum Mathematicarum

IS - 1-2

ER -