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, p. 805-815
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.
Classification:  30G35
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {4},
     year = {2009},
     pages = {805-815},
     zbl = {1201.30067},
     mrnumber = {2647912},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/ASNSP_2009_5_8_4_805_0/

[1] F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, “Analysis of Dirac Systems and Computational Algebra”, Vol. 39, Progress in Mathematical Physics, Birkhäuser Boston Inc., Boston, MA, 2004. | MR 2089988

[2] C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J. 32 (1965), 139–148. | MR 173012 | Zbl 0173.09001

[3] P. Dentoni and M. Sce, Funzioni regolari nell’algebra di Cayley, Rend. Sem. Mat. Univ. Padova 50 (1974), 251–267. | Numdam | MR 361113 | Zbl 0283.30039

[4] R. Fueter, Die Funktionentheorie der differentialgleichungen Θu=0 und ΘΘu=0 mit vier reellen variablen, Comment. Math. Helv. 7 (1934), 307–330. | JFM 61.1131.05 | MR 1509515

[5] R. Fueter, Über die analytische darstellung der regulären funktionen einer quaternionenvariablen, Comment. Math. Helv. 8 (1935), 371–378. | JFM 62.0120.04 | MR 1509533

[6] G. Gentili and C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable, Michigan Math. J. 56 (2008), 655–667. | MR 2490652 | Zbl 1184.30048

[7] G. Gentili and D. C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Paris 342 (2006), 741–744. | MR 2227751 | Zbl 1105.30037

[8] G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), 279–301. | MR 2353257 | Zbl 1124.30015

[9] K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter’s theorem, Methods Appl. Anal. 9 (2002), 273–289. | MR 1957490 | Zbl 1079.30066

[10] V. V. Kravchenko and M. V. Shapiro, “Integral Representations for Spatial Models of Mathematical Physics”, Vol. 351, Pitman Research Notes in Mathematics Series, Longman, Harlow, 1996. | MR 1429392 | Zbl 0872.35001

[11] G. Laville and E. Lehman, Analytic Cliffordian functions, Ann. Acad. Sci. Fenn. Math. 29 (2004), 251–268. | MR 2097231 | Zbl 1077.30046

[12] G. Laville and I. Ramadanoff, Holomorphic Cliffordian functions, Adv. Appl. Clifford Algebr. 8 (1998), 323–340. | MR 1697976 | Zbl 0940.30028

[13] A. Pogorui and M. Shapiro, On the structure of the set of zeros of quaternionic polynomials, Complex Var. Theory Appl. 49 (2004), 379–389. | MR 2073169 | Zbl 1160.30353

[14] T. Qian, Generalization of Fueter’s result to 𝐑 n+1 , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8 (1997), 111–117. | MR 1485323 | Zbl 0909.30036

[15] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 23 (1957), 220–225. | MR 97386 | Zbl 0084.28302

[16] A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–224. | MR 516081 | Zbl 0399.30038