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Extreme points of subordination and weak subordination families of harmonic mappings

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Abstract

The aim of the paper is to discuss the extreme points of subordination and weak subordination families of harmonic mappings. Several necessary conditions and sufficient conditions for harmonic mappings to be extreme points of the corresponding families are established.

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References

  1. Y. Abu-Muhanna: On extreme points of subordination families. Proc. Amer. Math. Soc. 87 (1983), 439–443.

    Article  MathSciNet  Google Scholar 

  2. Y. Abu-Muhanna and D. J. Hallenbeck: Subordination families and extreme points. Trans. Amer. Math. Soc. 308 (1988), 83–89.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. A. Dihan: Some subordination results and coefficient estimates for certain classes of analytic functions. Mathematica 49 (2007), 3–12.

    MathSciNet  Google Scholar 

  4. P. Duren: Theory of H p Spaces. Academic Press, New York, San Francisco, London, 1970.

    MATH  Google Scholar 

  5. P. Duren: Univalent Functions. Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.

    MATH  Google Scholar 

  6. P. Duren: Harmonic Mappings in the Plane. Cambridge university Press, New York, 2004.

    Book  MATH  Google Scholar 

  7. D. J. Hallenbeck and K.T. Hallenbeck: Classes of analytic functions subordinate to convex functions and extreme points. J. Math. Anal. Appl. 282 (2003), 792–800.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. J. Hallenbeck and T.H. Macgregor: Subordination and extreme-point theory. Pacific J. Math. 50 (1974), 455–468.

    MATH  MathSciNet  Google Scholar 

  9. T. H. MacGregor: Applications of extreme-point theory to univalent functions. Michigan Math. J. 19 (1972), 361–376.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Muir: Weak subordination for convex univalent harmonic functions. J. Math. Anal. Appl. 348 (2008), 862–871.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. V. Ryff: Subordinate H p functions. Duke Math. J. 33 (1966), 347–354.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. E. Schaubroeck: Subordination of planar harmonic functions. Complex Variables 41 (2000), 163–178.

    MATH  MathSciNet  Google Scholar 

  13. H. M. Srivastava and E. Sevtap Sümer: Some applications of a subordination theorem for a class of analytic functions. Appl. Math. Lett. 21 (2008), 394–399.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Tkaczyńska: On extreme points of subordination families with a convex majorant. J. Math. Anal. Appl. 145 (1990), 216–231.

    Article  MATH  MathSciNet  Google Scholar 

  15. Z. Lin: Subordination of analytic functions. Acta. Sci. Natur. Univ. Sunyatseni 29 (1990), 44–48. (In Chinese.)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Changsha, China

    Jinjing Qiao & Xiantao Wang

  2. Department of Mathematics, Hunan Normal University, Changsha, Hunan, 410081, P.R.China

    Jinjing Qiao & Xiantao Wang

Authors
  1. Jinjing Qiao
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  2. Xiantao Wang
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Corresponding author

Correspondence to Xiantao Wang.

Additional information

The research was partly supported by NSFs of China (No. 10771059 and No. 11071063) and the program for Science and Technology Innovative Research Team in Higher Educational Instituions of Hunan Province.

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Qiao, J., Wang, X. Extreme points of subordination and weak subordination families of harmonic mappings. Czech Math J 61, 145–155 (2011). https://doi.org/10.1007/s10587-011-0003-4

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  • DOI: https://doi.org/10.1007/s10587-011-0003-4

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