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Fekete–Szegö Problems for Spirallike Mappings and Close-to-Quasi-Convex Mappings on the Unit Ball of a Complex Banach Space

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Abstract

In the first part of this paper, we will give the Fekete–Szegö inequality for various subfamilies of spirallike mappings of type \(\beta \) on the unit ball of a complex Banach space. Our results give extensions of those given by Lai and Xu (Results Math 76(4), Paper No. 191, 2021) and Elin and Jacobzon (Results Math 77(3), Paper No. 137, 2022). We next give the Fekete–Szegö inequality for close-to-quasi-convex mappings of type B on the unit ball of a complex Banach space. Our results give extensions of that given by Xu et al. (Complex Var Elliptic Equ. https://doi.org/10.1080/17476933.2021.1975115).

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References

  1. Długosz, R., Liczberski, P.: Some results of Fekete-Szegö type for Bavrin’s families of holomorphic functions in \({\mathbb{C} }^n\). Ann. Mat. Pura Appl. 4(200), 1841–1857 (2021)

    Article  MATH  Google Scholar 

  2. Długosz, R., Liczberski, P.: Fekete-Szegö problem for Bavrin’s functions and close-to-starlike mappings in \({\mathbb{C} }^n\). Anal. Math. Phys. 12(4), Paper No. 103, 16 (2022)

  3. Elin, M., Jacobzon, F.: Note on the Fekete-Szegö problem for spirallike mappings in Banach spaces. Results Math. 77(3), Paper No. 137, 6 (2022)

  4. Fekete, M., Szegö, G.: Eine Bemerkung Uber Ungerade Schlichte Funktionen. J. Lond. Math. Soc. 8, 85–89 (1933). ((in German))

    Article  MATH  MathSciNet  Google Scholar 

  5. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 255. Marcel Dekker, Inc., New York (2003)

    Google Scholar 

  6. Hamada, H., Kohr, G.: Subordination chains and the growth theorem of spirallike mappings. Mathematica 42(65)(2), 153–161 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Hamada, H., Kohr, G.: \(\Phi \)-like and convex mappings in infinite dimensional spaces. Rev. Roumaine Math. Pures Appl. 47(3), 315–328 (2002)

    MATH  MathSciNet  Google Scholar 

  8. Hamada, H., Kohr, G., Kohr, M.: The Fekete–Szegö problem for starlike mappings and nonlinear resolvents of the Carathéodory family on the unit balls of complex Banach spaces. Anal. Math. Phys. 11(3), Paper No. 115 (2021)

  9. Hamada, H., Kohr, G., Kohr, M.: Fekete–Szegö problem for univalent mappings in one and higher dimensions. J. Math. Anal. Appl. 516, 126526 (2022)

    Article  MATH  Google Scholar 

  10. Kanas, S.: An unified approach to the Fekete–Szegö problem. Appl. Math. Comput. 218, 8453–8461 (2012)

    MATH  MathSciNet  Google Scholar 

  11. Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  12. Koepf, W.: On the Fekete–Szegö problem for close-to-convex functions. Proc. Am. Math. Soc. 101, 89–95 (1987)

    MATH  MathSciNet  Google Scholar 

  13. Lai, Y., Xu, Q.H.: On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables. Results Math. 76(4), Paper No. 191 (2021)

  14. Luo, H., Xu, Q.H.: On the Fekete and Szegö inequality for a subclass of strongly starlike mappings of order \(\alpha \). Results Math. 72(1–2), 343–357 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)

    MATH  Google Scholar 

  16. Roper, K., Suffridge, T.J.: Convexity properties of holomorphic mappings in \({\mathbb{C} }^n\). Trans. Am. Math. Soc. 351, 1803–1833 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Xu, Q.H., Liu, T.: Study on estimation of the homogeneous expansion of subclasses of biholomorphic mappings by a unified method. Acta Math. Sin. (Chin. Ser.) 52(6), 1189–1198 (2009)

    MATH  MathSciNet  Google Scholar 

  18. Xu, Q.H., Liu, T.: On the Fekete and Szegö problem for the class of starlike mappings in several complex variables. Abstr. Appl. Anal. 2014, Art. ID 807026 (2014)

  19. Xu, Q.H., Liu, T., Liu, X.: The sharp estimates of homogeneous expansions for the generalized class of close-to-quasi-convex mappings. J. Math. Anal. Appl. 389, 781–791 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Xu, Q.H., Liu, T., Liu, X.: Fekete and Szegö problem in one and higher dimensions. Sci. China Math. 61, 1775–1788 (2018)

    Article  MATH  MathSciNet  Google Scholar 

  21. Xu, Q.H., Liu, T., Lu, J.: The Fekete and Szegö inequality for a class of holomorphic mappings on the unit polydisk in \({\mathbb{C}}^n\) and its application. Complex Var. Elliptic Equ. (to appear). https://doi.org/10.1080/17476933.2021.1975115

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Acknowledgements

The author would like to thank the referees for many valuable suggestions.

Funding

Hidetaka Hamada is partially supported by JSPS KAKENHI Grant Number JP22K03363.

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  1. Faculty of Science and Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku, Fukuoka, 813-8503, Japan

    Hidetaka Hamada

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Hamada, H. Fekete–Szegö Problems for Spirallike Mappings and Close-to-Quasi-Convex Mappings on the Unit Ball of a Complex Banach Space. Results Math 78, 109 (2023). https://doi.org/10.1007/s00025-023-01895-6

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