RESEARCH ARTICLE

Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables

  • Qinghua XU , 1 ,
  • Ting YANG 1 ,
  • Taishun LIU 2 ,
  • Huiming XU 3
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  • 1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China
  • 2. Department of Mathematics, Huzhou University, Huzhou 313000, China
  • 3. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China

Received date: 12 Oct 2014

Accepted date: 10 Dec 2014

Published date: 12 Oct 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that maxfK|a3λa22|max{1/3,|λ1|},λ, and the estimate is sharp for each λ. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in n. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.

Cite this article

Qinghua XU , Ting YANG , Taishun LIU , Huiming XU . Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables[J]. Frontiers of Mathematics in China, 2015 , 10(6) : 1461 -1472 . DOI: 10.1007/s11464-015-0496-5

1
Bhowmik B, Ponnusamy S, Wirths K J. On the Fekete-Szegö problem for concave univalent functions. J Math Anal Appl, 2011, 373: 432−438

DOI

2
Bieberbach L. Über die Koeffizienten der einigen Potenzreihen welche eine schlichte Abbildung des Einheitskreises vermitten. S B Preuss: Akad Wiss, 1916

3
Cartan H. Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes. In: Montel P, ed. Lecons sur les Fonctions Univalentes ou Multivalentes. Paris: Gauthier-Villars, 1933

4
de Branges L. A proof of the Bieberbach conjecture. Acta Math, 1985, 154(1−2): 137−152

DOI

5
Duren P L. Univalent Functions, Berlin: Springer-Verlag, 1983

6
Fekete M, Szegö G. Eine Bemerkunguber ungerade schlichte Funktionen. J Lond Math Soc, 1933, 8: 85−89

DOI

7
Gong S. The Bieberbach Conjecture. Providence: Amer Math Soc/International Press, 1999

8
Graham I, Hamada H, Kohr G. Parametric representation of univalent mappings in several complex variables. Canad J Math, 2002, 54: 324−351

DOI

9
Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker, 2003

10
Graham I, Kohr G, Kohr M. Loewner chains and parametric representation in several complex variables. J Math Anal Appl, 2003, 281: 425−438

DOI

11
Hamada H, Honda T. Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin Ann Math Ser B, 2008, 29(4): 353−368

DOI

12
Hamada H, Honda T, Kohr G. Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation. J Math Anal Appl, 2006, 317: 302−319

DOI

13
Kanas S. An unified approach to the Fekete-Szegö problem. Appl Math Comput, 2012, 218: 8453−8461

DOI

14
Keogh F R, Merkes E P. A coefficient inequality for certain classes of analytic functions. Proc Amer Math Soc, 1969, 20: 8−12

DOI

15
Kohr G. On some best bounds for coefficients of several subclasses of biholomorphic mappings in ℂn. Complex Variables, 1998, 36: 261−284

16
Kowalczyk B, Lecko A. Fekete-Szegö problem for close-to-convex functions with respect to the Koebe function. Acta Math Sci Ser B Engl Ed, 2014, 34(5): 1571−1583

DOI

17
London R R. Fekete-Szegö inequalities for close-to-convex functions. Proc Amer Math Soc, 1993, 117(4): 947−950

DOI

18
Liu X S, Liu T S. The refining estimation of homogeneous expansions for quasi-convex mappings. Adv Math (China), 2007, 36: 679−685

19
Liu X S, Liu T S. On the quasi-convex mappings on the unit polydisk in ℂn. J Math Anal Appl, 2007, 335: 43−55

20
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 Ser B, 2011, 32: 241−252

21
Pfluger A. The Fekete-Szegö inequality for complex parameter. Complex Var Theory Appl, 1986, 7: 149−160

DOI

22
Pommerenke C. Univalent Functions. Göttingen: Vandenhoeck & Ruprecht, 1975

23
Roper K, Suffridge T J. Convexity properties of holomorphic mappings in ℂn. Trans Amer Math Soc, 1999, 351: 1803−1833

24
Sheil-Small T. On convex univalent functions. J Lond Math Soc, 1969, 1: 483−492

DOI

25
Suffridge T J. Some remarks on convex maps of the unit disc. Duke Math J, 1970, 37: 775−777

DOI

26
Xu Q H, Liu T S. On coefficient estimates for a class of holomorphic mappings. Sci China Ser A, 2009, 52: 677−686

DOI

27
Xu Q H, Liu T S, Liu X S. The sharp estimates of homogeneous expansions for the generalized class of close-to-quasi-convex mappings. J Math Anal Appl, 2012, 389: 781−791

DOI

28
Zhang W J, Liu T S. The growth and covering theorems for quasi-convex mappings on the unit ball in complex Banach spaces. Sci China Ser A, 2002, 45: 1538−1547

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