Jingjing Muand Xingdi Chen

Department of Mathematics,Huaqiao University,Quanzhou,Fujian 362021,P.R.China.

Landau-Type Theorems for Solutions of a Quasilinear Differential Equation

Jingjing Mu?and Xingdi Chen

Department of Mathematics,Huaqiao University,Quanzhou,Fujian 362021,P.R.China.

.In this paper,we study solutions of the quasilinear differential equationˉz?ˉzf(z)+z?zf(z)+(1?|z|2)?z?ˉzf(z)=f(z).We utilize harmonic mappings to obtain an explicit representation of solutions of this equation.By this result,we give two versions of Landau-type theorem under proper normalization conditions.

AMS subject classifications:30C99,30C62

Chinese Library classifications:O175.27

Harmonic mapping,biharmonic mapping,Landau’stheorem,quasilinear differential equation.

1 Introduction

Letf(z)=u(x,y)+iv(x,y)be a twice continuously differentiable function of the unit diskD={z∈C:|z|＜1}.Iffsatisfies the Laplacian equation Δf(z)=4fzˉz=0,then it is said to be aharmonic mapping.A harmonic mapping defined on a simply connected domain has a canonical expression

whereh(z)andg(z)are analytic onD.If a four time scontinuously differentiable functionfsatis fies ΔΔf(z)=0,thenit is said tobe abiharmonic mapping,which has a representation with

wherep(z)andq(z)are harmonic mappings ofD(see[2]).

For a continuously differentiable function f(z),z∈D,we write

Lewy[5]showed that a harmonic mapping is locally univalent if and only if its Jacobian Jf(z)does not vanish for any z∈D.

If f(z)is a harmonic mapping of D satisfying that limr→1f(reiθ)=f?(eiθ)and f?(eiθ)is a Lebesgue integrable function on T={z∈C:|z|=1},then

De fine a kernel function K(z)by

Olofsson[13]introduced a quasilinear differential equation

and proved

Theorem A.Suppose that f(z)∈C2(D)satis fies Eq.(1.2)and limr→1f(reiθ)=f?(eiθ),z=reiθ.If f?(eiθ)is a Lebesgue integrable function on T,then

In this paper,we show that the kernel function K(z)is a biharmonic mapping,which is also a solution of Eq.(1.2)(see Lemma 2.1 in Section 2).Moreover,we utilize harmonic mappings to give an explicit representation of the solution of the Eq.(1.2)(see Theorem 2.1 in Section 2).

The classical Landau’s theorem[9]states that if f(z)is an analytic function on D satisfying that f(0)=f′(0)?1=0 and|f(z)|≤ M for z∈D,then f is univalent in the diskThis result is sharp for the function

Recently,Landau’s theorem has been introduced in other classes of mappings,Chen,Gauthier and Hengartner [3]obtained two versions of Landau-type theorems for bounded harmonic mappings.Under different normalization conditions,those papers[8,10,14]improved some results in[3].Abdulhadi and Muhanna[1]gave two versions of Landautype theorems for biharmonic mappings,and then their results were improved by the papers[4,11,12,14].Among them,Zhu and Liu[14]obtained

Theorem B.Suppose that f(z)=|z|2g(z)+h(z)is a biharmonic mapping in the unit disk D such that|Jf(0)|=1,|g(z)|≤M1and|h(z)|≤M2for all z∈D.

(1)If M2＞1,or M2=1 and M1＞0,then f is univalent in the disk Dr1,and f(Dr1)contains a schlicht disk Dσ1={z∈D:|z?f(0)|≤σ1},where r1=r1(M1,M2)is the minimum of positive roots of the following equation:

and

where λ0(M2)is given by(3.2).

(2)If M2=1and M1=0,then f is univalent in D and f(D)=D.

Theorem C.Suppose that f(z)=|z|2g(z)+h(z)is a biharmonic mapping of the unit disk D,such that λf(0)=1,|g(z)|≤ M1and|h(z)|≤M2for z∈D.

(1)If M2＞1,or M2=1 and M1＞0,then f is univalent in the disk Dr2,and f(Dr2)contains a schlicht disk Dσ2={z∈C:|z? f(0)|≤σ2},where r2=r2(M1,M2)is the minimum of positive roots of the following equation:

and

(2)If M2=1 and M1=0,then f is univalent in D and f(D)=D.

In this paper,utilizing the representation(2.2)at Theorem 2.1,we obtain two versions of Landau-type theorem under the analogous normalization conditions given at the above Theorem B and Theorem C,correspondingly.

Theorem 1.1.Let g(z)be a harmonic mapping in D.Suppose f(z)is a solution of(1.2)with a representation

If g(0)=|Jg(0)|?1=0 and|g(z)|≤M(M≥1)for all z∈D,then f(z)is univalent in the disk Dρ1and f(Dρ1)contains a disk DR1,where ρ1is the minimum of positive roots of the equation φ(ρ)=0,R1=ψ(ρ1),and

where λ0(M)is given by(3.2).

Theorem 1.2.Let g(z)be a harmonic mapping in D.Suppose f(z)is a solution of(1.2)with a representation

If g(0)=λg(0)?1=0 and|g(z)|≤M(M≥1)for all z∈D,then f(z)is univalent in the disk Dρ2and f(Dρ2)contains a disk DR2,where ρ2is the minimum of positive roots of the following equation,

and

2 Explicit representation of solutions of the quasilinear differential equation

In order to give a representation of a solution of the quasilinear differential equation(1.2),we first give some properties of the kernel function K(z).

Lemma 2.1.The kernel functionis a solution of Eq.(1.2)and is also a biharmonic mapping.

Proof.By some elementary calculations,we get

Using the above three relations,we have

That is,K(z)is a solution of Eq.(1.2).

Since

one shows that K(z)is a biharmonic mapping,this completes the proof.

Next we give an explicit representation of solution of Eq.(1.2).

Theorem 2.1.Suppose that f(z)∈ C2(D),satisfies limr→1? f(reiθ)=f?(eiθ)with z=reiθ.If f?(eiθ)is a Lebesgue integrable function on the unit circle T,then a solution of Eq.(1.2)has a representation

where g(z)is a harmonic mapping in D.

Proof.By(2.1),we have

From Theorem A and the relation(2.3),we obtain

which shows that

Hence,f(z)is a biharmonic mapping.

Next,we will prove that f(z)are solutions of Eq.(1.2).Since

we have

Thus,

This completes the proof.

3 Proof of Theorem 1.1 and Theorem 1.2

In order to prove Theorem 1.1 and Theorem 1.2,we need the following lemmas.

Lemma A.[4,7]Suppose that f(z)is a harmonic mapping of the unit disk D such that|f(z)|≤M for all z∈D.Then

Moreover,the inequality(3.1)is sharp.

(1)If f satis fies|f(z)|≤M for all z∈D and|Jf(0)|=1,then

(2)If f satis fies|f(z)|≤M for all z∈D and λf(0)=1,then

Next,we will give prove the proofs of Theorem 1.1 and Theorem 1.2.

Proof of Theorem 1.1.Since g(z)is a harmonic mapping in the unit disk,g(z)can be written aswith

where h1(z)and g1(z)are analytic in D.By(2.5),we have

For fixed ρ∈(0,1),we choose z1,z2with z1/=z2,|z1|＜ρ and|z1|＜ρ.Let[z1,z2]be the line segment from z1to z2.Then,it follows that

and

Using Lemma A and the Cauchy-Schwarz inequality,we obtain

The combination of Lemma B with Lemma C gives us that

this implies that there exists a ρ1satisfying|f(z2)? f(z1)|＞0.For any|z′|=ρ1,we have

this completes the proof.

Remark 3.1.For any ρ∈(0,1),φ(ρ)is a continuous function.Since limρ→0+φ(ρ)=λ0(M)and limρ→1?φ(ρ)=?∞,the equation φ(ρ)has a root in(0,1).Since R1=ψ(ρ1)＞ρ1φ(ρ1),we have R1＞0.

Proof of Theorem 1.2.Notice that λf(0)=1.We replace λ0(M)in(1.4)and(1.5)with λf(0).Using the same method as the proof of Theorem 1.1,we obtain the proof of Theorem 1.2 from the relation(3.4)at Lemma C. □

The authors are grateful to the referees for their useful comments and suggestion.This work is partially supported by NNSF of China(11101165),the Natural Science Founda-tion of Fujian Province of China(2014J01013),NCETFJ Fund(2012FJ-NCET-ZR05),Pro-motion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(ZQN-YX110).

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19 May 2014;Accepted 12 September 2014

?Corresponding author.Email addresses:mujingjing123@163.com(J.J.Mu),chxtt@hqu.edu.cn(X.D.Chen)