Univalent harmonic mappings and Hardy spaces

Univalent harmonic mappings and Hardy spaces

The main purpose of this paper is to establish a relationship between univalent harmonic mappings andHardy spaces. The main result obtained in this paper improves previously published results. Moreover, we generalizesome nice results in the analytic case to the harmonic case.

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  • [1] Abu-Muhanna Y. Bloch, BMO and harmonic univalent functions. Complex Variables Theory Appl 1996; 31: 271- 279.
  • [2] Abu-Muhanna Y, Lyzzaik A. The boundary behavior of harmonic univalent maps. Pac J Math 1990; 141: 1-20.
  • [3] Bshouty D, Hengartner W. Univalent harmonic mappings in the plane. In: Kuhnau R, editor. Handbook of Complex Analysis: Geometric Function Theory. Vol. 2. Amsterdam, the Netherlands: Elsevier, 2005, pp. 479-506.
  • [4] Clunie J, Sheil-Small T. Harmonic univalent functions. Ann Acad Sci Fenn Ser AI 1984; 9: 3-25.
  • [5] Duren P. Harmonic Mappings in the Plane. Tracts in Mathematics, 156. Cambridge, UK: Cambridge University Press, 2004.
  • [6] Hernandez R, Martin MJ. Stable geometric properties of analytic and harmonic functions. Math Proc Camb Phil Soc 2013; 155: 343-359.
  • [7] Kim YC. Some inequalities for uniformly locally univalent functions on the unit disk. Math Inequal Appl 2007; 10: 805-809.
  • [8] Kim YC, Sugawa T. Growth and coefficient estimates for uniformly locally univalent functions on the unit disk. Rocky Mountain J Math 2002; 32: 179-200.
  • [9] Lewy H. On the non-vanishing of the Jacobian in certain one-to-one mappings. B Am Math Soc 1936; 42: 689-692.
  • [10] Nowak M. Integral means of univalent harmonic maps. Ann Univ Mariae Curie-Sklodwska 1996; 50: 155-162.
  • [11] Pommerenke C. Linear-invariante Familien analytischer Funktionen I. Math Ann 1964; 155: 108-154 (in German).
  • [12] Ponnusamy S, Qiao J, Wang X. Uniformly locally univalent harmonic mappings. arXiv:1601.01139v1 [math.CV], 2016.
  • [13] Sheil-Small T. Some conformal mapping inequalities for starlike and convex functions. J London Math Soc 1969; 2-1: 577-587.
  • [14] Sheil-Small T. Constants for planar harmonic mappings. J London Math Soc 1990; 42: 237-248.