The basic results of a new theory of regular functions of a quaternionic variable have been recently stated, following an idea of Cullen. In this paper we prove the minimum modulus principle and the open mapping theorem for regular functions. The proofs involve some peculiar geometric properties of such functions which are of independent interest.
@article{ASNSP_2009_5_8_4_805_0,
author = {Gentili, Graziano and Stoppato, Caterina},
title = {The open mapping theorem for regular quaternionic functions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {805--815},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {4},
year = {2009},
mrnumber = {2647912},
zbl = {1201.30067},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2009_5_8_4_805_0/}
}
TY - JOUR AU - Gentili, Graziano AU - Stoppato, Caterina TI - The open mapping theorem for regular quaternionic functions JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2009 SP - 805 EP - 815 VL - 8 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2009_5_8_4_805_0/ LA - en ID - ASNSP_2009_5_8_4_805_0 ER -
%0 Journal Article %A Gentili, Graziano %A Stoppato, Caterina %T The open mapping theorem for regular quaternionic functions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2009 %P 805-815 %V 8 %N 4 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2009_5_8_4_805_0/ %G en %F ASNSP_2009_5_8_4_805_0
Gentili, Graziano; Stoppato, Caterina. The open mapping theorem for regular quaternionic functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 805-815. http://archive.numdam.org/item/ASNSP_2009_5_8_4_805_0/
[1] , , and , “Analysis of Dirac Systems and Computational Algebra”, Vol. 39, Progress in Mathematical Physics, Birkhäuser Boston Inc., Boston, MA, 2004. | MR
[2] , An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J. 32 (1965), 139–148. | MR | Zbl
[3] and , Funzioni regolari nell’algebra di Cayley, Rend. Sem. Mat. Univ. Padova 50 (1974), 251–267. | EuDML | Numdam | MR | Zbl
[4] , Die Funktionentheorie der differentialgleichungen und mit vier reellen variablen, Comment. Math. Helv. 7 (1934), 307–330. | EuDML | JFM | MR
[5] , Über die analytische darstellung der regulären funktionen einer quaternionenvariablen, Comment. Math. Helv. 8 (1935), 371–378. | EuDML | JFM | MR
[6] and , Zeros of regular functions and polynomials of a quaternionic variable, Michigan Math. J. 56 (2008), 655–667. | MR | Zbl
[7] and , A new approach to Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Paris 342 (2006), 741–744. | MR | Zbl
[8] and , A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), 279–301. | MR | Zbl
[9] , and , Generalizations of Fueter’s theorem, Methods Appl. Anal. 9 (2002), 273–289. | MR | Zbl
[10] and , “Integral Representations for Spatial Models of Mathematical Physics”, Vol. 351, Pitman Research Notes in Mathematics Series, Longman, Harlow, 1996. | MR | Zbl
[11] and , Analytic Cliffordian functions, Ann. Acad. Sci. Fenn. Math. 29 (2004), 251–268. | EuDML | MR | Zbl
[12] and , Holomorphic Cliffordian functions, Adv. Appl. Clifford Algebr. 8 (1998), 323–340. | MR | Zbl
[13] and , On the structure of the set of zeros of quaternionic polynomials, Complex Var. Theory Appl. 49 (2004), 379–389. | MR | Zbl
[14] , Generalization of Fueter’s result to , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8 (1997), 111–117. | EuDML | MR | Zbl
[15] , Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 23 (1957), 220–225. | MR | Zbl
[16] , Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–224. | MR | Zbl
