Skip to main content
SpringerLink
Log in

Loewner Chains Associated with Close to Almost Starlike Mappings of Order \(\alpha \)

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

In this article, we give an equivalent characterization for close to almost starlike mappings of order \(\alpha \)\((0\le \alpha <1)\) in terms of Loewner chains. Next, the growth theorem and distortion theorem along a direction are obtained for this subclass of biholomorphic mappings. In particular, the results of Pfaltzgraff and Suffridge can be obtained when \(\alpha =0\).

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

References

  1. Loewner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann. 89, 103–121 (1923)

    Article  MathSciNet  Google Scholar 

  2. Kufarev, P.P.: A remark on integrals of the Loewner equation. Dokl. Akad. Nauk SSSR 57, 655–656 (1947)

    Google Scholar 

  3. Pfaltzgraff, J.A.: Subordination chains and univalence of holomorphic mappings in \(\mathbb{C}^n\). Math. Ann. 210, 55–68 (1974)

    Article  MathSciNet  Google Scholar 

  4. Poreda, T.: On the univalent subordination chains of holomorphic mappings in Banach spaces. Comment. Math. 28, 295–304 (1989)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Graham, I., Kohr, G., Kohr, M.: Loewner chains and parametric representation in several complex variables. J. Math. Anal. Appl. 281, 425–438 (2003)

    Article  MathSciNet  Google Scholar 

  7. Graham, I., Kohr, G., Kohr, M.: Basic properties of Loewner chains in several complex variables. In: FitzGerald, C., Gong, S. (eds.) Geometric Function Theory in Several Complex Variables, pp. 165–181. World Scientific Publishing, River Edge (2004)

    Chapter  Google Scholar 

  8. Abate, M., Bracci, F., Contreras, M., Diaz-Madrigal, S.: The evolution of Loewner’s differential equations. Eur. Math. Soc. Newsl. 78, 31–38 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Kinneberg, K.: Loewner chains and Holder geometry. Ann. Acad. Sci. Fenn. Math. 40, 803–835 (2015)

    Article  MathSciNet  Google Scholar 

  10. Pfaltzgraff, J.A., Suffridge, T.J.: Close-to-starlike holomorphic functions of several variables. Pac. J. Math. 57(1), 271–279 (1975)

    Article  MathSciNet  Google Scholar 

  11. Hamada, H., Kohr, G.: Subordination chains and the growth theorem of spirallike mappings. Mathematica (Cluj) 42(65), 153–161 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Xu, Q.H., Liu, T.S.: Loewner chains and a subclass of biholomorphic mappings. J. Math. Anal. Appl. 334, 1096–1105 (2007)

    Article  MathSciNet  Google Scholar 

  13. Liu, X.S., Liu, T.S.: The sharp distortion theorem for a subclass of close-to-convex mappings. Chin. Ann. Math. 33A, 91–100 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Xu, Q.H., Liu, T.S., Liu, X.S.: On a subclass of close-to-convex mappings. Complex Anal. Oper. Theory 9, 275–286 (2015)

    Article  MathSciNet  Google Scholar 

  15. Srivastava, H.M., Rafiullah Khan, M., Arif, M.: Some subclasses of close-to-convex mappings associated with conic regions. Appl. Math. Comput. 285, 94–102 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Graham, I., Kohr, G.: Geometric Function Theory in One and High Dimensions, Pure and Applied Mathematics, New York (2003)

  17. Feng, S.X., Liu, T.S.: The generalized Roper–Suffridge extension operator. Acta Math. Sci. 28B(1), 63–80 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Feng, S.X., Liu, T.S., Ren, G.B.: The growth and covering theorems for several mappings on the unit ball in complex Banach spaces. Chin. Ann. Math. 28A(2), 215–230 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Liu, T.S., Liu, X.S.: A refinement about estimation of expansion coefficients for normalized biholomorphic mappings. Sci. China Ser. A 48(7), 865–879 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11701307), the Key Scientific Research Projects in Universities of Henan Province (No. 18B110016), the Foster Foundation of Pingdingshan University (No. PXY-PYJJ2016007).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Pingdingshan University, Pingdingshan, 467000, China

    Xiaofei Zhang

Authors
  1. Xiaofei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaofei Zhang.

Additional information

Communicated by Stephan Ruscheweyh.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X. Loewner Chains Associated with Close to Almost Starlike Mappings of Order \(\alpha \). Comput. Methods Funct. Theory 19, 643–657 (2019). https://doi.org/10.1007/s40315-019-00289-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-019-00289-6

Keywords

Mathematics Subject Classification

Search

Navigation