Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T07:04:56.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Bohr theorems for slice regular functions over octonions

Part of: General nonassociative rings Generalized function theory Geometric function theory

Published online by Cambridge University Press:  21 October 2020

Zhenghua Xu
Zhenghua Xu*
Affiliation:
School of Mathematics, HeFei University of Technology, Hefei230601, China (zhxu@hfut.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we mainly investigate two versions of the Bohr theorem for slice regular functions over the largest alternative division algebras of octonions $\mathbb {O}$. To this end, we establish the coefficient estimates for self-maps of the unit ball of $\mathbb {O}$ and the Carathéodory class in this setting. As a further application of the coefficient estimate, the 1/2-covering theorem is also proven for slice regular functions with convex image.

Keywords

function of one hypercomplex variabledivision algebrasBohr theoremcovering theorem

MSC classification

Primary: 30G35: Functions of hypercomplex variables and generalized variables
Secondary: 17A35: Division algebras 30C25: Covering theorems in conformal mapping theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1595 - 1610
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abu-Muhanna, Y., Ali, R. M. and Ponnusamy, S.. On the Bohr inequality. In Progress in approximation theory and applicable complex analysis, Springer Optim. Appl., 117, pp. 269–300 (Cham: Springer, 2017).Google Scholar
Aizenberg, L.. Multidimensional analogues of Bohr's theorem on power series. Proc. Amer. Math. Soc. 128 (2000), 1147–1155.CrossRefGoogle Scholar
Alpay, D., Bolotnikov, V., Colombo, F., Sabadini, I.. Self-mappings of the quaternionic unit ball: multiplier properties, the Schwarz-Pick inequality, and the Nevanlinna-Pick interpolation problem. Indiana Univ. Math. J. 64 (2015), 151–180.Google Scholar
Baez, J. C.. The octonions. Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145–205.CrossRefGoogle Scholar
Bayart, F., Pellegrino, D. and Seoane-Sepúlveda, J. B.. The Bohr radius of the n-dimensional polydisk is equivalent to $\sqrt {(logn)/n}$. Adv. Math. 264 (2014), 726–746.CrossRefGoogle Scholar
Bisi, C. and Gentili, G.. Möbius transformations and the Poincaré distance in the quaternionic setting. Indiana Univ. Math. J. 58 (2009), 2729–2764.CrossRefGoogle Scholar
Bisi, C. and Stoppato, C.. The Schwarz-Pick lemma for slice regular functions. Indiana Univ. Math. J. 61 (2012), 297–317.CrossRefGoogle Scholar
Blasco, O.. The Bohr radius of a Banach space. In Vector measures, integration and related topics. Oper. Theory Adv. Appl., vol. 201, pp. 59–64 (Basel: Birkhäuser Verlag, 2010).Google Scholar
Boas, H. P. and Khavinson, D.. Bohr's power series theorem in several variables. Proc. Amer. Math. Soc. 125 (1997), 2975–2979.CrossRefGoogle Scholar
Bohr, H.. A theorem concerning power series. Proc. London Math. Soc. 13 (1914), 1–5.CrossRefGoogle Scholar
Colombo, F., Sabadini, I. and Struppa, D. C.. Slice monogenic functions. Israel J. Math. 171 (2009), 385–403.CrossRefGoogle Scholar
Defant, A., Maestre, M. and Schwarting, U.. Bohr radii of vector valued holomorphic functions. Adv. Math. 231 (2012), 2837–2857.CrossRefGoogle Scholar
Della Rocchetta, C., Gentili, G. and Sarfatti, G.. The Bohr theorem for slice regular functions. Math. Nachr. 285 (2012), 2093–2105.CrossRefGoogle Scholar
Dixon, P. G.. Banach algebras satisfying the non-unital von Neumann inequality. Bull. London Math. Soc. 27 (1995), 359–362.CrossRefGoogle Scholar
Gentili, G., Stoppato, C. and Struppa, D. C.. Regular functions of a quaternionic variable. Springer monographs in mathematics (Berlin-Heidelberg: Springer, 2013).Google Scholar
Gentili, G. and Struppa, D. C.. A new theory of regular functions of a quaternionic variable. Adv. Math. 216 (2007), 279–301.CrossRefGoogle Scholar
Gentili, G. and Struppa, D. C.. Regular functions on the space of Cayley numbers. Rocky Mt. J. Math. 40 (2010), 225–241.CrossRefGoogle Scholar
Ghiloni, R. and Perotti, A.. Slice regular functions on real alternative algebras. Adv. Math. 226 (2011), 1662–1691.CrossRefGoogle Scholar
Ghiloni, R., Perotti, A. and Stoppato, C.. Division algebras of slice functions. Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 2055–2082.CrossRefGoogle Scholar
Graham, I. and Kohr, G.. Geometric function theory in one and higher dimensions. In Monographs and textbooks in pure and applied mathematics, vol. 255 (New York: Marcel Dekker, Inc., 2003).Google Scholar
Gürlebeck, K. and Morais, J.. Bohr type theorem for monogenic power series. Comput. Methods Funct. Theory 9 (2009), 633–651.CrossRefGoogle Scholar
Gürlebeck, K., Morais, J. and Cerejeiras, P.. Borel-Carathéodory type theorem for monogenic functions. Complex Anal. Oper. Theory 3 (2009), 99–112.CrossRefGoogle Scholar
Hamada, H., Honda, T. and Kohr, G.. Bohr's theorem for holomorphic mappings with values in homogeneous balls. Israel J. Math. 173 (2009), 177–187.CrossRefGoogle Scholar
Kresin, G., Maz'ya, V.. Sharp real-part theorems: A unified approach. Lecture Notes in Mathematics, vol. 1903 (Berlin: Springer, 2007).Google Scholar
Liu, T. and Wang, J.. An absolute estimate of the homogeneous expansions of holomorphic mappings. Pacific J. Math. 231 (2007), 155–166.CrossRefGoogle Scholar
Paulsen, V. I., Popescu, G. and Singh, D.. On Bohr's inequality. Proc. London Math. Soc. (3) 85 (2002), 493–512.CrossRefGoogle Scholar
Paulsen, V. I. and Singh, D.. Extensions of Bohr's inequality. Bull. London Math. Soc. 38 (2006), 991–999.CrossRefGoogle Scholar
Popescu, G.. Multivariable Bohr inequalities. Trans. Am. Math. Soc. 359 (2007), 5283–5317.CrossRefGoogle Scholar
Popescu, G.. Bohr inequalities for free holomorphic functions on polyballs. Adv. Math. 347 (2019), 1002–1053.CrossRefGoogle Scholar
Popescu, G.. Bohr inequalities on noncommutative polydomains. Integr. Equ. Oper. Theory 91 (2019), 55pp, Art. 7.CrossRefGoogle Scholar
Ren, G. and Wang, X.. Carathéodory theorems for slice regular functions. Complex Anal. Oper. Theory 9 (2015), 1229–1243.CrossRefGoogle Scholar
Stoppato, C.. Regular Möbius transformations of the space of quaternions. Ann. Global Anal. Geom. 39 (2011), 387–401.CrossRefGoogle Scholar
Xu, Z.. Generalized Bohr radius for slice regular functions over quaternions. Ann. Mat. Pura Appl. (2020), DOI:10.1007/s10231-020-01034-w.Google Scholar
Xu, Z. and Ren, G.. Slice starlike functions over quaternions. J. Geom. Anal. 28 (2018), 3775–3806.CrossRefGoogle Scholar