Skip to main content
SpringerLink
Log in

The Fekete-Szegö problem for starlike mappings and nonlinear resolvents of the Carathéodory family on the unit balls of complex Banach spaces

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we first give a coefficient inequality for holomorphic functions on the unit disc \({\mathbb {U}}\) in \({\mathbb {C}}\) which are subordinate to a holomorphic function p on \({\mathbb {U}}\) with \(p'(0)\ne 0\). Next, as applications of this theorem, we will give the Fekete-Szegö inequality for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type B on the unit ball \({\mathbb {B}}\) of a complex Banach space. We also give the Fekete-Szegö inequality for \((1+r)J_r\), where \(J_r=J_r[f]\) is the nonlinear resolvent of a mapping f in the Carathéodory family \({{\mathscr {M}}}({\mathbb {B}})\). Various particular cases will be also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Aharonov, D., Elin, M., Reich, S., Shoikhet, D.: Parametric representation of semi-complete vector fields on the unit balls in \({\mathbb{C}}^n\) and in Hilbert space. Rend. Mat. Acc. Lincei. 10, 229–253 (1999)

    MATH  Google Scholar 

  2. Ali, R.M.: Starlikeness associated with parabolic regions. Int. J. Math. Math. Sci. 4, 561–570 (2005)

    Article  MathSciNet  Google Scholar 

  3. Bracci, F.: Shearing process and an example of a bounded support function in S0(B2). Comput. Methods Funct. Theory 15, 151–157 (2015)

  4. Bracci, F., Graham, I., Hamada, H., Kohr, G.: Variation of Loewner chains, extreme and support points in the class \(S^0\) in higher dimensions. Constr. Approx. 43, 231–251 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bracci, F., Roth, O.: Support points and the Bieberbach conjecture in higher dimension. In: Complex Analysis and Dynamical Systems, pp. 67–79. Trends in Mathematics, Springer, Cham (2018)

  6. Elin, M., Reich, S., Shoikhet, D.: Numerical Range of Holomorphic Mappings and Applications. Birkhäuser, Cham (2019)

    Book  Google Scholar 

  7. Elin, M., Shoikhet, D., Sugawa, T.: Geometric properties of the nonlinear resolvent of holomorphic generators. J. Math. Anal. Appl. 483, 123614 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Gong, S.: The Bieberbach Conjecture. American Mathematical Society, Cambridge (1999)

    Book  Google Scholar 

  10. Graham, I., Hamada, H., Honda, T., Kohr, G., Shon, K.H.: Growth, distortion and coefficient bounds for Carathéodory families in \({\mathbb{C}}^n\) and complex Banach spaces. J. Math. Anal. Appl. 416, 449–469 (2014)

    Article  MathSciNet  Google Scholar 

  11. Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Canad. J. Math. 54, 324–351 (2002)

    Article  MathSciNet  Google Scholar 

  12. Graham, I., Hamada, H., Kohr, G.: Extremal problems for mappings with \(g\)-parametric representation on the unit polydisc in \(Cn\). In: Complex Analysis and Dynamical Systems, pp. 141–167. Trends in Mathematics, Birkhüser, Cham (2018)

  13. Graham, I., Hamada, H., Kohr, G.: Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in \({\mathbb{C}}^n\). J. Math. Anal. Appl. 491, 124289 (2020)

    Article  MathSciNet  Google Scholar 

  14. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Support points and extreme points for mappings with \(A\)-parametric representation in \({\mathbb{C}}^n\). J. Geom. Anal. 26, 1560–1595 (2016)

    Article  MathSciNet  Google Scholar 

  15. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Bounded support points for mappings with \(g\)-parametric representation in \({\mathbb{C}}^2\). J. Math. Anal. Appl. 454, 1085–1105 (2017)

    Article  MathSciNet  Google Scholar 

  16. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker Inc., New York (2003)

    Book  Google Scholar 

  17. Gurganus, K.R.: \(\Phi \)-like holomorphic functions in \({\mathbf{C}}^{n}\) and Banach spaces. Trans. Am. Math. Soc. 205, 389–406 (1975)

    MathSciNet  MATH  Google Scholar 

  18. Hamada, H., Honda, T.: Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin. Ann. Math. Ser. B. 29(4), 353–368 (2008)

    Article  MathSciNet  Google Scholar 

  19. Hamada, H., Honda, T., Kohr, G.: Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation. J. Math. Anal. Appl. 317, 302–319 (2006)

    Article  MathSciNet  Google Scholar 

  20. Hamada, H., Honda, T., Kohr, G.: Parabolic starlike mappings in several complex variables. Manuscripta Math. 123, 301–324 (2007)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Hamada, H., Kohr, G.: Support points for families of univalent mappings on bounded symmetric domains. Sci. China Math. 63, 2379–2398 (2020)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  26. Liu, T., Liu, X.: A refinement about estimation of expansion coefficients for normalized biholomorphic mappings. Sci. China Ser. A. 48, 865–879 (2005)

    Article  MathSciNet  Google Scholar 

  27. Liu, X., Liu, T.: The refining estimation of homogeneous expansions for quasi-convex mappings. Adv. Math. (China) 36, 679–685 (2007)

    MathSciNet  Google Scholar 

  28. Liu, X., Liu, T.: The sharp estimates of all homogeneous expansions for a class of quasi-convex mappings on the unit polydisk in \({\mathbb{C}}^n\). Chin. Ann. Math. Ser. B. 32, 241–252 (2011)

    Article  MathSciNet  Google Scholar 

  29. Muir, J.R.: Open problems related to a Herglotz-type formula for vector-valued mappings. In: Geometric function theory in higher dimension, pp. 107–115. Springer, Cham (2017)

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

    MATH  Google Scholar 

  31. Reich, S., Shoikhet, D.: Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstr. Appl. Anal. 1, 1–44 (1996)

    Article  MathSciNet  Google Scholar 

  32. Reich, S., Shoikhet, D.: Metric domains, holomorphic mappings and nonlinear semigroups. Abstr. Appl. Anal. 3, 203–228 (1998)

    Article  MathSciNet  Google Scholar 

  33. Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces. Imperial College Press, London (2005)

    Book  Google Scholar 

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

    Article  Google Scholar 

  35. Schleissinger, S.: On the parametric representation of univalent functions on the polydisc. Rocky Mountain J. Math. 48, 981–1001 (2018)

    Article  MathSciNet  Google Scholar 

  36. Suffridge, T.J.: Starlike and convex maps in Banach spaces. Pacific J. Math. 46, 474–489 (1973)

    Article  MathSciNet  Google Scholar 

  37. Xu, Q.H., Liu, T.: On coefficient estimates for a class of holomorphic mappings. Sci. China Ser. A. 52, 677–686 (2009)

    Article  MathSciNet  Google Scholar 

  38. Xu, Q.H., Liu, T.: Coefficient bounds for biholomorphic mappings which have a parametric representation. J. Math. Anal. Appl. 355, 126–130 (2009)

    Article  MathSciNet  Google Scholar 

  39. Xu, Q.H., Liu, T.: Biholomorphic mappings on bounded starlike circular domains. J. Math. Anal. Appl. 366, 153–163 (2010)

    Article  MathSciNet  Google Scholar 

  40. Xu, Q.H., Liu, T.: On the Fekete and Szegö problem for the class of starlike mappings in several complex variables. Abstr. Appl. Anal. 807026, 6 (2014)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Xu, Q.H., Liu, T., Liu, X.: The coefficient inequalities for a class of holomorphic mappings in several complex variables. Chin. Ann. Math. Ser. B 41, 37–48 (2020)

    Article  MathSciNet  Google Scholar 

  43. Xu, Q.H., Yang, T., Liu, T., Xu, H.: Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables. Front. Math. China. 10, 1461–1472 (2015)

    Article  MathSciNet  Google Scholar 

  44. Zhang, W., Liu, T.: On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space. Sci. China Ser. A. 45, 1538–1547 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Hidetaka Hamada

  2. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Gabriela Kohr & Mirela Kohr

Authors
  1. Hidetaka Hamada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriela Kohr
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mirela Kohr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hidetaka Hamada.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

H. Hamada was partially supported by JSPS KAKENHI Grant Number JP19K03553.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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, 115 (2021). https://doi.org/10.1007/s13324-021-00557-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-021-00557-6

Keywords

Mathematics Subject Classification

Search

Navigation