Geodesic
Loewner Chains and Biholomorphic Mappings in Cn and Reflexive Complex Banach Spaces
Publications de l'Institut Mathématique, _N_S_75 (2004) no. 89, p. 199 .

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This paper is a survey of very recent results about biholomorphic mappings of the ball in $\Bbb{C}^n$ and in reflexive complex Banach spaces. After recalling existence and regularity results in $\Bbb{C}^n$, we present certain applications including univalence criteria and quasiconformal extension results. We also consider nonuniqueness phenomena for solutions of the Loewner differential equation, and a geometric characterization of Loewner chains which satisfy a growth condition in $t$ based on a generalization of the Carathéodory convergence theorem. Finally we describe some properties of Loewner chains and the Loewner equation on the unit ball of a reflexive complex Banach space.
DOI : 10.2298/PIM0475199G
Classification : 32H02 30C45
Keywords: Carathéodory class, Loewner chain, Loewner differential equation, transition mapping, kernel convergence, starlike mapping, convex mapping, close-to-starlike mapping 
@article{10_2298_PIM0475199G,
     author = {Ian Graham and Gabriela Kohr and John A. Pfaltzgraff},
     title = {Loewner {Chains} and {Biholomorphic} {Mappings} in {Cn} and {Reflexive} {Complex} {Banach} {Spaces}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {199 },
     publisher = {mathdoc},
     volume = {_N_S_75},
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     year = {2004},
     doi = {10.2298/PIM0475199G},
     zbl = {1081.32009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM0475199G/}
}
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Ian Graham; Gabriela Kohr; John A. Pfaltzgraff. Loewner Chains and Biholomorphic Mappings in Cn and Reflexive Complex Banach Spaces. Publications de l'Institut Mathématique, _N_S_75 (2004) no. 89, p. 199 . doi : 10.2298/PIM0475199G. http://geodesic.mathdoc.fr/articles/10.2298/PIM0475199G/

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