Acta Mathematica Sinica, English Series

, Volume 33, Issue 4, pp 554–564 | Cite as

On the Fekete and Szegö problem for starlike mappings of order α

Article

Abstract

Let Sα* be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szegö inequality for the class Sα*, and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in Cn.

Keywords

Fekete–Szegö problem starlike mappings of order α sharp coefficient bound 

MR(2010) Subject Classification

32H02 30C45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bieberbach, L.: Über die Koeffizienten der einigen Potenzreihen welche eine schlichte Abbildung des Einheitskreises vermitten, S. B. Preuss. Akad. Wiss, 1916MATHGoogle Scholar
  2. [2]
    Bhowmik, B., Ponnusamy, S., Wirths, K. J.: On the Fekete–Szegö problem for concave univalent functions. J. Math. Anal. Appl., 373, 432–438 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Cartan, H.: Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes, in: P. Montel (Ed.), Lecons sur les Fonctions Univalentes ou Multivalentes, Gauthier-Villars, Paris, 1933Google Scholar
  4. [4]
    de-Branges, L.: A proof of the Bieberbach conjecture. Acta Math., 154(1–2), 137–152 (1985)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Fekete, M., Szegö, G.: Eine Bemerkunguber ungerade schlichte Funktionen. J. Lond. Math. Soc., 8, 85–89 (1933)CrossRefMATHGoogle Scholar
  6. [6]
    Gong, S.: The Bieberbach Conjecture, Amer. Math. Soc., International Press, Providence, RI, 1999Google Scholar
  7. [7]
    Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Canadian J. Math., 54, 324–351 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Graham, I., Kohr, G., Kohr, M.: Loewner chains and parametric representation in several complex variables. J. Math. Anal. Appl., 281, 425–438 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003MATHGoogle Scholar
  10. [10]
    Graham, I., Hamada, H., Honda, T., et al.: Growth, distortion and coefficient bounds for Carathéodory families in Cn and complex Banach spaces. J. Math. Anal. Appl., 416, 449–469 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Hamada, H., Kohr, G., Liczberski, P.: Starlike mappings of order a on the unit ball in complex Banach spaces. Glas. Mat. Ser., 36(3), 39–48 (2001)MathSciNetMATHGoogle Scholar
  12. [12]
    Hamada, H., Honda, T., Kohr, G.: Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representationt. J. Math. Anal. Appl., 317, 302–319 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Hamada, H., Honda, T.: Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin. Ann. Math., 29B(4), 353–368 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Kohr, G.: Certain partial differential inequalities and applications for holomorphic mappings defined on the unit ball of Cn. Ann. Univ. Mariae Curie Skl., Sect. A, 50, 87–94 (1996)MATHGoogle Scholar
  15. [15]
    Kohr, G.: On some best bounds for coefficients of several subclasses of biholomorphic mappings in Cn. Complex Variables, 36, 261–284 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Kanas, S.: An unified approach to the Fekete–Szegö problem. Appl. Math. Comput., 218, 8453–8461 (2012)MathSciNetMATHGoogle Scholar
  17. [17]
    Keogh, F. R., Merkes, E. P.: A coefficient inequality for certain classes of analytic functions. Proc. Amer. Math. Soc., 20, 8–12 (1969)MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    London, R. R.: Fekete–Szegö inequalities for close-to-convex functions. Proc. Amer. Math. Soc., 117(4), 947–950 (1993)MathSciNetMATHGoogle Scholar
  19. [19]
    Liu, X. S., Liu, T. S.: The sharp estimates of all homogeneous expansions for a class of quasi-convex mappings on the unit polydisk in Cn. Chin. Ann. Math., 32B, 241–252 (2011)CrossRefMATHGoogle Scholar
  20. [20]
    Pfluger, A.: The Fekete–Szegö inequality for complex parameter. Complex Var. Theory Appl., 7, 149–160 (1986)CrossRefMATHGoogle Scholar
  21. [21]
    Xu, Q. H., Liu, T. S.: On coefficient estimates for a class of holomorphic mappings. Sci. China Ser. A-Math., 52, 677–686 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Xu, Q. H., Liu, T. S., Liu, X. S.: The sharp estimates of homogeneous expansions for the generalized class ofclose-to-quasi-convex mappings. J. Math. Anal. Appl., 389, 781–791 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Xu, Q. H., Liu, T. S.: On the Fekete and Szegö problem for the class of starlike mappings in several complex variables. Abstr. Appl. Anal., ID 807026, 6 pp (2014)Google Scholar
  24. [24]
    Xu, Q. H., Ting, Y., Liu, T. S., et al.: Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables. Front. Math. China, 10, 1461–1472 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceJiangXi Normal UniversityNanchangP. R. China
  2. 2.Department of MathematicsHuzhou UniversityHuzhouP. R. China

Personalised recommendations