On a result by Clunie and Sheil-Small

In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping F in the unit disk D, if F (D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2 ∈ D. Here H and G are holomorphic mappings in D determined by F = H + G, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in C and improve it provided F is additionally a quasiconformal mapping in Ω. Introduction. Let Ω be a nonempty domain in C. Throughout the paper we always assume that F : Ω → C is a sense-preserving injective harmonic mapping in Ω of the following form (0.1) F (z) = H(z) +G(z) , z ∈ Ω , where H and G are holomorphic mappings in Ω. Note that if Ω is a simply connected domain, then each harmonic mapping F in Ω has a decomposition (0.1) up to a constant function; cf. e.g. [4]. From the classical Lewy’s theorem it follows that the Jacobian J[F ] does not vanish on Ω; cf. [3]. Since 2000 Mathematics Subject Classification. Primary 30C55, 30C62.


Introduction.
Let Ω be a nonempty domain in C. Throughout the paper we always assume that F : Ω → C is a sense-preserving injective harmonic mapping in Ω of the following form (0.1) where H and G are holomorphic mappings in Ω.Note that if Ω is a simply connected domain, then each harmonic mapping F in Ω has a decomposition (0.1) up to a constant function; cf.e.g.[4].From the classical Lewy's theorem it follows that the Jacobian J[F ] does not vanish on Ω; cf.[3].Since where ∂ := 1 2 (∂ x − i∂ y ) and ∂ := 1 2 (∂ x + i∂ y ) are the so-called formal derivatives operators.Therefore the complex dilatation (0.3) µ F (z) := ∂F (z) is well defined and for every nonempty set E ⊂ Ω, put (0.4) Since G /H is a holomorphic mapping, we conclude from the maximum principle that for every nonempty compact set E ⊂ Ω, (0.5) Let D(a, r) stand for the Euclidean disk with the center at a ∈ C and the radius r > 0, i.e.D(a, r) := {z ∈ C : |z − a| < r}.In particular D := D(0, 1) is the unit disk.
The classical result by J. Clunie and T. Sheil-Small [2, Corollary 5.8] reads as follows.A little bit stronger version of Theorem A was proved in [1, Lemma 2.1].Under the additional assumption that F is a quasiconformal mapping, the conclusion (0.6) can be improved; cf.Theorem 2.1 and Corollary 2.3 in Section 2 which are main results of this paper.To this end we show in Section 1 several auxiliary properties involving the functions H and G with the function F .In Section 3 we present several applications of the results from the previous sections.They deal with the quasiconformality of the function F and Lipschitz type relationships between the functions F and H.
All results in this paper are strictly related to the ones presented by the second named author during the XVI-th Conference on Analytic Functions and Related Topics, June 26-29, 2011 Chełm (Poland).
A more general case where the convexity of F (D) is replaced by the socalled α-convexity of F (D) is studied in [5].

Auxiliary properties of harmonic mappings.
In this section we study the holomorphic mappings H and G associated with the mapping F by the equality (0.1).Lemma 1.1.Suppose that z 1 , z 2 ∈ Ω are points such that z 1 = z 2 and for a certain compact set E ⊂ Ω, i.e. the line segment with endpoints F (z 1 ) and where k := µ F E .Moreover, the following inequalities hold Proof.Take arbitrary distinct points z 1 , z 2 ∈ Ω satisfying (1.1).Then the function has the following properties: Then by (0.5) we see that Write w := F (z 2 ) − F (z 1 ).Using the following formulas Combining (1.11) with (1.9), we get which yields the second inequality in (1.2).On the other hand, we conclude from (1.11) and (1.9) that which yields the first inequality in (1.2). .
This together with (1.2) yields the inequalities (1.3) and the equality in (1.4).From (1.11) we conclude that which yields the inequality in (1.4).Applying (1.11) once again, we see that which leads to (1.5).Using the formulas (1.10), we have ds .
Hence and by (1.9) we see that which leads to (1.6), and the proof is complete.Proof.Let z 1 , z 2 ∈ Ω be arbitrarily chosen points such that z 1 = z 2 .Since the set F (Ω) is convex and the inverse mapping F −1 is continuous, we deduce that

Variants of J. Clunie and T. Sheil-Small inequality.
As an application of Lemma 1.1 we can derive the following improvement of Theorem A by J. Clunie and T. Sheil-Small.
where U := F −1 (V ) and S : [0; 1] → R is the function defined by the formula Proof.Fix z 1 , z 2 ∈ U .If z 1 = z 2 , then the inequality in (2.1) is obvious.Therefore we may assume that z 1 = z 2 .Let γ be the function defined by (1.7).By the convexity of V , Since the inverse mapping F −1 is continuous and γ([0; 1]) is a compact set, the set we deduce from (0.1) that a + b = 1, From Lemma 1.1 it follows that (2.5) Combining this with the last inequality in (2.5), we get Hence and by the first inequality in (2.5) we have From the first inequality in (2.5) it follows that Combining this with (2.6) and (2.2), we see that Applying now (2.4), we obtain By (0.4) and the second inclusion in (2.3), k = µ F E ≤ µ F U .Since S is an increasing function, we see that S(k) ≤ S( µ F U ).This together with (2.7) yields (2.1), which is the desired conclusion.
Remark 2.2.From the formula (2.2) it follows easily that S is a strictly increasing continuous function in [0; 1] and Proof.Take arbitrary points z 1 , z 2 ∈ Ω such that z 1 = z 2 .As in the proof of Theorem 1.1 we see that the function γ defined by (1.7) satisfies the equality in (2.3) with the compact set Hence H(z 2 ) = H(z 1 ), and therefore H is an injective mapping.Then the function as z = z 1 , is well defined.Since H is an injective mapping, we see that ω is a holomorphic function in Ω.By Theorem 2.1, Suppose that |ω(z 2 )| = S(k).Then by the maximum principle for holomorphic functions we deduce that ω is a constant function, and so ω(z) = λ for a certain λ ∈ C satisfying |λ| = S(k).Taking into account (2.10), we conclude that ) for z ∈ Ω, and consequently Combining this with (2.8) we see that k = 0 or k = 1.If k = 1, then for every z ∈ Ω, which is impossible.Therefore k = 0, and then (2.11) yields ω(z) = 0 as z ∈ Ω. Hence G(z) − G(z 1 ) = 0 as z ∈ Ω.This means that G is a constant function, which contradicts the assumption.Thus |ω(z 2 )| = S(k), which together with (2.11) leads to |ω(z 2 )| < S(k).Then (2.9) follows from (2.10), which completes the proof.
Corollary 2.5.If F (Ω) is a convex domain, then F is a quasiconformal mapping if and only if there exists a constant L such that 0 ≤ L < 1 and Moreover, if the condition (2.12) holds for some L ∈ [0; 1), then F is a quasiconformal mapping with µ F Ω ≤ L.
Proof.If F is a quasiconformal mapping, then k := µ F Ω < 1, and Theorem 2.1 implies the condition (2.12) with L := S(k) < 1. Conversely, fix z ∈ Ω.From the condition (2.12) it follows that A passage to the limit with ζ tending to z implies that Since z is an arbitrary point in Ω, we see that Hence µ F Ω ≤ L < 1, and consequently F is a quasiconformal mapping, which completes the proof.

Examples of applications.
In what follows we derive several applications of the results from the previous sections, dealing with the quasiconformality of the function F and Lipschitz type relationships between the functions F and H.
Proof.Suppose that F (Ω) is a convex domain and fix z 1 , z 2 ∈ Ω.If z 1 = z 2 , then the inequalities in (3.1) and (3.2) are obvious.Therefore we may assume that z 1 = z 2 .As in the proof of Theorem 2.1 we see that the function γ defined by (1.7) satisfies the equality in (2.3) with the compact set E := F −1 (γ([0; 1])).Moreover, µ F E ≤ µ F Ω = k.Then from the first inequality in (1.3) it follows that This yields the inequality in (3.1) for any k ≤ 1 and the first inequality in (3.2) as k < 1. Assume now that k < 1.Then the inequality (1.6 which yields the second inequality in (3.2).
Let us recall that for all L 1 , L 2 > 0, a mapping f : Ω → C is: (iii) L 2 , L 1 -biLipschitz if f is simultaneously a L 2 -Lipschitz and L 1 -co-Lipschitz mapping.A mapping f : Ω → C is said to be: Lipschitz, coLipschitz and biLipschitz provided f is respectively: L 2 -Lipschitz for a certain L 2 > 0, L 1 -coLipschitz for a certain L 1 > 0 and L 2 , L 1 -biLipschitz for some L 1 , L 2 > 0.
From Corollary 2.5 we can see that F is a biLipschitz mapping if and only if H is a biLipschitz mapping provided F is a quasiconformal mapping and F (Ω) is a convex domain.However, from Theorem 3.1 we can derive the following more precise result.Corollary 3.2.Suppose that k := µ F Ω < 1 and F (Ω) is a convex domain.Then for every L > 0: In particular, F is a biLipschitz mapping if and only if H is a biLipschitz mapping.
Proof.The implications (i)-(iv) follow directly from the conditions (3.3), (3.4) and (3.2).The last statement is a direct conclusion from these implications.
Theorem 3.3.Suppose that F (Ω) is a convex domain.Then the following four conditions are equivalent to each other: (i) F is a quasiconformal mapping; (ii) there exists a constant L 1 such that 1 ≤ L 1 < 2 and Proof.Suppose that the condition (i) holds, i.e. k := µ F Ω < 1.By Theorem 3.1 the first inequality in (3.2) holds, and consequently the condition (3.5) holds with L 1 := 1 + k < 2. Applying Theorem 3.1 once more, we see that the second inequality in (3.2) holds, and consequently the condition Conversely, the condition (v) clearly implies the one (iv).Next the condition (iv) yields the one (iii).It remains to prove the implications (ii) =⇒ (i) and (iii) =⇒ (i).
Fix z ∈ D, r > 0 and θ ∈ R and set w := z + re iθ .Assume first that the condition (3.6) holds for a certain L 2 ≥ 1.Then Remark 3.4.Note that the proofs of implications (ii) =⇒ (i) and (iii) =⇒ (i) have a local character and do not require the assumption that F (Ω) is a convex domain.Therefore each of conditions (ii) and (iii) implies that F is a quasiconformal mapping without the convexity of the image F (Ω).