On normal and non-normal holomorphic functions on complex Banach manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 1, p. 1-15

Let X be a complex Banach manifold. A holomorphic function f:X is called a normal function if the family f ={fφ:φ𝒪(Δ,X)} forms a normal family in the sense of Montel (here 𝒪(Δ,X) denotes the set of all holomorphic maps from the complex unit disc into X). Characterizations of normal functions are presented. A sufficient condition for the sum of a normal function and non-normal function to be non-normal is given. Criteria for a holomorphic function to be non-normal are obtained. These results are used to draw one interesting conclusion on the boundary behavior of normal holomorphic functions in a convex bounded domain D in a complex Banach space V. Let {x n } be a sequence of points in D which tends to a boundary point ξD such that lim n f(x n )=L for some L ¯. Sufficient conditions on a sequence {x n } of points in D and a normal holomorphic function f are given for f to have the admissible limit value L, thus extending the result obtained by Bagemihl and Seidel.

Classification:  32A18
@article{ASNSP_2009_5_8_1_1_0,
     author = {Dovbush, Peter},
     title = {On normal and non-normal holomorphic functions on complex Banach manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {1},
     year = {2009},
     pages = {1-15},
     zbl = {1183.32004},
     mrnumber = {2512198},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_1_1_0}
}
Dovbush, Peter. On normal and non-normal holomorphic functions on complex Banach manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 1, pp. 1-15. http://www.numdam.org/item/ASNSP_2009_5_8_1_1_0/

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