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

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. 1. A harmonic mapping f in the unit disk D = {z : |z| < 1} has a representation: (1.1) f(z) = h(z) + g(z) where h and g are holomorphic functions in D. We assume that f is locally univalent and sense-preserving in D, which is equivalent to Jf (z) > 0, z ∈ D, where Jf (z) denotes the Jacobian of f : (1.2) Jf (z) = ∣∣h′(z)∣∣2 − ∣∣g′(z)∣∣2 . 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 then used efficiently by Schaubroeck in [5]. 2000 Mathematics Subject Classification. Primary 30C55, Secondary 31A05.


1.
A harmonic mapping f in the unit disk D = {z : |z| < 1} has a representation: where h and g are holomorphic functions in D.
We assume that f is locally univalent and sense-preserving in D, which is equivalent to J f (z) > 0, z ∈ D, where J f (z) denotes the Jacobian of f : 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 then used efficiently by Schaubroeck in [5].
For any holomorphic automorphism ϕ of D (ϕ ∈ Aut(D)) we denote
The transformations (1.3) and (1.4) are called the Koebe transform and the affine transform of a locally univalent harmonic function f = h + g. Put In what follows L denotes a family of locally univalent and sense-preserving harmonic functions f = h + g in D which have the expansion: A family L is called an affine and linear invariant ALIF if for any f ∈ L the function T ϕ (f ) and A ε (f ) belong to L for all ϕ ∈ Aut(D) and all |ε| < 1.
A family L is called LIF (linear invariant family) if for any f ∈ L and all ϕ ∈ Aut(D) the function T ϕ (f ) ∈ L.
Example. The best known ALIF family is the class S H of univalent harmonic mappings in D preserving orientation, as well as the subclasses K H of convex and C H of close-to-convex mapping [2].
A simple example of a family which is LIF but not ALIF is the family of locally univalent holomorphic functions in D.
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.
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.
These relations depend on the upper bound for the Jacobian J f (z) in the terms of ord L and ord L.

Remark 1.1.
As we see from the proof, the operators T ϕ and A ε do not

2.
We start with a slight improvement of Theorem 3.3 from [5].
The bounds in (2.1) are sharp and the sign of equality holds for the function Observe that f (z) is univalent for α ∈ (0, 2], which follows from univalence of k α (z) for these α and the invariance of univalent harmonic functions w.r.t. operator A ε .
The next theorem is in some sense inverse to Theorem 2.1. We do not assume even the linear invariance of the family L. Theorem 2.2. Let f ∈ L and assume that the upper bound in (2.1) holds for some α > 0. Then there exists ε, |ε| < 1 such that The inequality (2.3) is sharp, which means that in the right side of the inequality |ε| < 1 we can not write any constant smaller than 1.
Proof. We will apply the same ideas from [3]. By the assumption, f ∈ L satisfies (2.1) which implies that for z = re iθ , For r = 0 the above inequality after differentiation gives Therefore, by (2.4) for every real θ we have due to the fact that h (0) = 1. The above inequality is equivalent to Let us put Because ω 0 (0) = 0 and |ω 0 (z)| < 1 we have On the other hand, (2.5) can be rewritten as In order to prove the sharpness consider the function We have a 2 (f 0 ) = α + 1 2 and Moreover, and the minimum is attained for θ = 0. Therefore, which implies that inequality (2. 3) for f 0 can be written in the form This makes the result of Theorem 2.2 sharp.

Remark 2.1. Theorem 2.2 is also valid for holomorphic functions f (z).
In this case we have to put in the proof ε = 0.

Remark 2.2.
From the above proof we see that in the statement of Theorem 2.2 it is sufficient to assume that f (z) satisfies only the right-(or left-) hand side of inequality (2.1).

Corollary 2.1.
If the family L is LIF and for any f ∈ L inequality (2.1) holds, then ord L ≤ α + 1 2 .
3. Now we introduce the definition of new order ord L (we will call the strong order) of a linear invariant family (LIF ) of harmonic mappings L.
In terms of this new order one can formulate iff version of Theorem 2.1 without assuming family L to be affine.

Definition 3.1.
Let L be LIF . The strong order ord L of a family L of harmonic mappings f is defined by the formula

Remark 3.1.
In the case when f is holomorphic in D, definition (3.1) coincides with that introduced by Pommerenke in [4].

Definition 3.2.
For any fixed f ∈ L we define Of course, if L is LIF , ord L = α, then ord f ≤ α for any f ∈ L.
Proof. Assume first that ord f ≤ α, ϕ(z) = z+a 1+az , z, a ∈ D and T ψ (f ) = F = H + G, ψ ∈ Aut(D). Consider F a (z) = T ϕ (F ) = H a + G a , where H, H a , G and G a are holomorphic functions in D. By direct calculations we find and by the definition of the order Therefore, from (3.4) we obtain Choosing first θ = a, a = 0, we obtain (a = e iθ ) Applying (3.5), we get Choosing now θ = π + a, we obtain Writing together the above inequalities, we get Integration over the interval [0, r] implies (3.3). Assume now that the right-hand side of (3.3) holds for some α > 0 and r ∈ (0, 1). We have to prove now that ord f ≤ α.
Because for the function F = T ϕ (f ) the inequality (3.3) holds, therefore we have This implies, as in the proof of Theorem 2.2, the validity of equality (2.5) for the function F = T ϕ (f ), and we have ord f ≤ α.
From the proof of the above theorem we obtain the following corollary. Corollary 3.1. If L is LIF , then ord L ≤ α if and only if the right-hand side of (3.3) holds for any f ∈ L and for every r ∈ (0, δ), 0 < δ < 1.
If L is LIF , then by Theorem 3.1 we can give an equivalent definition of the ord L. Namely, for any f ∈ L and any z ∈ D .
From Theorem 2.1 and Corollary 2.1 we derive the next corollaries.
If L is ALIF , then From (3.7) and the fact that L A(L) = L if L is an ALIF , we have (3.8).

Remark 3.2.
The equality ord L = ord L is possible. Take for example L = L A(k α ), where k α is given by (2.2). However, the inequality ord L = ord L can hold as well, as shows the following.
Example. Let k α (z) be the generalized Koebe function given by (2.2).
for every z ∈ D, then ord f 1 = ord f 2 .
Proof. We conclude the proof of Theorem 3.2 by Corollary 3. where L (f ) is defined as above in the Example.

Definition 3.3. Put
U H α = {L : L is LIF and ord L ≤ α} and call it the universal LIF of strong order α.

Remark 3.3. From the definition it is obvious that
We have to prove that for any f ∈ U H α and |ε| < 1, the function we can find that Indeed, we have for any harmonic function f = u + iv, where u and v are real functions: which implies (3.10). By Corollary 3.2 putting |z| = r, we have But from (3.10) Therefore, because f ∈ U H α . Corollary 3.5. If L is LIF , then ord L ≥ 1.
Proof. Assume on the contrary that ord L < 1. Then from the left-hand side of (3.3) it follows that This implies that the numerator of the above expression, which is |h (z)| 2 − |g (z)| 2 → +∞, and therefore, |h (z)| → +∞. This is in contradiction with the minimum principle because we would have min |z|=r<1 |h (z)| ≤ 1, due to the fact that h (0) = 1.
Acknowledgements. The authors would like to express their great thanks to the Referee, whose detailed and careful review helped to improve the presentation of the results. The second author has been partially supported by the RFBR Grant No. 11-01-00952a.