Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian

@article{SobczakKne2011OldAN,
  title={Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian},
  author={Magdalena Sobczak-Kne{\'c} and V. V. Starkov and Jan Szynal},
  journal={Annales Umcs, Mathematica},
  year={2011},
  volume={65},
  pages={191-202}
}
The relation between the Jacobian and the orders of a linear invariant family of locally univalent harmonic mapping in the plane is studied. The new order (called the strong order) of a linear invariant family is defined and the relations between order and strong order are established. 

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References

SHOWING 1-10 OF 13 REFERENCES

Order of linearly invariant family of mappings in

In this paper we suggest a new definition of the order of a linearly invariant family of locally biholomorphic mappings of the unit ball in . This definition is equivalent to the one given by

Subordination of planar harmonic functions

In this paper, we initiate a systematic study of subordination of harmonic functions. We establish results about coefficient relationships, integral means, and majorization of the Jacobian. In

Harmonic Mappings in the Plane

1. Preliminaries 2. Local properties of harmonic mappings 3. Harmonic mappings onto convex regions 4. Harmonic self-mappings of the disk 5. Harmonic univalent functions 6. Extremal problems 7.

Subordination of planar harmonic functions, Complex Variables

  • Theory Appl
  • 2000

Constants for Planar Harmonic Mappings

Univalent harmonic mappings in the plane, Ann

  • Univ. Mariae Curie-Skłodowska Sect. A
  • 1994

Order of linearly invariant mappings in C n

  • Complex Variables Theory Appl
  • 2000