Taiwanese Journal of Mathematics

ON SHARP INEQUALITIES OF HOMOGENEOUS EXPANSIONS FOR STARLIKE MAPPINGS OF ORDER $\alpha$ IN SEVERAL COMPLEX VARIABLES

Xiaosong Liu, Taishun Liu, and Qinghua Xu

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Abstract

In this paper, we establish sharp inequalities of homogeneous expansions for starlike mappings and starlike mappings of order $\alpha$ defined on the unit ball of Banach complex spaces. As corollaries, we also obtain the sharp estimates of the third homogeneous expansions for the above mappings defined on the unit polydisk in $\mathbb{C}^n$ with some restricted conditions.

Article information

Source
Taiwanese J. Math., Volume 17, Number 3 (2013), 801-813.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705984

Digital Object Identifier
doi:10.11650/tjm.17.2013.2415

Mathematical Reviews number (MathSciNet)
MR3072262

Zentralblatt MATH identifier
1280.32007

Subjects
Primary: 32A30: Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) {For functions of several hypercomplex variables, see 30G35} 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions

Keywords
starlike mapping starlike mapping of order $\alpha$ inequality of homogeneous expansions estimate of homogeneous expansion

Citation

Liu, Xiaosong; Liu, Taishun; Xu, Qinghua. ON SHARP INEQUALITIES OF HOMOGENEOUS EXPANSIONS FOR STARLIKE MAPPINGS OF ORDER $\alpha$ IN SEVERAL COMPLEX VARIABLES. Taiwanese J. Math. 17 (2013), no. 3, 801--813. doi:10.11650/tjm.17.2013.2415. https://projecteuclid.org/euclid.twjm/1499705984


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References

  • L. Bieberbach, Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, S.-B. Preuss, Akad. Wiss., 1916.
  • S. Gong, The Bieberbach Conjecture, Amer. Math. Soc., International Press, Providence, RI, 1999.
  • I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, New York, Marcel Dekker, 2003.
  • H. Hamada and T. Honda, Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables, Chin. Ann. Math., 29(B) (2008), 353-368.
  • H. Hamada, G. Kohr and P. Liczberski, Starlike mappings of order $\alpha$ on the unit ball in complex Banach spaces, Glas Mat Ser III, 36 (2001), 39-48.
  • J. A. Hummel, The coefficient regions of starlike functions, Pacif. J. Math., 7 (1957), 1381-1389.
  • G. Kohr, On some best bounds for coefficients of several subclasses of biholomorphic mappings in $\mathbb{C}^n$, Complex Var. Theory Appl., 36 (1998), 261-284.
  • X. S. Liu, On the quasi-convex mappings on the unit polydisc in $\Cn$, J. Math. Anal. Appl., 335 (2007), 43-55.
  • X. S. Liu and M. S. Liu, Quasi-convex mappings of order $\alpha$ on the unit polydisc in $\Cn$, Rocky Mountain J. of Math., 40 (2010), 1619-1644.
  • X. S. Liu and T. S. Liu, An inequality of homogeneous expansion for biholomorphic quasi-convex mappings on the unit polydisc and its application, Acta Math. Sci., 19B (2009), 201-209.
  • X. S. Liu and T. S. Liu, The sharp estimates of all homogeneous expansions for a class of quasi-convex mappings on the unit polydisc in $\mathbb{C}^{n}$, Chin. Ann. Math., 32B (2011), 241-252.
  • X. S. Liu, T. S. Liu, The sharp estimates for each item in the homogeneous polynomial expansions of a subclass of close-to-convex mappings, Sci. China Math., 40 (2010), 1079-1090 (in Chinese).
  • X. S. Liu and T. S. Liu, The sharp estimate of the third homogeneous expansion for a class of starlike mappings of order $\alpha$ on the unit polydisc in $\mathbb{C}^n$, Acta Math. Sci., 32B (2012), 752-764.
  • T. S. Liu and X. S. Liu, A refinement about estimation of expansion coefficients for normalized biholomorphic mappings, Sci. China Ser. A-Math., 48 (2005), 865-879.
  • Q. H. Xu and T. S. Liu, On coefficient estimates for a class of holomorphic mappings, Sci. China Ser. A-Math., 52 (2009), 677-686.
  • T. J. Suffridge, Starlike and convex maps in Banach spaces, Pacif. J. of Math., 46 (1973), 575-589.