Subordination of planar harmonic functions

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

“…The proof is exactly the same as in [5], only the value J f (0) = 1 − |a −1 (f )| 2 has to be taken into account. Namely, using the inequality from [5]:…”

“…In this paper we give an improvement of one result from [5] (Theorem 2.1) and establish the relations between ord L and the new order called the strong order ord L defined below. Introduction of the new order ord L, allow us to prove Theorem 3.1 for arbitrary family L which is an extension of Theorem 2.1, while L is ALIF family.…”

“…The properties of the transformation (1.3)-(1.5) have been used in [5] to obtain some bounds for the Jacobian J f (z) in terms of the order of a linear invariant family.…”

“…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 then used efficiently by Schaubroeck in [5].…”

Weak subordination for convex univalent harmonic functions

“…A natural extension of subordination to complex-valued harmonic functions f and F in Δ with f (0) = F (0) = 0 is to say f is subordinate to F if f (z) = F (φ(z)) where φ is analytic in Δ, |φ(z)| < 1, z ∈ Δ, and φ(0) = 0. See [8] for results relating to this definition. There are a few limitations to this definition because φ must be analytic to preserve harmonicity and, even if f (Δ) ⊂ F (Δ) and F is one-to-one, such a φ may not exist as is the case for analytic functions.…”