Abstract
Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian
Highlights
For the properties of harmonic mappings we can refer to surveys [1] and [2]
The notion of an affine and linear invariant family of univalent harmonic functions was proposed by Sheil-Small [6], and extended to local univalent mappings and used efficiently by Schaubroeck in [5]
We introduce the definition of new order ord L of a linear invariant family (LIF ) of harmonic mappings L
Summary
For the properties of harmonic mappings we can refer to surveys [1] and [2]. The notion of an affine and linear invariant family of univalent harmonic functions was proposed by Sheil-Small [6], and extended to local univalent mappings and used efficiently by Schaubroeck in [5]. In what follows L denotes a family of locally univalent and sense-preserving harmonic functions f = h + g in D which have the expansion: (1.6) A family L is called an affine and linear invariant ALIF if for any f ∈ L the function Tφ(f ) and Aε(f ) belong to L for all φ ∈ Aut(D) and all |ε| < 1.
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