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Journal article
The Julia-Caratheodory theorem on the bidisk revisited
Acta scientiarum mathematicarum (Szeged), v 83(1-2), pp 165-175
01 Jan 2017
Abstract
The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Caratheodory 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.
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Details
- Title
- The Julia-Caratheodory theorem on the bidisk revisited
- Creators
- John E. McCarthy - Washington Univ, One Brookings Dr, St Louis, MO 63130 USAJames E. Pascoe - Washington Univ, One Brookings Dr, St Louis, MO 63130 USA
- Publication Details
- Acta scientiarum mathematicarum (Szeged), v 83(1-2), pp 165-175
- Publisher
- Univ Szeged, Bolyai Institute
- Number of pages
- 11
- Grant note
- DMS 1565243; DMS1606260 / National Science Foundation; National Science Foundation (NSF)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000405987400013
- Scopus ID
- 2-s2.0-85021042583
- Other Identifier
- 991021879784304721
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- Mathematics