Abstract
In this paper, we study the analogue of spirallikeness for slice regular functions of one quaternionic variable. In particular, we introduce the concept of slice \(\gamma \)-spirallike functions of order \(\alpha \) and investigate its geometric function theory, such as coefficient estimates, growth and covering theorems. As a byproduct, the Robertson’s result concerning the radii of starlikeness for holomorphic spirallike functions is generalized into slice regular functions by a very concise method, but new even for the classical case.
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References
Arora, V., Ponnusamy, S., Sahoo, S. K.: Successive coefficients for spirallike and related functions. Rev. R. Acad. Cienc. Exactas Físicas. Nat. Ser. A Mat. RACSAM 113(4), 2969-2979 (2019)
Barnard, R.W., FitzGerald, C.H., Gong, S.: The growth and \(1/4\) theorems for starlike mapping in \({\mathbb{C} }^{n}\). Pacific J. Math. 150, 13–22 (1991)
Bracci, F., Levenshtein, M., Reich, S., Shoikhet, D.: Growth estimates for the numerical range of holomorphic mappings and applications. Comput. Methods Funct. Theory 16(3), 457–487 (2016)
Clunie, J.: On meromorphic schlicht functions. J. London Math. Soc. 34, 215–216 (1959)
Colombo, F., Sabadini, I., Struppa, D. C.: Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, Progress in Mathematics, vol. 289. Birkhäuser/Springer, Basel, (2011)
de Branges, L.: A proof of the Bieberbach conjecture. Acta Mathematica 154, 137–152 (1985)
Gal, S.G., González-Cervantes, J.O., Sabadini, I.: On some geometric properties of slice regular functions of a quaternion variable. Complex Var. Elliptic Equ. 60(10), 1431–1455 (2015)
Gal, S.G., González-Cervantes, J.O., Sabadini, I.: Univalence results for slice regular functions of a quaternion variable. Complex Var. Elliptic Equ. 60(10), 1346–1365 (2015)
Gentili, G., Stoppato, C.: The open mapping theorem for regular quaternionic functions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)8, 805-815 (2009)
Gentili, G., Stoppato, C., Struppa, D.C.: Regular functions of a quaternionic variable. Springer Monogr. Math. Springer, Cham (2022)
Gentili, G., Struppa, D.C.: A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris. 342(10), 741–744 (2006)
Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)
Gong, S.: Convex and starlike mappings in several complex variables, Mathematics and its Applications, 435. Dordrecht; Science Press, Beijing, Kluwer Academic Publishers (1998)
Gori, A., Vlacci, F.: Starlikeness for functions of a hypercomplex variable. Proc. Amer. Math. Soc. 145(2), 791–804 (2017)
Graham, I., Kohr, G.: Geometric function theory in one and higher dimensions, Monographs and Textbooks in Pure and Applied Mathematics, 255. Marcel Dekker Inc, New York (2003)
Hamada, H.: Approximation properties on spirallike domains of \({\mathbb{C} }^{n}\). Adv. Math. 268, 467–477 (2015)
Libera, R.J.: Univalent \(\alpha \)-spiral functions. Canadian J. Math. 19, 449–456 (1967)
Ren, G., Wang, X.: Growth and distortion theorems for slice monogenic functions. Pacific J. Math. 290(1), 169–198 (2017)
Robertson, M.S.: Radii of star-likeness and close-to-covexity. Proc. Amer. Math. Soc. 16, 847–852 (1965)
Sim, Y.J., Thomas, D.K.: A note on spirallike functions. Bull. Aust. Math. Soc. 105(1), 117–123 (2022)
Wang, L.: The tilted Carathéodory class and its applications. J. Korean Math. Soc. 49(4), 671–686 (2012)
Wang, X.: On geometric aspects of quaternionic and octonionic slice regular functions. J. Geom. Anal. 27(4), 2817–2871 (2017)
Xu, Z., Ren, G.: Slice starlike functions over quaternions. J. Geom. Anal. 28(4), 3775–3806 (2018)
Acknowledgements
The authors would like express their hearty thanks to Xiaofei Zhang (Pingdingshan University) for helpful discussions.
Funding
This work was supported by the Natural Science Foundation of Anhui Province (No. 2308085MA04), the National Natural Science Foundation of China (Nos. 11801125, 62002095 and 12201159) and by the Hainan Provincial Natural Science Foundation of China (No. 120QN177).
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Z. X., D. Z. and J. S. wrote the main manuscript text and Y. L. prepared Fig. 1. All authors reviewed the manuscript.
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Communicated by Irene Sabadini.
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Xu, Z., Zhang, D., Liu, Y. et al. Slice Spirallike Functions over Quaternions. Complex Anal. Oper. Theory 17, 113 (2023). https://doi.org/10.1007/s11785-023-01410-3
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DOI: https://doi.org/10.1007/s11785-023-01410-3