The Julia-Carathéodory theorem on the bidisk revisited

McCarthy John E. and Pascoe James E.: The Julia-Carathéodory theorem on the bidisk revisited. In: Acta scientiarum mathematicarum, (83) 1-2. pp. 165-175. (2017)

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Abstract

The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. Th e JuliaCaratheodory theorem on the bidisk states tha t 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 tha t case. In this note, we show tha t 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.

Item Type:	Article
Journal or Publication Title:	Acta scientiarum mathematicarum
Date:	2017
Volume:	83
Number:	1-2
ISSN:	0001 6969
Page Range:	pp. 165-175
Language:	English
Publisher:	Bolyai Institute, University of Szeged
Place of Publication:	Szeged
Official URL:	http://www.acta.hu
Related URLs:	http://acta.bibl.u-szeged.hu/48269/
DOI:	10.14232/actasm-016-311-x
Uncontrolled Keywords:	Matematika
Additional Information:	Bibliogr.: 175 p. ; összefoglalás angol nyelven
Subjects:	01. Natural sciences 01. Natural sciences > 01.01. Mathematics
Date Deposited:	2017. Jul. 11. 10:43
Last Modified:	2026. Feb. 24. 08:10
URI:	http://acta.bibl.u-szeged.hu/id/eprint/48923

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