Arch. Math. 101 (2013), 285–291c© 2013 Springer Basel

0003-889X/13/030285-7

published online September 11, 2013DOI 10.1007/s00013-013-0548-6 Archiv der Mathematik

Covering and distortion theorems for planar harmonic univalentmappings

Shaolin Chen, Saminathan Ponnusamy, and Xiantao Wang

Abstract. In this paper, we investigate Clunie and Sheil-Small’scovering theorems for sense-preserving planar harmonic univalent map-pings defined in the unit disk. Our results significantly improve the earlierknown result. Also, we obtain a distortion theorem for fully starlike har-monic mappings in the unit disk.

Mathematics Subject Classification (2010). Primary 30H05, 30H10,30H30; Secondary 30C20, 30C45, 30C62, 31C05.

Keywords. Harmonic univalent mapping, Covering theorem,Distortion theorem.

1. Introduction and main results. The present article is mainly concerned withthe class S0

H of sense-preserving planar harmonic univalent mappings f = h+gdefined on the unit disk D = {z : |z| < 1}, where h and g are holomorphicfunctions in D normalized in a standard form: h(0) = g(0) = g′(0) = 0 andh′(0) = 1, see [8]. If g(z) is identically zero on the decomposition of f(z),then the class S0

H in this case reduces to the classical family S of normalizedholomorphic univalent functions h(z) = z +

∑∞n=2 anz

n in D.Recall that the Jacobian Jf of a harmonic function f = h+ g is given by

Jf = |fz|2 − |fz|2 = |h′|2 − |g′|2,

This research of the first author was partly supported by the NSF of China (No. 11071063).This work was also supported in part by the Construct Program of the Key Discipline inHunan Province (No. [2011] 76) and the Start Project of Hengyang Normal University (No.12B34).The second author is currently on leave from the Department of Mathematics, Indian Insti-tute of Technology Madras, Chennai-600 036, India.

286 Sh. Chen et al. Arch. Math.

and so f is locally univalent and sense-preserving in D if and only if |g′(z)| <|h′(z)| in D (see [11]). We refer to [1,3,8,9,11–13] for the theory of planarharmonic mappings.

For a ∈ C, let D(a, r) = {z : |z−a| < r}. In particular, we use Dr to denotethe disk D(0, r) and D for the unit disk D1. For a harmonic function f definedin D, we use the following standard notations:

Λf (z) = max0≤θ≤2π

|fz(z) + e−2iθfz(z)| = |fz(z)| + |fz(z)|

and

λf (z) = min0≤θ≤2π

|fz(z) + e−2iθfz(z)| =∣∣ |fz(z)| − |fz(z)|

∣∣.

Thus, for a sense-preserving harmonic function f = h + g, one has Jf (z) =Λf (z)λf (z) and the dilatation ω of f defined by ω = g′/h′ is analytic in D

such that |ω(z)| < 1 in D.The following conjecture is in [13] (see also [9, p. 97]).

Conjecture 1.1. If f ∈ S0H , then the range f(D) contains a disk D 1

6.

Also, the following result is known from [8, Theorem 4.4].

Theorem A. Each function f ∈ S0H satisfies the inequality

|f(z)| ≥ 14

|z|(1 + |z|)2 , z ∈ D.

In particular, the range f(D) contains the disk D 116.

Let ψ be a non-constant holomorphic function in D. For z0 ∈ D, a diskD(ψ(z0), r) is called a univalent disk contained in ψ(D) if there is a subdomainD of D such that ψ is univalent in D and ψ(D) = D(ψ(z0), r). The radiusof the largest univalent disk with center ψ(z0) is denoted by r(z0, ψ). LetBψ = sup{r(z, ψ) : z ∈ D}. The locally univalent Bloch constant is defined by

Bloc = inf{Bψ : ψ′(0) = 1 and ψ′(z) �= 0 for z ∈ D}.Let BHloc denote the class of all locally biholomorphic functions in D satisfyingthe conditions

ψ(0) = ψ′(0) − 1 = 0 and (1 − |z|2)|ψ′(z)| ≤ 1 for z ∈ D.

In [10], Landau proved that

Bloc = inf{Bψ : ψ ∈ BHloc}.In [2, Theorem 1], the authors proved Bloc > 1

2 +2×10−8, but the best boundof Bloc is unknown.

The first aim of this paper is to consider Conjecture 1.1, and we improveTheorem A in the following form.

Theorem 1.2. If f ∈ S0H , then f(D) contains a disk with the radius R ≥ Bloc

4 >ρ04 , where ρ0 = 1

2 + 2 × 10−8.

Vol. 101 (2013) Planar harmonic univalent mappings 287

For convenience, we denote by H(D) the set of all univalent and sense-preserving harmonic mappings f = h + g in D satisfying the normalizationh(0) = g(0) = 0, where h and g are holomorphic in D. The functions h andg are referred to by us as the holomorphic and anti-holomorphic parts off , respectively. Let SH denote the subclass of H(D) with h′(0) = 1. Thus,S0H = {f = h + g ∈ SH : g′(0) = 0}. It is well-known that S0

H is compact(see [8]).

A function f ∈ H(D) is said to be a fully starlike harmonic mapping if itmaps every circle |z| = r < 1 onto a curve bounds a domain starlike withrespect to the origin (see [6, p. 138]).

In [7], Chuaqui and Hernandez discussed the relationship between theimages of the linear connectivity of D under the harmonic mappings f = h+gand under their corresponding holomorphic counterparts h, where h and gare holomorphic in D. For the extensive discussions on this topic, see [4,5].The second aim of the article is to discuss the relationship between f and itsholomorphic counterpart h, where f = h+ g ∈ H(D).

Theorem 1.3. Let f = h+g ∈ H(D) be a fully starlike harmonic mapping withfz(0) = 0, where h and g are holomorphic in D. Then for all r ∈ (0, 1) andz ∈ D,

11 + r

|f(rz)| ≤ |h(rz)| ≤ 11 − r

|f(rz)|. (1.1)

In particular, one has

|f(z)| ≤ 2|h(z)|, z ∈ D, (1.2)

and if h′(0) = 1, then

|h(z)| ≥ 18

|z|(1 + |z|)2 , z ∈ D. (1.3)

Note that (1.2) follows from (1.1), whereas (1.3) is a simple consequence of(1.2) and Theorem A. Thus, it suffices to prove (1.1). The proofs of Theorems1.2 and 1.3 will be presented in Section 2.

2. The proofs of the main results. First, we recall that (cf. [9, p. 5]) a sense-preserving univalent harmonic mapping is K-quasiconformal, K ∈ [1,∞), ifΛf (z) ≤ Kλf (z) for z ∈ D.

Proof of Theorem 1.2. Let f ∈ S0H . For z ∈ D, let ω(z) = fz(z)/fz(z). Then ω

is holomorphic in D, ω(0) = 0, and |ω(z)| < 1 for z ∈ D. Thus, by the Schwarzlemma, |ω(z)| ≤ |z| for z ∈ D. Therefore, for any fixed r ∈ (0, 1) and z ∈ Dr,we have

Λf (z)λf (z)

=1 + |ω(z)|1 − |ω(z)| ≤ 1 + r

1 − r= Kr.

For ζ ∈ D, let

F (ζ) = r−1f(rζ) = H(ζ) +G(ζ),

288 Sh. Chen et al. Arch. Math.

where

H(ζ) = r−1h(rζ) and G(ζ) = r−1g(rζ).

It is easy to see that F is a Kr-quasiconformal harmonic mapping in D sat-isfying Fζ(0) = H ′(0) = 1 and Fζ(0) = H(0) = G(0) = 0. Since F is sense-preserving, we see that H is a locally biholomorphic function. For all ρ < Bloc,there is a disk D(ξ0, ρ) which is the biholomorphic image of a subdomain Π ofD underH. For ξ ∈ D(ξ0, ρ), let Φ(ξ) = F

(H−1(ξ)

). Then Φ(D(ξ0, ρ)) = F (Π).

By calculations, we have

Φξ =FζH ′ and Φξ =

Fζ

H ′ ,

which implies that Φ is also a Kr-quasiconformal harmonic mapping inD(ξ0, ρ). Since Φξ = Fζ/H

′ ≡ 1 and |Φξ| ≤ r, we find that for each ξ′ ∈ D(ξ0, ρ)with ξ′ �= ξ0,

|Φ(ξ′) − Φ(ξ0)| =

∣∣∣∣∣∣∣

∫

[ξ0, ξ′]

Φξ(ξ)dξ + Φξ(ξ)dξ

∣∣∣∣∣∣∣

≥

∣∣∣∣∣∣∣

∫

[ξ0, ξ′]

Φξ(ξ)dξ

∣∣∣∣∣∣∣−

∣∣∣∣∣∣∣

∫

[ξ0, ξ′]

Φξ(ξ)dξ

∣∣∣∣∣∣∣

≥ (1 − r)|ξ′ − ξ0|,where [ξ0, ξ′] denotes the line segment with endpoints ξ0 and ξ′. If ξ′ tendsto the boundary ∂D(ξ0, ρ), then f(D) contains a disk with the radius R ≥ρr(1 − r). Since

max0<r<1

[r(1 − r)

]=

14,

by letting ρ tend to Bloc, we conclude that f(D) contains a disk with theradius R ≥ Bloc

4 > ρ04 , where ρ0 = 1

2 + 2 × 10−8. The proof of this theorem iscomplete. �

Proof of Theorem 1.3. For z ∈ D and a fixed r ∈ (0, 1), let

F (z) = r−1f(rz) = H(z) +G(z),

where H(z) = r−1h(rz) and G(z) = r−1g(rz). Then F is also fully starlike andFz(0) = G(0) = H(0) = 0. For z ∈ D, let w(z) = Fz(z)/Fz(z). Then w(0) = 0and for z ∈ D,

|w(z)| =|Fz(z)||Fz(z)| =

|fz(rz)||fz(rz)| < r. (2.1)

This implies that f is a Kr-quasiconformal harmonic mapping in Dr, whereKr = (1+r)/(1−r). Differentiating both sides of the equation F−1(F (z)) = zyields the relations

(F−1)ζH ′ + (F−1)ζG′ = 1 and (F−1)ζG′ + (F−1)ζH ′ = 0,

Vol. 101 (2013) Planar harmonic univalent mappings 289

where ζ = F (z). Solving these two equations gives

(F−1)ζ =H ′

JFand (F−1)ζ = −G′

JF, (2.2)

where JF denotes the Jacobian of F . Since F (D) is fully starlike with respectto the origin, for each point z ∈ D and t ∈ [0, 1], ϕ(t) = F (z)t ∈ F (D). Letγ = F−1 ◦ ϕ and F (z) = |F (z)|eiθ0 . Define

H(F−1(ς)) = ς −G(F−1(ς)),

where ς ∈ F (D). By the chain rule, we have

τς = G′ · (F−1)ς and τς = G′ · (F−1)ς , (2.3)

where τ = G ◦ F−1. Applying (2.1) and (2.2) to (2.3), we obtain

|τς | + |τς | =|w|

1 − |w| ≤ r

1 − r.

Hence we have

|H(z)| ≤∫

ϕ

(1 + |Gς(F−1(ς))| + |Gς(F−1(ς))|) |dς|

=∫

ϕ

(1 + |τς(ς)| + |τς(ς)|) |dς|

≤ |F (z)|1∫

0

(

1 +r

1 − r

)

dt

=1

1 − r|F (z)|,

which implies

|H(z)| ≤ 11 − r

|F (z)|, i.e. |h(rz)| ≤ 11 − r

|f(rz)|.

Next, we prove the first inequality in (1.1). Applying (2.2), we see thatRe[e−iθ0G(z)

]

= Re

⎡

⎣e−iθ0

⎛

⎝∫

γ

G′(z) dz

⎞

⎠

⎤

⎦

= Re

⎡

⎢⎣e

−iθ0

⎛

⎜⎝

1∫

0

G′(γ(t))d

dtγ(t) dt

⎞

⎟⎠

⎤

⎥⎦

= Re

⎧⎪⎨

⎪⎩e−iθ0

⎡

⎢⎣

1∫

0

G′(γ(t))(ϕ′(t)

∂

∂ζF−1(ϕ(t)) + ϕ′(t)

∂

∂ζF−1(ϕ(t))

)dt

⎤

⎥⎦

⎫⎪⎬

⎪⎭

290 Sh. Chen et al. Arch. Math.

= |F (z)|Re

⎧⎪⎨

⎪⎩e−iθ0

⎡

⎢⎣

1∫

0

G′(γ(t))H ′(γ(t))eiθ0 − |G′(γ(t))|2e−iθ0

JF (γ(t))dt

⎤

⎥⎦

⎫⎪⎬

⎪⎭

≤ |F (z)|1∫

0

|H ′(γ(t))G′(γ(t))e−2iθ0 | − |G′(γ(t))|2JF (γ(t))

dt

≤ |F (z)|1∫

0

|w(γ(t))|1 + |w(γ(t))| dt

≤ r|F (z)|1 + r

,

which gives

Re

{G(z)F (z)

}

≤ r

1 + r.

Hence, we conclude that

|H(z)||F (z)| ≥ Re

{H(z)F (z)

}

= 1 − Re

{G(z)F (z)

}

≥ 11 + r

,

and therefore

|h(rz)| ≥ 11 + r

|f(rz)|.

The proof of this theorem is complete. �

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Sh. Chen

Department of Mathematics and Computational Science,Hengyang Normal University,Hengyang, Hunan 421008,People’s Republic of Chinae-mail: mathechen@126.com

S. Ponnusamy

Indian Statistical Institute (ISI), Chennai Centre,SETS (Society for Electronic Transactions and security),MGR Knowledge City, CIT Campus, Taramani,Chennai 600113,Indiae-mail: samy@isichennai.res.in;

samy@iitm.ac.in

X. Wang

Department of Mathematics,Hunan Normal University,Changsha, Hunan 410081,People’s Republic of Chinae-mail: xtwang@hunnu.edu.cn

Received: 29 January 2013