Starlikeness for functions of a hypercomplex variable

Gori, Anna; Vlacci, Fabio

doi:10.1090/proc/13256

Starlikeness for functions of a hypercomplex variable
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by Anna Gori and Fabio Vlacci PDF

Proc. Amer. Math. Soc. 145 (2017), 791-804 Request permission

Abstract:

In this paper we introduce new notions of starlikeness for a class of functions of a hypercomplex variable. We then obtain equivalent formulations for starlikeness which resemble the analogous ones in the holomorphic case such as Nevanlinna’s criterion. Furthermore we give a characterization of algebraic starlikeness in terms of non-vanishing of a suitable analog of the Hadamard product.

References

Roger W. Barnard, A generalization of Study’s theorem on convex maps, Proc. Amer. Math. Soc. 72 (1978), no. 1, 127–134. MR 503546, DOI 10.1090/S0002-9939-1978-0503546-9
Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
Sorin G. Gal, Elements of geometric theory for functions of quaternionic variable, Adv. Appl. Clifford Algebras 10 (2000), no. 1, 91–106. MR 1803809, DOI 10.1007/BF03042011
Sorin G. Gal, Starlike, convex and alpha-convex functions of hyperbolic complex and of dual complex variable, Studia Univ. Babeş-Bolyai Math. 46 (2001), no. 2, 23–40. MR 1954251
Sorin G. Gal, J. Oscar González-Cervantes, and Irene Sabadini, On some geometric properties of slice regular functions of a quaternion variable, Complex Var. Elliptic Equ. 60 (2015), no. 10, 1431–1455. MR 3376178, DOI 10.1080/17476933.2015.1024670
Graziano Gentili and Caterina Stoppato, The open mapping theorem for regular quaternionic functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 4, 805–815. MR 2647912
Graziano Gentili, Simon Salamon, and Caterina Stoppato, Twistor transforms of quaternionic functions and orthogonal complex structures, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 11, 2323–2353. MR 3283400, DOI 10.4171/JEMS/488
Graziano Gentili, Caterina Stoppato, and Daniele C. Struppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics, Springer, Heidelberg, 2013. MR 3013643, DOI 10.1007/978-3-642-33871-7
Graziano Gentili and Daniele C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), no. 1, 279–301. MR 2353257, DOI 10.1016/j.aim.2007.05.010
Reinhold Remmert, Theory of complex functions, Graduate Texts in Mathematics, vol. 122, Springer-Verlag, New York, 1991. Translated from the second German edition by Robert B. Burckel; Readings in Mathematics. MR 1084167, DOI 10.1007/978-1-4612-0939-3
Herb Silverman, Evelyn Marie Silvia, and Donald Telage, Convolution conditions for convexity starlikeness and spiral-likeness, Math. Z. 162 (1978), no. 2, 125–130. MR 506577, DOI 10.1007/BF01215069
T. J. Suffridge, The principle of subordination applied to functions of several variables, Pacific J. Math. 33 (1970), 241–248. MR 261040