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On certain quasiconformal and elliptic mappings
Journal Pre-proof On certain quasiconformal and elliptic mappings
Shaolin Chen, Saminathan Ponnusamy
PII:
S0022-247X(20)30082-2
DOI:
https://doi.org/10.1016/j.jmaa.2020.123920
Reference:
YJMAA 123920
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
3 September 2019
Please cite this article as: S. Chen, S. Ponnusamy, On certain quasiconformal and elliptic mappings, J. Math. Anal. Appl. (2020), 123920, doi: https://doi.org/10.1016/j.jmaa.2020.123920.
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ON CERTAIN QUASICONFORMAL AND ELLIPTIC MAPPINGS SHAOLIN CHEN AND SAMINATHAN PONNUSAMY Abstract. Let D be the closure of the unit disk D in the complex plane C and g be a continuous function in D. In this paper, we discuss some characterizations of elliptic mappings f satisfying the Poisson’s equation Δf = g in D, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.
1. Preliminaries and main results Let D = {z : |z| < 1} denote the open unit disk in the complex plane C and let T = ∂D be the unit circle. Furthermore, we denote by C m (Ω) the set of all complex-valued m-times continuously differentiable functions from Ω into C, where Ω is a subset of C and m ∈ {0, 1, 2, . . .}. In particular, C(Ω) := C 0 (Ω) denotes the set of all continuous functions in Ω. Let G be a domain of C with G be its closure. We use dG (z) to denote the Euclidean distance from z to the boundary ∂G of G. Especially, we always use d(z) for dD (z). For a real 2 × 2 matrix A, the matrix norm and the matrix function are defined by A = sup{|Az| : |z| = 1}, and l(A) = inf{|Az| : |z| = 1}, respectively. For z = x + iy ∈ C, the formal derivative of a complex-valued function f = u + iv is given by u x uy Df := , vx vy so that Df = |fz | + |fz | and l(Df ) = |fz | − |fz |, where (1.1) fz = fx − ify /2, and fz = fx + ify /2. We use Jf := det Df = |fz |2 − |fz |2 to denote the Jacobian of f . A sense-preserving homeomorphism f from a domain Ω onto Ω , contained in the 1,2 Sobolev class Wloc (Ω), is said to be a K-quasiconformal mapping if, for z ∈ Ω, Df (z)2 ≤ K det Df (z), i.e., Df (z) ≤ Kl(Df (z)), where K ≥ 1. File: YJMAA123920_source.tex, printed:
30-1-2020, 10.44
2000 Mathematics Subject Classification. Primary: 31A05; Secondary: 30H30. Key words and phrases. Elliptic mapping, (K, K )-quasiconformal mapping, Poisson’s equation.
1
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Sh. Chen, S. Ponnusamy
A mapping f ∈ C 1 (Ω) is said to be an elliptic mapping (or (K, K )-elliptic mapping) if there are constants K ≥ 1 and K ≥ 0 such that f satisfies the following partial differential inequality Df (z)2 ≤ KJf (z) + K in the domain Ω ⊂ C (see [11, 27]). Let Ω1 and Ω2 be two subdomains of C. A sensepreserving homeomorphism f : Ω1 → Ω2 is said to be a (K, K )-quasiconformal mapping if f is absolutely continuous on lines in Ω1 , and there are constants K ≥ 1 and K ≥ 0 such that Df (z)2 ≤ KJf (z) + K , z ∈ Ω1 . Obviously, a (K, K )-quasiconformal mapping f ∈ C 1 (Ω) is an elliptic mapping in Ω. In particular, if K = 0, then the (K, K )-quasiconformal mappings are Kquasiconformal (cf. [2, 21]). Moreover, if a (K, K )-quasiconformal mapping is harmonic, then it is said to be harmonic (K, K )-quasiconformal. In 1968, Martio [25] has discussed the conditions for the K-quasiconformality of harmonic mappings from the closed unit disk onto itself. The harmonic Kquasiconformal mappings of Riemannian manifolds have been considered by Goldberg and Ishihara (see [13, 12]). Tam and Wan [32] have investigated some properties of harmonic quasiconformal diffeomorphism and the universal Teichmuller space. In 2002, Pavlovi´c [31] has generalized the corresponding results of Martio [25]. Kalaj [16], Partyka and Sakan [28] have investigated the K-quasiconformality of harmonic mappings from the unit disk onto bounded convex domains. See [22, 18, 24, 29, 33] and the references therein for detailed discussions on this topic. Recent papers [2], [21] and [20] bring much attention on the topic of (K, K )-quasiconformal mappings in the plane. This paper continues the study of previous work of [4, 5] and is mainly motivated by the articles of Finn and Serrin [11], and Kalaj and Mateljevi´c [21]. In order to state our main results, we need to recall some basic definitions and some results which motivate the present work. For θ ∈ [0, 2π] and z, w ∈ D with z = w, let 1 − zw 1 − |z|2 iθ and P (z, e ) = G(z, w) = log z−w |1 − ze−iθ |2 denote the Green function and the (harmonic) Poisson kernel, respectively. Let ψ : T → C be a bounded integrable function, and g ∈ C(D). The solution to the Poisson equation Δf = g f = ψ ∈ L1 (T)
in D, in T,
is given by f (z) = P [ψ](z) − G[g](z), where (1.2)
1 G[g](z) = 2π
1 G(z, w)g(w)dA(w), P [ψ](z) = 2π D
2π
P (z, eit )ψ(eit )dt, 0
On certain quasiconformal and elliptic mappings
3
and dA(w) denotes the Lebesgue measure on D. It is well known that if ψ and g are continuous in T and in D, respectively, then f = P [ψ] − G[g] has a continuous extension f˜ to the boundary, and f˜ = ψ in T (see [15, pp. 118-120] and [22, 17]). A continuous increasing function ω : [0, ∞) → [0, ∞) with ω(0) = 0 is called a majorant if ω(t)/t is non-increasing for t > 0 (see [9, 10, 30]). For α > 0 and a majorant ω, we use Bωα (D) to denote the generalized Bloch-type space of all functions f ∈ C 1 (D) with f Bωα (D) < ∞, where f Bωα (D) = |f (0)| + sup {Df (z)ω(dα (z))} . z∈D
For a given bounded integrable function ψ ∈ L1 (T) and a given g ∈ C(D), let Fg,ψ (D) = {f ∈ C 2 (D) : Δf = g in D and f = ψ in T}. Clearly, functions in F0,ψ (D) are harmonic in D. Furthermore, if f ∈ Fg,ψ (D), then f + G[g] is harmonic in D, and thus, has the representation (1.3)
f + G[g] = h1 + h2 ,
where h1 and h2 are analytic in D (cf. [8]), and G[g] is defined in (1.2). In [21], Kalaj and Mateljevi´c proved that a harmonic diffeomorphism between two bounded Jordan domains with C 2 boundaries is a harmonic (K, K )-quasiconformal mapping for some constants K ≥ 1 and K ≥ 0 if and only if it is Lipschitz continuous (see [21, Theorem 1.1 and Corollary 1.3]). This result can be considered as an extension of the corresponding results of Martio [25], Pavlovi´c [31], and Partyka and Sakan [29]. For related investigations on this topic, we refer to [28, 33]. In the following, we will give some characterizations of elliptic mappings without the C 2 boundary hypothesis of the image domains. Theorem 1.1. Suppose that ω is a given majorant. For a given bounded integrable function ψ ∈ L1 (T) and a given g ∈ C(D), let f ∈ Fg,ψ (D) be a univalent and sense-preserving mapping, and f (D) be a convex domain. If there are two positive constants C1 , C2 and α ∈ [0, 1] such that for any z1 , z2 ∈ D with z1 = z2 ,
1−α 2 ω (1 + |z1 |)(1 + |z2 |) C1 |f (z1 ) − f (z2 )| ≤ (1.4) ≤ 1−α
C1 |z1 − z2 | ω d(z )d(z ) 2 1
and
1 0
2
dt ≤ C2 , ω ((d(Φ(t)))1−α )
then f is an elliptic mapping, where Φ(t) := f −1 (f (z1 ) + t(f (z2 ) − f (z1 ))). Remark 1.2. In particular, if α = 1 in (1.4), then f is bi-Lipschitz. In this situation, Theorem 1.1 is trivial because all univalent and sense-preserving bi-Lipschitz mappings are quasiconformal mappings (see Chapter 14.78 in [14]). However, quasiconformal mappings are not necessarily bi-Lipschitz, not even Lipschitz (see Example 1.3).
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Example 1.3. Let
⎧ e ⎨z logα |z|2 f (z) = ⎩ 0
for z ∈ D \ {0}, for z = 0,
where α ∈ (0, 1/2) is a constant. Then f is a quasiconformal self-homeomorphism of D. However, f is not Lipschitz at the origin (cf. [23]). Proposition 1.4. For a given bounded integrable function ψ ∈ L1 (T) and a given g ∈ C(D), let f = h1 + h2 − G[g] ∈ Fg,ψ (D), where h1 and h2 are analytic in D. If there is a constant C3 ∈ [1, 2) such that for any z1 , z2 ∈ D, |f (z1 ) − f (z2 )| ≤ C3 |h1 (z1 ) − h1 (z2 )|,
(1.5)
then f is an elliptic mapping. We remark that the inverse of Proposition 1.4 does not necessarily hold (see Example 1.5). Example 1.5. For z ∈ D, let f (z) = 3z|z|2 − z|z|8 . Then (1) f is a (1, 729/(216/3 ))-quasiconformal mapping in D; (2) f is not a K-quasiconformal mapping for any K ≥ 1; (3) f does not satisfy the inequality (1.5). Proof. We first prove the univalence of f . Suppose on the contrary that f is not univalent. Then there are two distinct points z1 , z2 ∈ D such that f (z1 ) = f (z2 ), which implies that z1 |z1 |2 (3 − |z1 |6 ) = z2 |z2 |2 (3 − |z2 |6 ).
(1.6)
Case 1. If |z1 | = |z2 |, then, by (1.6), we have z1 = z2 . This is a contradiction with the assumption. Case 2. If |z1 | = |z2 |, then (1.6) reduces to |z1 |3 (3 − |z1 |6 ) = |z2 |3 (3 − |z2 |6 ), and consequently (|z1 |3 − |z2 |3 )(3 − |z1 |6 − |z2 |6 − |z1 |3 |z2 |3 ) = 0. This implies |z1 | = |z2 | = 1, which violates the hypothesis. Hence f is univalent. Also, for z ∈ D, elementary calculations lead to fz (z) = |z|2 (6 − 5|z|6 ) and fz (z) = z 2 (3 − 4|z|6 ), which give that (1.7)
Df (z)2 − Jf (z) ≤ Df (z)2 = max
|z|∈[0,1)
81|z|4 (1 − |z|6 )2 = 729/(216/3 )
and (1.8)
lim−
|z|→1
|fz (z)| = 1. |fz (z)|
The first assertion and the second assertion easily follows from (1.7) and (1.8), respectively. The last assertion is obvious.
On certain quasiconformal and elliptic mappings
5
For r ∈(0, 1) and f ∈ C 1 (D), the perimeter of the curve C(r) = w = f (reiθ ) : θ ∈ [0, 2π] , with counting multiplicity, is defined by (cf. [3, 5, 7]) 2π 2π iθ fz (reiθ ) − e−2iθ fz (reiθ ) dθ. |df (re )| = r f (r) = 0
0
In particular, let f (1) = sup0
(1.10)
Kf (1) , 2nπ √ f (1) K Df (z) ≤ 2π(1 − |z|) |an | + |bn | ≤
and f ∈ Bω1 (D), where ω(t) = t. In particular, if K = 1, then the above two estimates are sharp, and the extreme function is f (z) = z. Concerning the generalizedform of inequality Mateljevi´c [26] proved the ∞ (1.9), ∞ n n following result: Let f (z) = n=0 an z + n=1 bn z be a harmonic mapping with f (1) < ∞. Then, for n ≥ 1, the inequality f (1) nπ holds (see [26, Theorem 10]). Moreover, Kalaj [19] improved the inequality (1.10) and obtained a sharp inequality for harmonic diffeomorphisms of D. It reads as follows. (1.11)
|an | + |bn | ≤
Theorem B. If f is a harmonic sense-preserving diffeomorphism of D onto a Jordan domain Ω with rectifiable boundary of length 2πR, then the sharp inequality (1.12)
|fz (z)| ≤
R , z∈D 1 − |z|2
holds, where R is a positive constant. If the equality in (1.12) is attained for some a, then Ω is convex, and there is a holomorphic function μ : D → D and a constant θ ∈ [0, 2π], such that z z z+a dt μ(t)dt −iθ =R + . (1.13) F (z) := e f 2 2 1 + za 0 1 + t μ(t) 0 1 + t μ(t) Moreover, every function f defined by (1.13) is a harmonic diffeomorphism and maps D to a Jordan domain bounded by a convex curve of length 2πR and the inequality (1.12) is attained for z = a. The following result is also a generalization of Theorem A.
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n Theorem 1.6. Let K ≥ 1, K ≥ 0 and R > 0 be constants. If f (z) = ∞ n=0 an z + ∞ n n=1 bn z is a harmonic (K, K )-quasiconformal mapping of D onto a Jordan domain Ω with rectifiable boundary of length 2πR, then for n ≥ 1, √ K + KR , (1.14) |an | + |bn | ≤ n √ 1 −R + K + KK + K 2 R2 (1.15) Df (z) ≤ R + , for z ∈ D, 1+K 1 − |z|2 and f ∈ Bω1 (D). In particular, if K = K − 1 = 0 and Ω = D, then the estimates of (1.14) is sharp, and the extreme function is f (z) = z for z ∈ D. Moreover, if K = K − 1 = 0 and Ω = D, then the equal sign occurs in (1.15) for some fixed z = a if and only if z−a f (z) = eit 1−za , where t ∈ [0, 2π]. We remark that if K = 0 and K ∈ [1, 2], then the inequality (1.14) is better than (1.11). Let f ∈ C 1 (D). Then, for θ ∈ [0, 2π], the radial length of the curve Cθ (r) = w = f (ρeiθ ) : 0 ≤ ρ ≤ r < 1 , with counting multiplicity, is defined by ∗f (r, θ)
r
r
|df (ρe )| = iθ
= 0
fz (ρeiθ ) + e−2iθ fz (ρeiθ ) dρ.
0
In particular, let ∗f (1, θ) = sup ∗f (r, θ). 0≤r<1
We refer the reader to [5, 6] for some discussion of the radial length. In particular, the following result establishes the Fourier coefficient estimates of K-quasiconformal harmonic mappings with the finite radial length. ∞ n n Theorem C. ([3, Theorem 4]) Let f (z) = ∞ n=0 an z + n=1 bn z be a harmonic ∗ ∗ K-quasiconformal mapping in D. If f (1) = supθ∈[0,2π] f (1, θ) < ∞, then (1.16)
|an | + |bn | ≤ K∗f (1) for n ≥ 1.
Moreover, if K = 1, then the estimate (1.16) is sharp and the extreme function is f (z) = ∗f (1)z for z ∈ D. We improve Theorem C into the following form.
∞ n n Theorem 1.7. For K ≥ 1 and K ≥ 0, let f (z) = ∞ n=0 an z + n=1 bn z be a ∗ ∗ harmonic (K, K )-quasiconformal mapping in D. If f (1) = supθ∈[0,2π] f (1, θ) < ∞, then for n ≥ 1, (1.17)
|an | + |bn | ≤
√
K + K∗f (1).
In particular, if K = K − 1 = 0, then the estimate (1.17) is sharp, and the extreme function is f (z) = ∗f (1)z for z ∈ D.
On certain quasiconformal and elliptic mappings
7
The proof of Theorems 1.1, 1.6 and 1.7, and Proposition 1.4 will be presented in Section 2.
2. The proofs of the main results In this section, we shall prove Theorems 1.1 and 1.7, and Propositions 1.4 and 1.6. We start with some useful Lemmas.
Lemma D. ([4, Lemma 6]) Let ω be a majorant and ν ∈ [0, 1]. Then for t ∈ [0, ∞], ω(νt) ≥ νω(t). Lemma 2.1. Let ω be a majorant and α ∈ [0, 1] be a constant. Suppose that f ∈ C 1 (D) is univalent, and f (D) is a convex domain. Then the following two statements are equivalent: (a) For any z1 , z2 with z1 = z2 , there exists a constant C4 such that (1−α)/2 |f (z1 ) − f (z2 )| C4 1 ≤ ω (1 + |z1 |)(1 + |z2 |) ≤ (1−α)/2 . C4 |z1 − z2 | ω d(z1 )d(z2 ) (b) For any z ∈ D, there is a constant C5 > 0 such that 1 C5 . ω (1 + |z|)1−α ≤ l(Df (z)) ≤ Df (z) ≤ C5 ω ((d(z))1−α ) Proof. We first prove (a) ⇒ (b). For θ ∈ [0, 2π] and z = x + iy ∈ D, elementary computations lead to (see (1.1)) fz (z) + fz (z) = fx (z) and i(fz (z) − fz (z)) = fy (z) and therefore, fx (z) cos θ + fy (z) sin θ = fz (z)eiθ + fz (z)e−iθ .
(2.1)
For r ∈ [0, 1 − |z|), let ξ = z + reiθ . Then, by (a) and (2.1), we obtain
|f (z) − f (ξ)| lim+ r→0 |z − ξ|
max |fx (z) cos θ + fy (z) sin θ| = max θ∈[0,2π] ⎧ ⎫ ⎨ ⎬ C4 C4
≤ max lim+ = 1−α 1−α θ∈[0,2π] ⎩r→0 ω d(z)d(z + reiθ ) 2 ⎭ ω ((d(z)) )
Df (z) =
θ∈[0,2π]
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Sh. Chen, S. Ponnusamy
and Df (z) ≥ l(Df (z)) = min |fx (z) cos θ + fy (z) sin θ| θ∈[0,2π] |f (z) − f (ξ)| = min lim θ∈[0,2π] r→0+ |z − ξ|
⎫ ⎧ 1−α iθ 2 ⎨ ⎬ ω (1 + |z|)(1 + |z + re |) ≥ min lim ⎭ θ∈[0,2π] ⎩r→0+ C4 ω ((1 + |z|)1−α ) . C4
=
Now we prove (b) ⇒ (a). For t ∈ [0, 1] and z1 , z2 ∈ D with z1 = z2 , let χ(t) = z1 t + (1 − t)z2 . Since d(χ(t)) ≥ 1 − t|z1 | − (1 − t)|z2 | ≥ 1 − t − (1 − t)|z2 | = (1 − t)d(z2 ) and d(χ(t)) ≥ 1 − t|z1 | − (1 − t)|z2 | ≥ 1 − t|z1 | − (1 − t) = td(z1 ), we see that (2.2)
d(χ(t))
1−α
1−α ≥ t(1 − t)d(z1 )d(z2 ) 2 .
By calculations, we have (2.3)
|f (z1 ) − f (z2 )| =
1
(χ(t))χ (t)
fw (χ(t))χ (t) + fw 1 Df (χ(t))dt, ≤ |z1 − z2 | 0
dt
0
where w = χ(t). Now, we estimate the integral on the right. By (b) and (2.2), we have 1 1 1 C5 dt C5 dt Df (χ(t))dt ≤ 1−α ≤ 1−α , 0 0 ω 0 ω d(χ(t)) t(1 − t)d(z1 )d(z2 ) 2 which, together with Lemma D, implies that 1 1 dt C5
(2.4) Df (χ(t))dt ≤ 1−α 1−α 1−α 0 0 t 2 (1 − t) 2 ω d(z1 )d(z2 ) 2 Γ2 1+α C5 2
, = Γ(1 + α) ω d(z )d(z ) 1−α 2 1
where Γ denotes the usual Gamma function.
2
On certain quasiconformal and elliptic mappings
9
It follows from (2.3) and (2.4) that
Γ2 1+α C5 |f (z1 ) − f (z2 )| 2
. ≤ |z1 − z2 | Γ(1 + α) ω d(z )d(z ) 1−α 2 1 2
On the other hand, for any z1 , z2 ∈ D with z1 = z2 , let w1 = f (z1 ) and w2 = f (z2 ). For t ∈ [0, 1], let (2.5)
γ(t) = tw1 + (1 − t)w2
be the straight line segment connecting w1 and w2 . Since f (D) is a convex domain, we see that γ(t) ⊂ f (D) and η(t) = f −1 (γ(t)) ⊂ D for t ∈ [0, 1]. It is not difficult to know that 1 + |z1 | 1 + |z2 | 1 + |η(t)| ≥ and 1 + |η(t)| ≥ , 2 2 which, together with Lemma D, implies that ⎞ ⎛ 1−α 2 (1 + |z1 |)(1 + |z2 |) ⎠ ω (1 + |η(t)|)1−α ≥ ω ⎝ (2.6) 21−α 1−α
ω (1 + |z1 |)(1 + |z2 |) 2 ≥ . 21−α By (b) and (2.6), we obtain 1 1 1 (2.7) l(Df (η(t)))|η (t)|dt ≥ ω (1 + |η(t)|)1−α |η (t)|dt C5 0 0
1−α 2 1 ω (1 + |z1 |)(1 + |z2 |) |η (t)|dt ≥ 21−α C5
0 1−α 2 ω (1 + |z1 |)(1 + |z2 |) |z1 − z2 |, ≥ 21−α C5 It follows from (2.5) that 1 |γ (t)|dt |w1 − w2 | = |γ (t)| = 0 1 = fζ (η(t))η (t) + fζ (η(t))η (t) dt 0 1 l(Df (η(t)))|η (t)|dt, ≥ 0
which, together with (2.7), yields that
1−α 2 |)(1 + |z |) ω (1 + |z 1 2 |w1 − w2 | . ≥ |z1 − z2 | 21−α C5 The proof of this lemma is finished.
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Sh. Chen, S. Ponnusamy
Lemma E. ([17, Lemma 2.7]) If g ∈ C(D), then ∂ ∂ 1 max G[g](z) , G[g](z) ≤ g∞ for z ∈ D. ∂z ∂z 3 Lemma 2.2. Let f ∈ C 1 (D) be a sense-preserving mapping. Then f is an elliptic mapping if and only if there exist constants k1 ∈ [0, 1) and k2 ∈ [0, ∞) such that |fz (z)| ≤ k1 |fz (z)| + k2 for z ∈ D.
(2.8)
Proof. We first prove the sufficiency. By (2.8), for z ∈ D, we have 1 + k1 2k2 (2.9) Df (z) ≤ l(Df (z)) + 1 − k1 1 − k1 2 1 + k1 1 + k1 4k22 ≤ l(Df (z)) + l2 (Df (z)) + . 1 − k1 1 − k1 (1 − k1 )2 1 + k1 Case 1. For all z ∈ D, Df (z) ≤ l(Df (z)). 1 − k1 In this case, it is easy to know that 1 + k1 2 Jf (z), Df (z) ≤ 1 − k1 which implies that f is an elliptic mapping. Case 2. There is a subset E of D such that 1 + k1 Df (z) > l(Df (z)) for z ∈ E. 1 − k1 In this case, it follows from (2.9) that
Df (z) −
1 + k1 1 − k1
2 l(Df (z))
which implies that (2.10)
≤
Df (z) ≤ 2 2
1 + k1 1 − k1
1 + k1 1 − k1
2 l2 (Df (z)) +
Jf (z) +
4k22 for z ∈ E, (1 − k1 )2
4k22 . (1 − k1 )2
On the other hand, for z ∈ D\E, we have 1 + k1 2 Df (z) ≤ Jf (z), 1 − k1 which, together with (2.10), implies that f is also an elliptic mapping in D. Next, we show the necessity. If f is an elliptic mapping, then there exist constants K ≥ 1 and K ≥ 0 such that Df (z)2 ≤ KJf (z) + K for z ∈ D.
On certain quasiconformal and elliptic mappings
This gives that Df (z) ≤
Kl(Df (z)) +
Kl(Df (z))
2
2
+ 4K
≤ Kl(Df (z)) +
11
√
K ,
and consequently
√ K −1 K |fz (z)| + . |fz (z)| ≤ K +1 1+K The proof of this lemma is complete.
The proof of Theorem 1.1. Differentiating both sides of the equation f −1 (f (z)) = z and then simplifying the resulting relations lead to the formulae (2.11)
(f −1 )w =
fz fz and (f −1 )w = − , Jf Jf
where w = f (z). Next, for z1 , z2 ∈ D with z1 = z2 , we let ϕ(t) = t(f (z1 ) − f (z2 )) + f (z2 ), where t ∈ [0, 1]. Since f (D) is a convex domain, we see that ϕ(t) ⊂ f (D) and Φ(t) := f −1 (ϕ(t)) ⊆ D for t ∈ [0, 1]. For z ∈ D, let F (z) = f (z) + G[g](z). Then F is harmonic in D, and F can be represented by F = h1 + h2 in D, where hj (j = 1, 2) are analytic in D. With z = Φ(t), we have Φ(0) = z2 , Φ(1) = z1 and thus, by (2.11), we obtain z1 1 h1 (z1 ) − h1 (z2 ) = h1 (z)dz = h1 (Φ(t))Φ (t)dt z2 1
0
h1 (Φ(t)) fw−1 (ϕ(t))ϕ (t) + fw−1 (ϕ(t))ϕ (t) dt 0 1 f (Φ(t)) (Φ(t)) f z z ϕ (t) − h1 (Φ(t)) ϕ (t) dt, = J (Φ(t)) J (Φ(t)) f f 0 =
which implies that
1 h (Φ(t)) f (Φ(t)) − f (Φ(t)) ϕ (t) z z h1 (z1 ) − h1 (z2 ) 1 ϕ (t) (2.12) = dt f (z1 ) − f (z2 ) J (Φ(t)) f 0 1 |h1 (Φ(t))|Df (Φ(t)) dt ≤ Jf (Φ(t)) 0 1 l(Df (Φ(t))) + |G[g]z (Φ(t))| + |fz (Φ(t))| ≤ dt. l(Df (Φ(t))) 0
It follows from Lemma 2.1 that there exists a positive constant C6 such that
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Sh. Chen, S. Ponnusamy
|G[g]z (Φ(t))| + Df (Φ(t)) |G[g]z (Φ(t))| + |fz (Φ(t))| ≤ l(Df (Φ(t))) l(Df (Φ(t))) ≤
C6 |G[g]z (Φ(t))| +
C6 ω((d(Φ(t)))1−α ) |Φ(t)|)1−α )
C6 ≤ ω(1)
ω ((1 + |G[g]z (Φ(t))| +
1 ω ((d(Φ(t)))1−α )
,
which, together with (2.12) and Lemma E, gives that 1 h1 (z1 ) − h1 (z2 ) C6 (2.13) |G[g]z (Φ(t))|dt f (z1 ) − f (z2 ) ≤ 1 + ω(1) 0 1 dt ≤ μ, + 1−α 0 ω (d(Φ(t))) where μ := 1 +
C6 g∞ C6 C2 + ≥ 1. 3ω(1) ω(1)
For θ ∈ [0, 2π] and any fixed point z ∈ D, let ς = z + reiθ , where r ∈ [0, 1 − |z|). By (2.13), we have h1 (ς) − h1 (z) 1 ≤ lim μ r→0+ ς −z
h1 (ς) − h1 (z) h2 (ς) − h2 (z) G[g](z) − G[g](ς) + + lim r→0+ ς −z ς −z ς −z = h1 (z)eiθ + h2 (z)e−iθ − (G[g]z (z)eiθ + G[g]z (z)e−iθ )
which yields that (2.14)
1 |h (z)| ≤ μ 1 =
min h1 (z) − G[g]z (z) + (h2 (z) − G[g]z (z))e−2iθ θ∈[0,2π] min fz (z) + fz (z)e−2iθ = l(Df (z)). θ∈[0,2π]
Since |h1 | ≥ |h1 − G[g]z | − |G[g]z | = |fz | − |G[g]z |, by (2.14) and Lemma E, we see that (2.15)
|fz | ≤
(μ − 1) 1 |fz | + g∞ . μ 3μ
It follows from (2.15) and Lemma 2.2 that f is an elliptic mapping. The proof of this theorem is complete.
On certain quasiconformal and elliptic mappings
13
The proof of Proposition 1.4. For θ ∈ [0, 2π] and any fixed point z ∈ D, let ς = z + reiθ , where r ∈ [0, 1 − |z|). Then by (2.1) and (1.5), we have f (ς) − f (z) lim Df (z) = max |fx (z) cos θ + fy (z) sin θ| = max θ∈[0,2π] θ∈[0,2π] r→0+ ς −z h1 (ς) − h1 (z) = C3 |h1 (z)| ≤ C3 lim+ r→0 ς −z ≤ C3 |fz (z)| + C3 |G[g]z (z)|, which, together with Lemma E, yields that C3 g∞ . 3 It follows from (2.16) and Lemma 2.2 that f is an elliptic mapping. The proof of this proposition is complete. |fz (z)| ≤ (C3 − 1)|fz (z)| +
(2.16)
The proof of Theorem 1.6. We first prove (1.14). Since Df (z)2 ≤ KJf (z) + K , we see that (2.17)
Df (z) ≤
Kl(Df (z)) +
Kl(Df (z))
2
+ 4K
2 where z ∈ D. It follows from (2.17) that
fz (reiθ ) − e−2iθ fz (reiθ ) dθ ≥ r f (r) = r 0 √ 2π Df (reiθ ) − K dθ, ≥ r K 0 which gives that (2.18)
2π
2π
r
≤ Kl(Df (z)) +
√
2π
l(Df (reiθ ))dθ 0
√ Df (reiθ )dθ ≤ 2πr K + Kf (r).
0
For n ≥ 1 and r ∈ (0, 1), it follows from Cauchy’s integral formula that, ∂f (z) dz ∂f (z) dz 1 1 nan = and nbn = , 2πi |z|=r ∂z z n 2πi |z|=r ∂z zn which, together with the inequality (2.18), implies that 2π 1 n(|an | + |bn |) ≤ rDf (reiθ )dθ 2πrn 0
√ 1 + K (r) K 2πr ≤ f 2πrn
1 √ ≤ K + K (1) . 2π f 2πrn
K ,
14
Sh. Chen, S. Ponnusamy
Consequently,
√ |an | + |bn | ≤
(2.19)
K Kf (1) + = n 2nπ
√
K KR + . n n
This proves (1.14) Next, we prove (1.15). For z ∈ D, it follows from the inequality Df (z)2 ≤ KJf (z) + K that |fz (z)| ≤
K + KK + K 2 |fz (z)|2 , 1+K
−|fz (z)| +
which, together with (1.12), yields that √ 1 −R + K + KK + K 2 R2 Df (z) ≤ R + . 1+K 1 − |z|2
The proof of the theorem is complete. Theorem F. ([1, Theorem 2]) Let φ be subharmonic in D. If for all r ∈ [0, 1), r φ(ρeiθ ) dρ ≤ 1, A(r) = sup θ∈[0,2π]
0
then A(r) ≤ r. The proof of Theorem 1.7. Since f is a univalent (K, K )-elliptic mapping, we see that Df (z)2 ≤ KDf (z)l(Df (z)) + K for z ∈ D. This gives that (2.20)
Df (z) ≤
Kl(Df (z)) +
Kl(Df (z))
2
+ 4K
≤ Kl(Df (z)) +
2 It follows from (2.20) that, for θ ∈ [0, 2π] and r ∈ (0, 1), r r ∗ iθ −2iθ iθ |fz (ρe ) + e fz (ρe )| dρ ≥ l(Df (ρeiθ )) dρ f (r, θ) = 0 0
√ 1 r ≥ Df (ρeiθ ) − K dρ K 0 and consequently r √ √ (2.21) Df (ρeiθ ) dρ ≤ K r + K∗f (r, θ) ≤ K + K∗f (1). 0
Inequality (2.21) and Theorem F lead to r √
iθ ∗ (2.22) Df (ρe ) dρ ≤ K + Kf (1) r. 0
√
K .
On certain quasiconformal and elliptic mappings
15
The Cauchy integral formula shows that, for ρ ∈ (0, 1), ∂f (z) dz 1 1 ∂f (z) dz and nbn = , nan = 2πi |z|=ρ ∂z z n 2πi |z|=ρ ∂z zn which yields that 1 n(|an | + |bn |) ≤ 2πρn
(2.23)
2π
ρDf (ρeiθ )dθ. 0
Then combining (2.22) and (2.23) gives the final estimate for |an | + |bn |, namely, r 2π r n−1 iθ ρ dρ ≤ Df (ρe ) dρ dθ 2πn(|an | + |bn |) 0 0 0 2π √
∗ K + Kf (1) r dθ ≤ 0
and consequently
√
|an | + |bn | ≤ inf
r∈(0,1)
K + K∗f (1) rn−1
The proof of this theorem is complete.
=
√ K + K∗f (1).
Acknowledgements: We thank anonymous referees for drawing our attention to references [19, 26] and providing us with a lot of valuable comments. This research was partly supported by the Hunan Provincial Education Department Outstanding Youth Project (No. 18B365), the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the exchange project for the third regular session of the China-Montenegro Committee for Cooperation in Science and Technology (No. 3-13), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469), and the Open Fund Project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang Normal University (No. IIPA18K06). References 1. E. F. Beckenbach, A relative of the lemma of Schwarz, Bull. Amer. Math. Soc., 44 (1938), 698–707. 2. J. Chen, P. Li, S. K. Sahoo and X. Wang, On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domain, Israel J. Math., 220 (2017), 453–478. 3. Sh. Chen, G. Liu and S. Ponnusamy, Linear measure and K-quasiconformal harmonic mappings (in Chinese), Sci. Sin. Math., 47 (2017), 565–574. 4. Sh. Chen, S. Ponnusamy and A. Rasila, On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces, Math. Z., 279 (2015), 163–183. 5. Sh. Chen, S. Ponnusamy and A. Rasila, Lengths, area and Lipschitz-type spaces of planar harmonic mappings, Nonlinear Anal., 115 (2015), 62–80. 6. Sh. Chen and S. Ponnusamy, Radial length, radial John diaks and K-quasiconformal harmonic mappings, Potential. Anal., 50 (2019), 415–437.
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7. Sh. Chen, Lengths, area and modulus of continuity of some classes of complex-valued functions, Results. Math., 74 (2019), 1–16. 8. P. Duren, Harmonic mappings in the plane, Cambridge Univ. Press, 2004. 9. K. M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math., 178 (1997), 143–167. 10. K. M. Dyakonov, Holomorphic functions and quasiconformal mappings with smooth moduli, Adv. Math., 187 (2004), 146–172. 11. R. Finn and J. Serrin, On the H¨ older continuity of quasiconformal and elliptic mappings, Trans. Amer. Math. Soc., 89 (1958), 1–15. 12. S. I. Goldberg and T. Ishihara, Harmonic quasiconformal mappings of Riemannian manifolds, Bull. Amer. Math. Soc., 80 (1974), 562–566. 13. S. I. Goldberg and T. Ishihara, Harmonic quasiconformal mappings of Riemannian manifolds, Amer. J. Math., 98 (1976), 225–240. 14. J. Heinonen, T. Kilpel¨ ainen and O. Martio, Nonlinear Potential Theory of Degenerate Ellipitic Equations. Dover Publications Inc, Mineola (2006). 15. L. H¨ormander, Notions of convexity, Progress in Mathematics, Vol. 127, Birkh¨ auser Boston Inc, Boston 1994. 16. D. Kalaj, On harmonic diffeomorphisms of the unit disc onto a convex domains, Complex Var. Elliptic Equ., 48 (2003), 175–187. 17. D. Kalaj, Cauchy transform and Poisson’s equation, Adv. Math., 231 (2012), 213–242. 18. D. Kalaj, Muckenhoupt weights and Lindel¨ of theorem for harmonic mappings, Adv. Math., 280 (2015), 301–321. 19. D. Kalaj, A sharp inequality for harmonic diffeomorphisms of the unit disk, J. Geom. Anal., 29 (2019), 392–401. 20. D. Kalaj, Lipschitz property of minimisers between double connected surfaces, J. Geom. Anal., (2019). https://doi.org/10.1007/s12220-019-00235-x 21. D. Kalaj and M. Mateljevi´c, (K, K )-quasiconformal harmonic mappings, Potential Anal., 36 (2012), 117–135. 22. D. Kalaj and M. Pavlovi´c, On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation, Trans. Amer. Math. Soc., 363 (2011), 4043–4061. 23. D. Kalaj and E. Saksman, Quasiconformal maps with controlled Laplacian, J. Anal. Math., 137 (2019), 251–268. 24. V. Markovi´c, Harmonic maps and the Schoen conjecture, J. Amer. Math. Soc., 30 (2017), 799–817. 25. O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I, 425 (1968), 3–10. 26. M. Mateljevi´c, Schwarz lemma and distortion for harmonic functions via length and area, arXiv:1805.02979. 27. L. Nirenberg, On nonlinear elliptic partial differential equations and H¨older continuity, Commun. Pure. Appl. Math., 6 (1953), 103–156. 28. D. Partyka and K. Sakan, Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 811–830. 29. D. Partyka and K. Sakan, On bi-Lipschitz type inequalities for quasicopnformal harmonic mappings, Ann. Acad. Sci. Fenn. Math., 32 (2007), 579–594. 30. M. Pavlovi´c, On Dyakonov’s paper Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math., 183 (1999), 141–143. 31. M. Pavlovi´c, Boundary correspondence under harmonic qusiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn. Math., 27 (2002), 365–372. 32. L.F. Tam and Tom Y.-H. Wan Quasiconformal harmonic diffeomorphism and the universal Teichmuller space, J. Differential Geom., 42 (1995), 368–410. 33. J. Zhu, Quasiconformal harmonic mappings with unbounded gradient (in Chinese), Sci. Sin. Math., 48 (2018), 273–280.
On certain quasiconformal and elliptic mappings
17
Sh. Chen, College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan 421008, People’s Republic of China. Email address: [email protected] S. Ponnusamy, Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India. Email address: [email protected]
Shaolin Chen, Saminathan Ponnusamy
PII:
S0022-247X(20)30082-2
DOI:
https://doi.org/10.1016/j.jmaa.2020.123920
Reference:
YJMAA 123920
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
3 September 2019
Please cite this article as: S. Chen, S. Ponnusamy, On certain quasiconformal and elliptic mappings, J. Math. Anal. Appl. (2020), 123920, doi: https://doi.org/10.1016/j.jmaa.2020.123920.
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ON CERTAIN QUASICONFORMAL AND ELLIPTIC MAPPINGS SHAOLIN CHEN AND SAMINATHAN PONNUSAMY Abstract. Let D be the closure of the unit disk D in the complex plane C and g be a continuous function in D. In this paper, we discuss some characterizations of elliptic mappings f satisfying the Poisson’s equation Δf = g in D, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.
1. Preliminaries and main results Let D = {z : |z| < 1} denote the open unit disk in the complex plane C and let T = ∂D be the unit circle. Furthermore, we denote by C m (Ω) the set of all complex-valued m-times continuously differentiable functions from Ω into C, where Ω is a subset of C and m ∈ {0, 1, 2, . . .}. In particular, C(Ω) := C 0 (Ω) denotes the set of all continuous functions in Ω. Let G be a domain of C with G be its closure. We use dG (z) to denote the Euclidean distance from z to the boundary ∂G of G. Especially, we always use d(z) for dD (z). For a real 2 × 2 matrix A, the matrix norm and the matrix function are defined by A = sup{|Az| : |z| = 1}, and l(A) = inf{|Az| : |z| = 1}, respectively. For z = x + iy ∈ C, the formal derivative of a complex-valued function f = u + iv is given by u x uy Df := , vx vy so that Df = |fz | + |fz | and l(Df ) = |fz | − |fz |, where (1.1) fz = fx − ify /2, and fz = fx + ify /2. We use Jf := det Df = |fz |2 − |fz |2 to denote the Jacobian of f . A sense-preserving homeomorphism f from a domain Ω onto Ω , contained in the 1,2 Sobolev class Wloc (Ω), is said to be a K-quasiconformal mapping if, for z ∈ Ω, Df (z)2 ≤ K det Df (z), i.e., Df (z) ≤ Kl(Df (z)), where K ≥ 1. File: YJMAA123920_source.tex, printed:
30-1-2020, 10.44
2000 Mathematics Subject Classification. Primary: 31A05; Secondary: 30H30. Key words and phrases. Elliptic mapping, (K, K )-quasiconformal mapping, Poisson’s equation.
1
2
Sh. Chen, S. Ponnusamy
A mapping f ∈ C 1 (Ω) is said to be an elliptic mapping (or (K, K )-elliptic mapping) if there are constants K ≥ 1 and K ≥ 0 such that f satisfies the following partial differential inequality Df (z)2 ≤ KJf (z) + K in the domain Ω ⊂ C (see [11, 27]). Let Ω1 and Ω2 be two subdomains of C. A sensepreserving homeomorphism f : Ω1 → Ω2 is said to be a (K, K )-quasiconformal mapping if f is absolutely continuous on lines in Ω1 , and there are constants K ≥ 1 and K ≥ 0 such that Df (z)2 ≤ KJf (z) + K , z ∈ Ω1 . Obviously, a (K, K )-quasiconformal mapping f ∈ C 1 (Ω) is an elliptic mapping in Ω. In particular, if K = 0, then the (K, K )-quasiconformal mappings are Kquasiconformal (cf. [2, 21]). Moreover, if a (K, K )-quasiconformal mapping is harmonic, then it is said to be harmonic (K, K )-quasiconformal. In 1968, Martio [25] has discussed the conditions for the K-quasiconformality of harmonic mappings from the closed unit disk onto itself. The harmonic Kquasiconformal mappings of Riemannian manifolds have been considered by Goldberg and Ishihara (see [13, 12]). Tam and Wan [32] have investigated some properties of harmonic quasiconformal diffeomorphism and the universal Teichmuller space. In 2002, Pavlovi´c [31] has generalized the corresponding results of Martio [25]. Kalaj [16], Partyka and Sakan [28] have investigated the K-quasiconformality of harmonic mappings from the unit disk onto bounded convex domains. See [22, 18, 24, 29, 33] and the references therein for detailed discussions on this topic. Recent papers [2], [21] and [20] bring much attention on the topic of (K, K )-quasiconformal mappings in the plane. This paper continues the study of previous work of [4, 5] and is mainly motivated by the articles of Finn and Serrin [11], and Kalaj and Mateljevi´c [21]. In order to state our main results, we need to recall some basic definitions and some results which motivate the present work. For θ ∈ [0, 2π] and z, w ∈ D with z = w, let 1 − zw 1 − |z|2 iθ and P (z, e ) = G(z, w) = log z−w |1 − ze−iθ |2 denote the Green function and the (harmonic) Poisson kernel, respectively. Let ψ : T → C be a bounded integrable function, and g ∈ C(D). The solution to the Poisson equation Δf = g f = ψ ∈ L1 (T)
in D, in T,
is given by f (z) = P [ψ](z) − G[g](z), where (1.2)
1 G[g](z) = 2π
1 G(z, w)g(w)dA(w), P [ψ](z) = 2π D
2π
P (z, eit )ψ(eit )dt, 0
On certain quasiconformal and elliptic mappings
3
and dA(w) denotes the Lebesgue measure on D. It is well known that if ψ and g are continuous in T and in D, respectively, then f = P [ψ] − G[g] has a continuous extension f˜ to the boundary, and f˜ = ψ in T (see [15, pp. 118-120] and [22, 17]). A continuous increasing function ω : [0, ∞) → [0, ∞) with ω(0) = 0 is called a majorant if ω(t)/t is non-increasing for t > 0 (see [9, 10, 30]). For α > 0 and a majorant ω, we use Bωα (D) to denote the generalized Bloch-type space of all functions f ∈ C 1 (D) with f Bωα (D) < ∞, where f Bωα (D) = |f (0)| + sup {Df (z)ω(dα (z))} . z∈D
For a given bounded integrable function ψ ∈ L1 (T) and a given g ∈ C(D), let Fg,ψ (D) = {f ∈ C 2 (D) : Δf = g in D and f = ψ in T}. Clearly, functions in F0,ψ (D) are harmonic in D. Furthermore, if f ∈ Fg,ψ (D), then f + G[g] is harmonic in D, and thus, has the representation (1.3)
f + G[g] = h1 + h2 ,
where h1 and h2 are analytic in D (cf. [8]), and G[g] is defined in (1.2). In [21], Kalaj and Mateljevi´c proved that a harmonic diffeomorphism between two bounded Jordan domains with C 2 boundaries is a harmonic (K, K )-quasiconformal mapping for some constants K ≥ 1 and K ≥ 0 if and only if it is Lipschitz continuous (see [21, Theorem 1.1 and Corollary 1.3]). This result can be considered as an extension of the corresponding results of Martio [25], Pavlovi´c [31], and Partyka and Sakan [29]. For related investigations on this topic, we refer to [28, 33]. In the following, we will give some characterizations of elliptic mappings without the C 2 boundary hypothesis of the image domains. Theorem 1.1. Suppose that ω is a given majorant. For a given bounded integrable function ψ ∈ L1 (T) and a given g ∈ C(D), let f ∈ Fg,ψ (D) be a univalent and sense-preserving mapping, and f (D) be a convex domain. If there are two positive constants C1 , C2 and α ∈ [0, 1] such that for any z1 , z2 ∈ D with z1 = z2 ,
1−α 2 ω (1 + |z1 |)(1 + |z2 |) C1 |f (z1 ) − f (z2 )| ≤ (1.4) ≤ 1−α
C1 |z1 − z2 | ω d(z )d(z ) 2 1
and
1 0
2
dt ≤ C2 , ω ((d(Φ(t)))1−α )
then f is an elliptic mapping, where Φ(t) := f −1 (f (z1 ) + t(f (z2 ) − f (z1 ))). Remark 1.2. In particular, if α = 1 in (1.4), then f is bi-Lipschitz. In this situation, Theorem 1.1 is trivial because all univalent and sense-preserving bi-Lipschitz mappings are quasiconformal mappings (see Chapter 14.78 in [14]). However, quasiconformal mappings are not necessarily bi-Lipschitz, not even Lipschitz (see Example 1.3).
4
Sh. Chen, S. Ponnusamy
Example 1.3. Let
⎧ e ⎨z logα |z|2 f (z) = ⎩ 0
for z ∈ D \ {0}, for z = 0,
where α ∈ (0, 1/2) is a constant. Then f is a quasiconformal self-homeomorphism of D. However, f is not Lipschitz at the origin (cf. [23]). Proposition 1.4. For a given bounded integrable function ψ ∈ L1 (T) and a given g ∈ C(D), let f = h1 + h2 − G[g] ∈ Fg,ψ (D), where h1 and h2 are analytic in D. If there is a constant C3 ∈ [1, 2) such that for any z1 , z2 ∈ D, |f (z1 ) − f (z2 )| ≤ C3 |h1 (z1 ) − h1 (z2 )|,
(1.5)
then f is an elliptic mapping. We remark that the inverse of Proposition 1.4 does not necessarily hold (see Example 1.5). Example 1.5. For z ∈ D, let f (z) = 3z|z|2 − z|z|8 . Then (1) f is a (1, 729/(216/3 ))-quasiconformal mapping in D; (2) f is not a K-quasiconformal mapping for any K ≥ 1; (3) f does not satisfy the inequality (1.5). Proof. We first prove the univalence of f . Suppose on the contrary that f is not univalent. Then there are two distinct points z1 , z2 ∈ D such that f (z1 ) = f (z2 ), which implies that z1 |z1 |2 (3 − |z1 |6 ) = z2 |z2 |2 (3 − |z2 |6 ).
(1.6)
Case 1. If |z1 | = |z2 |, then, by (1.6), we have z1 = z2 . This is a contradiction with the assumption. Case 2. If |z1 | = |z2 |, then (1.6) reduces to |z1 |3 (3 − |z1 |6 ) = |z2 |3 (3 − |z2 |6 ), and consequently (|z1 |3 − |z2 |3 )(3 − |z1 |6 − |z2 |6 − |z1 |3 |z2 |3 ) = 0. This implies |z1 | = |z2 | = 1, which violates the hypothesis. Hence f is univalent. Also, for z ∈ D, elementary calculations lead to fz (z) = |z|2 (6 − 5|z|6 ) and fz (z) = z 2 (3 − 4|z|6 ), which give that (1.7)
Df (z)2 − Jf (z) ≤ Df (z)2 = max
|z|∈[0,1)
81|z|4 (1 − |z|6 )2 = 729/(216/3 )
and (1.8)
lim−
|z|→1
|fz (z)| = 1. |fz (z)|
The first assertion and the second assertion easily follows from (1.7) and (1.8), respectively. The last assertion is obvious.
On certain quasiconformal and elliptic mappings
5
For r ∈(0, 1) and f ∈ C 1 (D), the perimeter of the curve C(r) = w = f (reiθ ) : θ ∈ [0, 2π] , with counting multiplicity, is defined by (cf. [3, 5, 7]) 2π 2π iθ fz (reiθ ) − e−2iθ fz (reiθ ) dθ. |df (re )| = r f (r) = 0
0
In particular, let f (1) = sup0
(1.10)
Kf (1) , 2nπ √ f (1) K Df (z) ≤ 2π(1 − |z|) |an | + |bn | ≤
and f ∈ Bω1 (D), where ω(t) = t. In particular, if K = 1, then the above two estimates are sharp, and the extreme function is f (z) = z. Concerning the generalizedform of inequality Mateljevi´c [26] proved the ∞ (1.9), ∞ n n following result: Let f (z) = n=0 an z + n=1 bn z be a harmonic mapping with f (1) < ∞. Then, for n ≥ 1, the inequality f (1) nπ holds (see [26, Theorem 10]). Moreover, Kalaj [19] improved the inequality (1.10) and obtained a sharp inequality for harmonic diffeomorphisms of D. It reads as follows. (1.11)
|an | + |bn | ≤
Theorem B. If f is a harmonic sense-preserving diffeomorphism of D onto a Jordan domain Ω with rectifiable boundary of length 2πR, then the sharp inequality (1.12)
|fz (z)| ≤
R , z∈D 1 − |z|2
holds, where R is a positive constant. If the equality in (1.12) is attained for some a, then Ω is convex, and there is a holomorphic function μ : D → D and a constant θ ∈ [0, 2π], such that z z z+a dt μ(t)dt −iθ =R + . (1.13) F (z) := e f 2 2 1 + za 0 1 + t μ(t) 0 1 + t μ(t) Moreover, every function f defined by (1.13) is a harmonic diffeomorphism and maps D to a Jordan domain bounded by a convex curve of length 2πR and the inequality (1.12) is attained for z = a. The following result is also a generalization of Theorem A.
6
Sh. Chen, S. Ponnusamy
n Theorem 1.6. Let K ≥ 1, K ≥ 0 and R > 0 be constants. If f (z) = ∞ n=0 an z + ∞ n n=1 bn z is a harmonic (K, K )-quasiconformal mapping of D onto a Jordan domain Ω with rectifiable boundary of length 2πR, then for n ≥ 1, √ K + KR , (1.14) |an | + |bn | ≤ n √ 1 −R + K + KK + K 2 R2 (1.15) Df (z) ≤ R + , for z ∈ D, 1+K 1 − |z|2 and f ∈ Bω1 (D). In particular, if K = K − 1 = 0 and Ω = D, then the estimates of (1.14) is sharp, and the extreme function is f (z) = z for z ∈ D. Moreover, if K = K − 1 = 0 and Ω = D, then the equal sign occurs in (1.15) for some fixed z = a if and only if z−a f (z) = eit 1−za , where t ∈ [0, 2π]. We remark that if K = 0 and K ∈ [1, 2], then the inequality (1.14) is better than (1.11). Let f ∈ C 1 (D). Then, for θ ∈ [0, 2π], the radial length of the curve Cθ (r) = w = f (ρeiθ ) : 0 ≤ ρ ≤ r < 1 , with counting multiplicity, is defined by ∗f (r, θ)
r
r
|df (ρe )| = iθ
= 0
fz (ρeiθ ) + e−2iθ fz (ρeiθ ) dρ.
0
In particular, let ∗f (1, θ) = sup ∗f (r, θ). 0≤r<1
We refer the reader to [5, 6] for some discussion of the radial length. In particular, the following result establishes the Fourier coefficient estimates of K-quasiconformal harmonic mappings with the finite radial length. ∞ n n Theorem C. ([3, Theorem 4]) Let f (z) = ∞ n=0 an z + n=1 bn z be a harmonic ∗ ∗ K-quasiconformal mapping in D. If f (1) = supθ∈[0,2π] f (1, θ) < ∞, then (1.16)
|an | + |bn | ≤ K∗f (1) for n ≥ 1.
Moreover, if K = 1, then the estimate (1.16) is sharp and the extreme function is f (z) = ∗f (1)z for z ∈ D. We improve Theorem C into the following form.
∞ n n Theorem 1.7. For K ≥ 1 and K ≥ 0, let f (z) = ∞ n=0 an z + n=1 bn z be a ∗ ∗ harmonic (K, K )-quasiconformal mapping in D. If f (1) = supθ∈[0,2π] f (1, θ) < ∞, then for n ≥ 1, (1.17)
|an | + |bn | ≤
√
K + K∗f (1).
In particular, if K = K − 1 = 0, then the estimate (1.17) is sharp, and the extreme function is f (z) = ∗f (1)z for z ∈ D.
On certain quasiconformal and elliptic mappings
7
The proof of Theorems 1.1, 1.6 and 1.7, and Proposition 1.4 will be presented in Section 2.
2. The proofs of the main results In this section, we shall prove Theorems 1.1 and 1.7, and Propositions 1.4 and 1.6. We start with some useful Lemmas.
Lemma D. ([4, Lemma 6]) Let ω be a majorant and ν ∈ [0, 1]. Then for t ∈ [0, ∞], ω(νt) ≥ νω(t). Lemma 2.1. Let ω be a majorant and α ∈ [0, 1] be a constant. Suppose that f ∈ C 1 (D) is univalent, and f (D) is a convex domain. Then the following two statements are equivalent: (a) For any z1 , z2 with z1 = z2 , there exists a constant C4 such that (1−α)/2 |f (z1 ) − f (z2 )| C4 1 ≤ ω (1 + |z1 |)(1 + |z2 |) ≤ (1−α)/2 . C4 |z1 − z2 | ω d(z1 )d(z2 ) (b) For any z ∈ D, there is a constant C5 > 0 such that 1 C5 . ω (1 + |z|)1−α ≤ l(Df (z)) ≤ Df (z) ≤ C5 ω ((d(z))1−α ) Proof. We first prove (a) ⇒ (b). For θ ∈ [0, 2π] and z = x + iy ∈ D, elementary computations lead to (see (1.1)) fz (z) + fz (z) = fx (z) and i(fz (z) − fz (z)) = fy (z) and therefore, fx (z) cos θ + fy (z) sin θ = fz (z)eiθ + fz (z)e−iθ .
(2.1)
For r ∈ [0, 1 − |z|), let ξ = z + reiθ . Then, by (a) and (2.1), we obtain
|f (z) − f (ξ)| lim+ r→0 |z − ξ|
max |fx (z) cos θ + fy (z) sin θ| = max θ∈[0,2π] ⎧ ⎫ ⎨ ⎬ C4 C4
≤ max lim+ = 1−α 1−α θ∈[0,2π] ⎩r→0 ω d(z)d(z + reiθ ) 2 ⎭ ω ((d(z)) )
Df (z) =
θ∈[0,2π]
8
Sh. Chen, S. Ponnusamy
and Df (z) ≥ l(Df (z)) = min |fx (z) cos θ + fy (z) sin θ| θ∈[0,2π] |f (z) − f (ξ)| = min lim θ∈[0,2π] r→0+ |z − ξ|
⎫ ⎧ 1−α iθ 2 ⎨ ⎬ ω (1 + |z|)(1 + |z + re |) ≥ min lim ⎭ θ∈[0,2π] ⎩r→0+ C4 ω ((1 + |z|)1−α ) . C4
=
Now we prove (b) ⇒ (a). For t ∈ [0, 1] and z1 , z2 ∈ D with z1 = z2 , let χ(t) = z1 t + (1 − t)z2 . Since d(χ(t)) ≥ 1 − t|z1 | − (1 − t)|z2 | ≥ 1 − t − (1 − t)|z2 | = (1 − t)d(z2 ) and d(χ(t)) ≥ 1 − t|z1 | − (1 − t)|z2 | ≥ 1 − t|z1 | − (1 − t) = td(z1 ), we see that (2.2)
d(χ(t))
1−α
1−α ≥ t(1 − t)d(z1 )d(z2 ) 2 .
By calculations, we have (2.3)
|f (z1 ) − f (z2 )| =
1
(χ(t))χ (t)
fw (χ(t))χ (t) + fw 1 Df (χ(t))dt, ≤ |z1 − z2 | 0
dt
0
where w = χ(t). Now, we estimate the integral on the right. By (b) and (2.2), we have 1 1 1 C5 dt C5 dt Df (χ(t))dt ≤ 1−α ≤ 1−α , 0 0 ω 0 ω d(χ(t)) t(1 − t)d(z1 )d(z2 ) 2 which, together with Lemma D, implies that 1 1 dt C5
(2.4) Df (χ(t))dt ≤ 1−α 1−α 1−α 0 0 t 2 (1 − t) 2 ω d(z1 )d(z2 ) 2 Γ2 1+α C5 2
, = Γ(1 + α) ω d(z )d(z ) 1−α 2 1
where Γ denotes the usual Gamma function.
2
On certain quasiconformal and elliptic mappings
9
It follows from (2.3) and (2.4) that
Γ2 1+α C5 |f (z1 ) − f (z2 )| 2
. ≤ |z1 − z2 | Γ(1 + α) ω d(z )d(z ) 1−α 2 1 2
On the other hand, for any z1 , z2 ∈ D with z1 = z2 , let w1 = f (z1 ) and w2 = f (z2 ). For t ∈ [0, 1], let (2.5)
γ(t) = tw1 + (1 − t)w2
be the straight line segment connecting w1 and w2 . Since f (D) is a convex domain, we see that γ(t) ⊂ f (D) and η(t) = f −1 (γ(t)) ⊂ D for t ∈ [0, 1]. It is not difficult to know that 1 + |z1 | 1 + |z2 | 1 + |η(t)| ≥ and 1 + |η(t)| ≥ , 2 2 which, together with Lemma D, implies that ⎞ ⎛ 1−α 2 (1 + |z1 |)(1 + |z2 |) ⎠ ω (1 + |η(t)|)1−α ≥ ω ⎝ (2.6) 21−α 1−α
ω (1 + |z1 |)(1 + |z2 |) 2 ≥ . 21−α By (b) and (2.6), we obtain 1 1 1 (2.7) l(Df (η(t)))|η (t)|dt ≥ ω (1 + |η(t)|)1−α |η (t)|dt C5 0 0
1−α 2 1 ω (1 + |z1 |)(1 + |z2 |) |η (t)|dt ≥ 21−α C5
0 1−α 2 ω (1 + |z1 |)(1 + |z2 |) |z1 − z2 |, ≥ 21−α C5 It follows from (2.5) that 1 |γ (t)|dt |w1 − w2 | = |γ (t)| = 0 1 = fζ (η(t))η (t) + fζ (η(t))η (t) dt 0 1 l(Df (η(t)))|η (t)|dt, ≥ 0
which, together with (2.7), yields that
1−α 2 |)(1 + |z |) ω (1 + |z 1 2 |w1 − w2 | . ≥ |z1 − z2 | 21−α C5 The proof of this lemma is finished.
10
Sh. Chen, S. Ponnusamy
Lemma E. ([17, Lemma 2.7]) If g ∈ C(D), then ∂ ∂ 1 max G[g](z) , G[g](z) ≤ g∞ for z ∈ D. ∂z ∂z 3 Lemma 2.2. Let f ∈ C 1 (D) be a sense-preserving mapping. Then f is an elliptic mapping if and only if there exist constants k1 ∈ [0, 1) and k2 ∈ [0, ∞) such that |fz (z)| ≤ k1 |fz (z)| + k2 for z ∈ D.
(2.8)
Proof. We first prove the sufficiency. By (2.8), for z ∈ D, we have 1 + k1 2k2 (2.9) Df (z) ≤ l(Df (z)) + 1 − k1 1 − k1 2 1 + k1 1 + k1 4k22 ≤ l(Df (z)) + l2 (Df (z)) + . 1 − k1 1 − k1 (1 − k1 )2 1 + k1 Case 1. For all z ∈ D, Df (z) ≤ l(Df (z)). 1 − k1 In this case, it is easy to know that 1 + k1 2 Jf (z), Df (z) ≤ 1 − k1 which implies that f is an elliptic mapping. Case 2. There is a subset E of D such that 1 + k1 Df (z) > l(Df (z)) for z ∈ E. 1 − k1 In this case, it follows from (2.9) that
Df (z) −
1 + k1 1 − k1
2 l(Df (z))
which implies that (2.10)
≤
Df (z) ≤ 2 2
1 + k1 1 − k1
1 + k1 1 − k1
2 l2 (Df (z)) +
Jf (z) +
4k22 for z ∈ E, (1 − k1 )2
4k22 . (1 − k1 )2
On the other hand, for z ∈ D\E, we have 1 + k1 2 Df (z) ≤ Jf (z), 1 − k1 which, together with (2.10), implies that f is also an elliptic mapping in D. Next, we show the necessity. If f is an elliptic mapping, then there exist constants K ≥ 1 and K ≥ 0 such that Df (z)2 ≤ KJf (z) + K for z ∈ D.
On certain quasiconformal and elliptic mappings
This gives that Df (z) ≤
Kl(Df (z)) +
Kl(Df (z))
2
2
+ 4K
≤ Kl(Df (z)) +
11
√
K ,
and consequently
√ K −1 K |fz (z)| + . |fz (z)| ≤ K +1 1+K The proof of this lemma is complete.
The proof of Theorem 1.1. Differentiating both sides of the equation f −1 (f (z)) = z and then simplifying the resulting relations lead to the formulae (2.11)
(f −1 )w =
fz fz and (f −1 )w = − , Jf Jf
where w = f (z). Next, for z1 , z2 ∈ D with z1 = z2 , we let ϕ(t) = t(f (z1 ) − f (z2 )) + f (z2 ), where t ∈ [0, 1]. Since f (D) is a convex domain, we see that ϕ(t) ⊂ f (D) and Φ(t) := f −1 (ϕ(t)) ⊆ D for t ∈ [0, 1]. For z ∈ D, let F (z) = f (z) + G[g](z). Then F is harmonic in D, and F can be represented by F = h1 + h2 in D, where hj (j = 1, 2) are analytic in D. With z = Φ(t), we have Φ(0) = z2 , Φ(1) = z1 and thus, by (2.11), we obtain z1 1 h1 (z1 ) − h1 (z2 ) = h1 (z)dz = h1 (Φ(t))Φ (t)dt z2 1
0
h1 (Φ(t)) fw−1 (ϕ(t))ϕ (t) + fw−1 (ϕ(t))ϕ (t) dt 0 1 f (Φ(t)) (Φ(t)) f z z ϕ (t) − h1 (Φ(t)) ϕ (t) dt, = J (Φ(t)) J (Φ(t)) f f 0 =
which implies that
1 h (Φ(t)) f (Φ(t)) − f (Φ(t)) ϕ (t) z z h1 (z1 ) − h1 (z2 ) 1 ϕ (t) (2.12) = dt f (z1 ) − f (z2 ) J (Φ(t)) f 0 1 |h1 (Φ(t))|Df (Φ(t)) dt ≤ Jf (Φ(t)) 0 1 l(Df (Φ(t))) + |G[g]z (Φ(t))| + |fz (Φ(t))| ≤ dt. l(Df (Φ(t))) 0
It follows from Lemma 2.1 that there exists a positive constant C6 such that
12
Sh. Chen, S. Ponnusamy
|G[g]z (Φ(t))| + Df (Φ(t)) |G[g]z (Φ(t))| + |fz (Φ(t))| ≤ l(Df (Φ(t))) l(Df (Φ(t))) ≤
C6 |G[g]z (Φ(t))| +
C6 ω((d(Φ(t)))1−α ) |Φ(t)|)1−α )
C6 ≤ ω(1)
ω ((1 + |G[g]z (Φ(t))| +
1 ω ((d(Φ(t)))1−α )
,
which, together with (2.12) and Lemma E, gives that 1 h1 (z1 ) − h1 (z2 ) C6 (2.13) |G[g]z (Φ(t))|dt f (z1 ) − f (z2 ) ≤ 1 + ω(1) 0 1 dt ≤ μ, + 1−α 0 ω (d(Φ(t))) where μ := 1 +
C6 g∞ C6 C2 + ≥ 1. 3ω(1) ω(1)
For θ ∈ [0, 2π] and any fixed point z ∈ D, let ς = z + reiθ , where r ∈ [0, 1 − |z|). By (2.13), we have h1 (ς) − h1 (z) 1 ≤ lim μ r→0+ ς −z
h1 (ς) − h1 (z) h2 (ς) − h2 (z) G[g](z) − G[g](ς) + + lim r→0+ ς −z ς −z ς −z = h1 (z)eiθ + h2 (z)e−iθ − (G[g]z (z)eiθ + G[g]z (z)e−iθ )
which yields that (2.14)
1 |h (z)| ≤ μ 1 =
min h1 (z) − G[g]z (z) + (h2 (z) − G[g]z (z))e−2iθ θ∈[0,2π] min fz (z) + fz (z)e−2iθ = l(Df (z)). θ∈[0,2π]
Since |h1 | ≥ |h1 − G[g]z | − |G[g]z | = |fz | − |G[g]z |, by (2.14) and Lemma E, we see that (2.15)
|fz | ≤
(μ − 1) 1 |fz | + g∞ . μ 3μ
It follows from (2.15) and Lemma 2.2 that f is an elliptic mapping. The proof of this theorem is complete.
On certain quasiconformal and elliptic mappings
13
The proof of Proposition 1.4. For θ ∈ [0, 2π] and any fixed point z ∈ D, let ς = z + reiθ , where r ∈ [0, 1 − |z|). Then by (2.1) and (1.5), we have f (ς) − f (z) lim Df (z) = max |fx (z) cos θ + fy (z) sin θ| = max θ∈[0,2π] θ∈[0,2π] r→0+ ς −z h1 (ς) − h1 (z) = C3 |h1 (z)| ≤ C3 lim+ r→0 ς −z ≤ C3 |fz (z)| + C3 |G[g]z (z)|, which, together with Lemma E, yields that C3 g∞ . 3 It follows from (2.16) and Lemma 2.2 that f is an elliptic mapping. The proof of this proposition is complete. |fz (z)| ≤ (C3 − 1)|fz (z)| +
(2.16)
The proof of Theorem 1.6. We first prove (1.14). Since Df (z)2 ≤ KJf (z) + K , we see that (2.17)
Df (z) ≤
Kl(Df (z)) +
Kl(Df (z))
2
+ 4K
2 where z ∈ D. It follows from (2.17) that
fz (reiθ ) − e−2iθ fz (reiθ ) dθ ≥ r f (r) = r 0 √ 2π Df (reiθ ) − K dθ, ≥ r K 0 which gives that (2.18)
2π
2π
r
≤ Kl(Df (z)) +
√
2π
l(Df (reiθ ))dθ 0
√ Df (reiθ )dθ ≤ 2πr K + Kf (r).
0
For n ≥ 1 and r ∈ (0, 1), it follows from Cauchy’s integral formula that, ∂f (z) dz ∂f (z) dz 1 1 nan = and nbn = , 2πi |z|=r ∂z z n 2πi |z|=r ∂z zn which, together with the inequality (2.18), implies that 2π 1 n(|an | + |bn |) ≤ rDf (reiθ )dθ 2πrn 0
√ 1 + K (r) K 2πr ≤ f 2πrn
1 √ ≤ K + K (1) . 2π f 2πrn
K ,
14
Sh. Chen, S. Ponnusamy
Consequently,
√ |an | + |bn | ≤
(2.19)
K Kf (1) + = n 2nπ
√
K KR + . n n
This proves (1.14) Next, we prove (1.15). For z ∈ D, it follows from the inequality Df (z)2 ≤ KJf (z) + K that |fz (z)| ≤
K + KK + K 2 |fz (z)|2 , 1+K
−|fz (z)| +
which, together with (1.12), yields that √ 1 −R + K + KK + K 2 R2 Df (z) ≤ R + . 1+K 1 − |z|2
The proof of the theorem is complete. Theorem F. ([1, Theorem 2]) Let φ be subharmonic in D. If for all r ∈ [0, 1), r φ(ρeiθ ) dρ ≤ 1, A(r) = sup θ∈[0,2π]
0
then A(r) ≤ r. The proof of Theorem 1.7. Since f is a univalent (K, K )-elliptic mapping, we see that Df (z)2 ≤ KDf (z)l(Df (z)) + K for z ∈ D. This gives that (2.20)
Df (z) ≤
Kl(Df (z)) +
Kl(Df (z))
2
+ 4K
≤ Kl(Df (z)) +
2 It follows from (2.20) that, for θ ∈ [0, 2π] and r ∈ (0, 1), r r ∗ iθ −2iθ iθ |fz (ρe ) + e fz (ρe )| dρ ≥ l(Df (ρeiθ )) dρ f (r, θ) = 0 0
√ 1 r ≥ Df (ρeiθ ) − K dρ K 0 and consequently r √ √ (2.21) Df (ρeiθ ) dρ ≤ K r + K∗f (r, θ) ≤ K + K∗f (1). 0
Inequality (2.21) and Theorem F lead to r √
iθ ∗ (2.22) Df (ρe ) dρ ≤ K + Kf (1) r. 0
√
K .
On certain quasiconformal and elliptic mappings
15
The Cauchy integral formula shows that, for ρ ∈ (0, 1), ∂f (z) dz 1 1 ∂f (z) dz and nbn = , nan = 2πi |z|=ρ ∂z z n 2πi |z|=ρ ∂z zn which yields that 1 n(|an | + |bn |) ≤ 2πρn
(2.23)
2π
ρDf (ρeiθ )dθ. 0
Then combining (2.22) and (2.23) gives the final estimate for |an | + |bn |, namely, r 2π r n−1 iθ ρ dρ ≤ Df (ρe ) dρ dθ 2πn(|an | + |bn |) 0 0 0 2π √
∗ K + Kf (1) r dθ ≤ 0
and consequently
√
|an | + |bn | ≤ inf
r∈(0,1)
K + K∗f (1) rn−1
The proof of this theorem is complete.
=
√ K + K∗f (1).
Acknowledgements: We thank anonymous referees for drawing our attention to references [19, 26] and providing us with a lot of valuable comments. This research was partly supported by the Hunan Provincial Education Department Outstanding Youth Project (No. 18B365), the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the exchange project for the third regular session of the China-Montenegro Committee for Cooperation in Science and Technology (No. 3-13), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469), and the Open Fund Project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang Normal University (No. IIPA18K06). References 1. E. F. Beckenbach, A relative of the lemma of Schwarz, Bull. Amer. Math. Soc., 44 (1938), 698–707. 2. J. Chen, P. Li, S. K. Sahoo and X. Wang, On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domain, Israel J. Math., 220 (2017), 453–478. 3. Sh. Chen, G. Liu and S. Ponnusamy, Linear measure and K-quasiconformal harmonic mappings (in Chinese), Sci. Sin. Math., 47 (2017), 565–574. 4. Sh. Chen, S. Ponnusamy and A. Rasila, On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces, Math. Z., 279 (2015), 163–183. 5. Sh. Chen, S. Ponnusamy and A. Rasila, Lengths, area and Lipschitz-type spaces of planar harmonic mappings, Nonlinear Anal., 115 (2015), 62–80. 6. Sh. Chen and S. Ponnusamy, Radial length, radial John diaks and K-quasiconformal harmonic mappings, Potential. Anal., 50 (2019), 415–437.
16
Sh. Chen, S. Ponnusamy
7. Sh. Chen, Lengths, area and modulus of continuity of some classes of complex-valued functions, Results. Math., 74 (2019), 1–16. 8. P. Duren, Harmonic mappings in the plane, Cambridge Univ. Press, 2004. 9. K. M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math., 178 (1997), 143–167. 10. K. M. Dyakonov, Holomorphic functions and quasiconformal mappings with smooth moduli, Adv. Math., 187 (2004), 146–172. 11. R. Finn and J. Serrin, On the H¨ older continuity of quasiconformal and elliptic mappings, Trans. Amer. Math. Soc., 89 (1958), 1–15. 12. S. I. Goldberg and T. Ishihara, Harmonic quasiconformal mappings of Riemannian manifolds, Bull. Amer. Math. Soc., 80 (1974), 562–566. 13. S. I. Goldberg and T. Ishihara, Harmonic quasiconformal mappings of Riemannian manifolds, Amer. J. Math., 98 (1976), 225–240. 14. J. Heinonen, T. Kilpel¨ ainen and O. Martio, Nonlinear Potential Theory of Degenerate Ellipitic Equations. Dover Publications Inc, Mineola (2006). 15. L. H¨ormander, Notions of convexity, Progress in Mathematics, Vol. 127, Birkh¨ auser Boston Inc, Boston 1994. 16. D. Kalaj, On harmonic diffeomorphisms of the unit disc onto a convex domains, Complex Var. Elliptic Equ., 48 (2003), 175–187. 17. D. Kalaj, Cauchy transform and Poisson’s equation, Adv. Math., 231 (2012), 213–242. 18. D. Kalaj, Muckenhoupt weights and Lindel¨ of theorem for harmonic mappings, Adv. Math., 280 (2015), 301–321. 19. D. Kalaj, A sharp inequality for harmonic diffeomorphisms of the unit disk, J. Geom. Anal., 29 (2019), 392–401. 20. D. Kalaj, Lipschitz property of minimisers between double connected surfaces, J. Geom. Anal., (2019). https://doi.org/10.1007/s12220-019-00235-x 21. D. Kalaj and M. Mateljevi´c, (K, K )-quasiconformal harmonic mappings, Potential Anal., 36 (2012), 117–135. 22. D. Kalaj and M. Pavlovi´c, On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation, Trans. Amer. Math. Soc., 363 (2011), 4043–4061. 23. D. Kalaj and E. Saksman, Quasiconformal maps with controlled Laplacian, J. Anal. Math., 137 (2019), 251–268. 24. V. Markovi´c, Harmonic maps and the Schoen conjecture, J. Amer. Math. Soc., 30 (2017), 799–817. 25. O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I, 425 (1968), 3–10. 26. M. Mateljevi´c, Schwarz lemma and distortion for harmonic functions via length and area, arXiv:1805.02979. 27. L. Nirenberg, On nonlinear elliptic partial differential equations and H¨older continuity, Commun. Pure. Appl. Math., 6 (1953), 103–156. 28. D. Partyka and K. Sakan, Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 811–830. 29. D. Partyka and K. Sakan, On bi-Lipschitz type inequalities for quasicopnformal harmonic mappings, Ann. Acad. Sci. Fenn. Math., 32 (2007), 579–594. 30. M. Pavlovi´c, On Dyakonov’s paper Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math., 183 (1999), 141–143. 31. M. Pavlovi´c, Boundary correspondence under harmonic qusiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn. Math., 27 (2002), 365–372. 32. L.F. Tam and Tom Y.-H. Wan Quasiconformal harmonic diffeomorphism and the universal Teichmuller space, J. Differential Geom., 42 (1995), 368–410. 33. J. Zhu, Quasiconformal harmonic mappings with unbounded gradient (in Chinese), Sci. Sin. Math., 48 (2018), 273–280.
On certain quasiconformal and elliptic mappings
17
Sh. Chen, College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan 421008, People’s Republic of China. Email address: [email protected] S. Ponnusamy, Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India. Email address: [email protected]