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

Abstract

Abstract. 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.

Coefficients of univalent harmonic mappings

“…It is well known that ord K H = 2 and ord C H = 3. In 2004, Starkov [23] (for details see [22]) introduced the order of a linear invariant family (which is not necessarily affine invariant family) F H which is defined as follows:…”

The Schwarzian derivatives of harmonic functions and univalence conditions

“…where Φ(z) = (z + z 0 )/(1 + z 0 z) and z 0 runs over the disk D. Properties of the linear and affine invariant families of harmonic functions can be found in [21,22,12]. Note that the order of an univalent analytic or univalent sense preserving harmonic function is always finite (cf., [6,8]).…”

Absolutely convex, uniformly starlike and uniformly convex harmonic mappings

“…(b) For the proof of Case (b), we consider g 2 (z) = (α/2)(F (a, b; c; z) − 1) so that f 2 (z) = z + g 2 (z). Using (14) it follows that…”