# Landau-Type Theorems for Certain Biharmonic Mappings

Authors: Ming-Sheng Liu; Zhen-Xing Liu; Jun-Feng Xu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

LetF(z)=|z|2g(z)+h(z)(|z|<1) be a biharmonic mapping of the unit disk 𝔻, where g and h are harmonic in 𝔻. In this paper, the Landau-type theorems for biharmonic mappings of the form L(F) are provided. Here L represents the linear complex operator L=(z∂/∂z)-(z¯∂/∂z)¯ defined on the class of complex-valued C1 functions in the plane. The results, presented in this paper, improve the related results of earlier authors.


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## Body

## 1. Introduction

Suppose thatf(z)=u(x,y)+iv(x,y),z=x+iy is a four times continuously differentiable complex-valued function in a domain D⊂ℂ. If f satisfies the biharmonic equation Δ(Δf)=0, then we call that f is biharmonic, where Δ represents the Laplacian operator:
(1)Δ=4∂2∂z∂z¯=∂2∂x2+∂2∂y2.Biharmonic functions arise in many physical situations, particularly in fluid dynamics and elasticity problems, and have many important applications in engineering (see [1] for details). It is known that a mapping F is biharmonic in a simply connected domain D if and only if F has the following representation:
(2)F(z)=|z|2g(z)+h(z),
where g(z) and h(z) are complex-valued harmonic functions in D [1]. Also, it is known that g(z) and h(z) can be expressed as
(3)g(z)=g1(z)+g2¯(z),z∈D,h(z)=h1(z)+h2¯(z),z∈D,
where g1, g2, k1, and k2 are analytic in D [2, 3].For a continuously differentiable mappingf in D, we define
(4)Λf(z)=max⁡0⩽θ⩽2π|fz+e-2iθfz¯|=|fz|+|fz¯|,λf(z)=min⁡0⩽θ⩽2π|fz+e-2iθfz¯|=||fz|-|fz¯||.
We use Jf to denote the Jacobian of f(5)Jf(z)=|fz(z)|2-|fz¯(z)|2.
Then Jf=λfΛf if Jf⩾0.In [4], the authors considered the following differential operator L defined on the class of complex-valued C1 functions:
(6)L=z∂∂z-z¯∂∂z¯.Evidently,L is a complex linear operator and satisfies the usual product rule:
(7)L(af+bg)=aL(f)+bL(g),L(fg)=fL(g)+gL(f),
where a and b are complex constants; f and g are C1 functions. In addition, the operator L possesses a number of interesting properties. For instance, it is easy to see that the operator L preserves both harmonicity and biharmonicity. Many other basic properties are stated in [4].Landau’s theorem states that iff is an analytic function on the unit disk 𝔻 with f(0)=f′(0)-1=0 and |f(z)|<M for z∈𝔻, then f is univalent in the disk 𝔻r0={z∈ℂ:|z|<r0} with r0=1/(M+M2-1), and f(𝔻r0) contains a disk 𝔻R0 with R0=Mr02. This result is sharp, with the extremal function f(z)=Mz((1-Mz)/(M-z)). Recently, many authors considered the Landau-type theorems for harmonic mappings [5–9] and biharmonic mappings [1, 4, 10–13]. Chen et al. [10] obtained the Landau-type theorems for biharmonic mappings of the form L(F) as follows.Theorem A (see [10]).
LetF(z)=|z|2g(z)+h(z) be a biharmonic mapping of the unit disk 𝔻={z∈ℂ:|z|<1} such that F(0)=h(0)=0 and Jh(0)=1, where g(z) and h(z) are harmonic in 𝔻. Assume that both |g(z)| and |h(z)| are bounded by M. Then there is a constant ρ1(0<ρ1<1) such that L(F) is univalent in 𝔻ρ1, where ρ1 satisfies the following equation:
(8)π4M-6Mρ12(1-ρ1)2-4Mρ13(1-ρ1)3-16Mπ2m1arctanρ1-4Mρ1(1-ρ1)3=0,
where m1≈6.059 is the minimum value of the function
(9)2-x2+(4/π)arctanxx(1-x2),
for 0<x<1. The minimum is attained at x≈0.588. Moreover, the range L(F)(𝔻ρ1) contains a schlicht disk 𝔻R1, where
(10)R1=ρ1[π4M-2Mρ12(1-ρ1)2-16Mπ2m1arctanρ1].Theorem B (see [10]).
LetF(z)=|z|2g(z) be a biharmonic mapping in 𝔻 such that g(0)=0, Jg(0)=1, and |g(z)|<M, where g(z) is harmonic in 𝔻. Then there is a constant ρ2(0<ρ2<1) such that L(F) is univalent in 𝔻ρ2, where ρ2 satisfies the following equation:
(11)π4M-48Mπ2m1arctanρ2-2Mρ2(1-ρ2)3=0,
where m1 is defined as in Theorem A. Moreover, L(F)(𝔻ρ2) contains a disk 𝔻R2 with
(12)R2=ρ23[π4M-16Mπ2m1arctanρ2].However, these results are not sharp. The main object of this paper is to improve Theorems A and B. We get three versions of Landau-type theorems for biharmonic mappings of the formL(F), where F belongs to the class of biharmonic mappings, and Theorems 11 and 14 improve Theorems A and B. In order to establish our main results, we need to recall the following lemmas.Lemma 1 (see [6, 14]).
Suppose thatf(z) is a harmonic mapping of the unit disk 𝔻 such that |f(z)|⩽M for all 𝔻. Then
(13)Λf(z)⩽4Mπ(1-|z|2),z∈𝔻.
The inequality is sharp.Lemma 2 (see [9, 12, 15]).
Suppose thatf(z)=h(z)+g(z)¯ is a harmonic mapping of the unit disk 𝔻 such that |f(z)|⩽M for all z∈𝔻 with h(z)=∑n=0∞‍anzn and g(z)=∑n=1∞‍bnzn. Then |a0|⩽M and for any n⩾1(14)|an|+|bn|⩽4Mπ.
These estimates are sharp.Lemma 3 (see [8, 11]).
Suppose thatf is a harmonic mapping of 𝔻 with f(0)=λf(0)-1=0. If Λf⩽Λ for z∈𝔻; then
(15)|an|+|bn|⩽Λ2-1nΛ,n=2,3,….
These estimates are sharp.Lemma 4 (see [11]).
Suppose thatf(z)=h(z)+g(z)¯ is a harmonic mapping of the unit disk 𝔻 such that |f(z)|⩽M for all z∈𝔻 with h(z)=∑n=0∞‍anzn and g(z)=∑n=1∞‍bnzn. If |Jf(0)|=1; then λf(0)⩾λ0(M), where M0=π/22π2-164≈1.1296 and
(16)λ0(M)={2M2-1+M2+1,1⩽M⩽M0,π4M,M>M0.Lemma 5 (see [13]).
Suppose thatf(z)=h(z)+g(z)¯ is a harmonic mapping of the unit disk 𝔻 with h(z)=∑n=0∞‍anzn and g(z)=∑n=1∞‍bnzn. If f satisfies |f(z)|⩽M for all z∈𝔻 and |Jf(0)|=1, then
(17)(∑n=2∞‍(|an|+|bn|)2)1/2⩽M4-1·λf(0).Lemma 6.
Suppose thatM>0, Λ⩾1. Then the equation
(18)φ(r)=1-12Mr2π(1-r2)-8Mr3π(1-r)3-Λ2-1Λ·2r-r2(1-r)2=0
has a unique root in (0,1).Proof .
It is easy to prove that the functionφ is continuous and strictly decreasing on [0,1), φ(0)=1>0, and lim⁡r→1-φ(r)=-∞. Hence, the assertion follows from the mean value theorem. This completes the proof.Lemma 7.
Suppose thatM1>0, M2⩾1, and λ0(M2) is defined by (16). Then the equation
(19)λ0(M2)-12M1r2π(1-r2)-8M1r3π(1-r)3-λ0(M2)M24-1·[2r4r2+r4+1(1-r2)5/2+rr4-3r2+4(1-r2)3/2]=0
has a unique root in (0,1).Lemma 8.
LetM⩾1. Then the equation
(20)1-M4-1·[3rr4-3r2+4(1-r2)3/2+2r4r2+r4+1(1-r2)5/2]=0
has a unique root in (0,1).Lemma 9.
For anyz1≠z2 in 𝔻r(0<r<1), we have
(21)∫01‍|tz1+(1-t)z2|2dt⩾|z1|3+|z2|33(|z1|+|z2|)>0.


## 2. Main Results

We first establish a new version of the Landau-type theorem for biharmonic mappings on the unit disk𝔻 as follows.Theorem 10.
LetF(z)=|z|2g(z)+h(z) be a biharmonic mapping of the unit disk 𝔻, with F(0)=h(0)=λF(0)-1=0, |g(z)|⩽M, and Λh(z)⩽Λ for z∈𝔻, where M>0,Λ⩾1. Then L(F) is univalent in the disk 𝔻r0, where r0 is the unique root in (0,1) of the equation
(22)1-12Mr2π(1-r2)-8Mr3π(1-r)3-Λ2-1Λ·2r-r2(1-r)2=0,
and L(F)(𝔻r0) contains a schlicht disk 𝔻σ0, where
(23)σ0=r0[1-Λ2-1Λ·r01-r0-4Mr02π(1-r0)2].Proof .
LetF(z)=|z|2g(z)+h(z) satisfy the hypothesis of Theorem 10, where
(24)g(z)=g1(z)+g2¯(z)=∑n=0∞‍anzn+∑n=0∞‍bn¯z¯n,h(z)=h1(z)+h2¯(z)=∑n=1∞‍cnzn+∑n=1∞‍dn¯z¯n
are harmonic in 𝔻. As L is linear and L(|z|2)=0, we may set
(25)H:=L(F)=|z|2L(g)+L(h).
Then we have(26)Hz=2|z|2gz+|z|2zgzz-z¯2gz¯+hz+zhzz,Hz¯=-2|z|2gz¯-|z|2z¯gz¯z¯+z2gz-hz¯-z¯hz¯z¯.
Note thatλF(0)=||c1|-|d1||=λh(0)=1; by Lemma 3, we have
(27)|cn|+|dn|⩽Λ2-1nΛ,n=2,3,….
Thus, forz1≠z2 in 𝔻r(0<r<r0), we have
(28)|H(z1)-H(z2)|=|∫[z1,z2]‍Hz(z)dz+Hz¯(z)dz¯|⩾|∫[z1,z2]‍hz(0)dz-hz¯(0)dz¯|-2|∫[z1,z2]‍|z|2(gzdz-gz¯dz¯)|-|∫[z1,z2]‍|z|2(zgzzdz-z¯gz¯z¯dz¯)|-|∫[z1,z2]‍zhzzdz-z¯hz¯z¯dz¯|-|∫[z1,z2]‍z2gzdz¯-z¯2gz¯dz|-|∫[z1,z2]‍(hz-hz(0))dzhhhhhhh-(hz¯-hz¯(0))dz¯∫[z1,z2]‍|.
Let(29)I1=|∫[z1,z2]‍hz(0)dz-hz¯(0)dz¯|,I2=|∫[z1,z2]‍|z|2(gzdz-gz¯dz¯)|,I3=|∫[z1,z2]‍|z|2(zgzzdz-z¯gz¯z¯dz¯)|,I4=|∫[z1,z2]‍zhzzdz-z¯hz¯z¯dz¯|,I5=|∫[z1,z2]‍z2gzdz¯-z¯2gz¯dz|,I6=|∫[z1,z2]‍(hz-hz(0))dz-(hz¯-hz¯(0))dz¯|.
By Lemmas1, 2, and 3, elementary calculations yield that
(30)I1⩾∫[z1,z2]‍λh(0)|dz|=λh(0)|z1-z2|=|z1-z2|,I2⩽∫[z1,z2]‍|z|2(|gz||dz|+|gz¯||dz¯|)⩽r2|z1-z2|Λg(z)⩽|z1-z2|4Mr2π(1-r2),I3⩽|z1-z2|∑n=2∞‍n(n-1)(|an|+|bn|)rn+1⩽|z1-z2|8Mr3π(1-r)3,I4⩽|z1-z2|∑n=2∞‍n(n-1)(|cn|+|dn|)rn-1⩽|z1-z2|Λ2-1Λ·r(1-r)2,I5⩽∫[z1,z2]‍(|z|2|gz||dz¯|+|z¯|2|gz¯||dz|)⩽r2|z1-z2|Λg(z)⩽|z1-z2|4Mr2π(1-r2),I6⩽|z1-z2|∑n=2∞‍n(|cn|+|dn|)rn-1⩽|z1-z2|Λ2-1Λ·r1-r.
Using these estimates and Lemma 6, we obtain
(31)|H(z1)-H(z2)|⩾I1-2I2-I3-I4-I5-I6⩾|z1-z2|[1-12Mr2π(1-r2)hhhhhhhhhhhh-8Mr3π(1-r)3-Λ2-1Λ·2r-r2(1-r)2]>0,
which implies H(z1)≠H(z2).
For anyz such that z∈∂𝔻r0, by Lemmas 2, 4, and 5, we obtain
(32)|H(z)|=||z|2(zgz-z¯gz¯)+(zhz-z¯hz¯)|⩾|zhz(0)-z¯hz¯(0)|-|z(hz-hz(0))-z¯(hz¯-hz¯(0))|-||z|2(zgz-z¯gz¯)|⩾r0[1-∑n=2∞‍(|cn|+|dn|)nr0n-1hhhhhh-∑n=1∞‍(|an|+|bn|)nr0n+1]⩾r0[1-Λ2-1Λ·r01-r0-4Mr02π(1-r0)2]=σ0.
This completes the proof.Next we improve Theorem A as follows.Theorem 11.
LetF(z)=|z|2g(z)+h(z) be a biharmonic mapping of the unit disk 𝔻, with F(0)=h(0)=JF(0)-1=0, |g(z)|⩽M1, and |h(z)|⩽M2 for z∈𝔻, where M1>0, M2⩾1. Then L(F) is univalent in the disk 𝔻r3, where r3 is the unique root in (0,1) of the equation
(33)λ0(M2)-12M1r2π(1-r2)-8M1r3π(1-r)3-λ0(M2)M24-1·[2r4r2+r4+1(1-r2)5/2+rr4-3r2+4(1-r2)3/2]=0,
and L(F)(𝔻r3) contains a schlicht disk 𝔻σ3, where λ0(M2) is defined by (16) and
(34)σ3=r3[r3r34-3r32+4(1-r32)3/2λ0(M2)-λ0(M2)M24-1hhhhhh·r3r34-3r32+4(1-r32)3/2-4M1r32π(1-r3)2].Proof.
Note thatJF(0)=|c1|2-|d1|2=Jh(0)=1; by Lemma 4, we have
(35)λh(0)⩾λ0(M2).
We adopt the same method in Theorem10, for z1≠z2 in 𝔻r(0<r<r3); by Lemmas 1, 2, and 5, we get
(36)I1⩾∫[z1,z2]‍λh(0)|dz|=λh(0)|z1-z2|,I2⩽∫[z1,z2]‍|z|2(|gz||dz|+|gz¯||dz¯|)⩽r2|z1-z2|Λg(z)⩽|z1-z2|4M1r2π(1-r2),I3⩽|z1-z2|∑n=2∞‍n(n-1)(|an|+|bn|)rn+1⩽|z1-z2|8M1r3π(1-r)3,I4⩽|z1-z2|∑n=2∞‍n(n-1)(|cn|+|dn|)rn-1⩽|z1-z2|(∑n=2∞‍(|cn|+|dn|)2)1/2·(∑n=2∞‍n2(n-1)2r2(n-1))1/2⩽|z1-z2|λh(0)M24-1·2r4r2+r4+1(1-r2)5/2,I5⩽∫[z1,z2]‍(|z|2|gz||dz¯|+|z¯|2|gz¯||dz|)⩽r2|z1-z2|Λg(z)⩽|z1-z2|4M1r2π(1-r2),I6⩽|z1-z2|∑n=2∞‍n(|cn|+|dn|)rn-1⩽|z1-z2|(∑n=2∞‍(|cn|+|dn|)2)1/2·(∑n=2∞‍n2r2(n-1))1/2⩽|z1-z2|λh(0)M24-1·rr4-3r2+4(1-r2)3/2.
Using these estimates and Lemma 7, by (35), we obtain
(37)|H(z1)-H(z2)|⩾I1-2I2-I3-I4-I5-I6⩾|z1-z2|(λh(0)-12M1r2π(1-r2)hhhhhhhhhh-8M1r3π(1-r)3-λh(0)M24-1hhhhhhhhhh·[2r4r2+r4+1(1-r2)5/2+rr4-3r2+4(1-r2)3/2])⩾|z1-z2|(λ0(M2)-12M1r2π(1-r2)hhhhhhhhhh-8M1r3π(1-r)3-λ0(M2)M24-1hhhhhhhhhh·[2r4r2+r4+1(1-r2)5/2+rr4-3r2+4(1-r2)3/2])>0,
which implies H(z1)≠H(z2).
For anyz such that z∈∂𝔻r3, by (35) and Lemmas 2 and 5, we obtain
(38)|H(z)|⩾r3[λh(0)-∑n=2∞‍(|cn|+|dn|)nr3n-1hhhhhh-∑n=1∞‍(|an|+|bn|)nr3n+1]⩾r3[r3r34-3r32+4(1-r32)3/2λh(0)-λh(0)M24-1hhhhhh·r3r34-3r32+4(1-r32)3/2-4M1r32π(1-r3)2]⩾r3[r3r34-3r32+4(1-r32)3/2λ0(M2)-λ0(M2)M24-1hhhhhh·r3r34-3r32+4(1-r32)3/2-4M1r32π(1-r3)2]=σ3.
This completes the proof.SettingM1=M2=M in Theorem 11, we have the following corollary.Corollary 12.
LetF(z)=|z|2g(z)+h(z) be a biharmonic mapping of the unit disk 𝔻, with F(0)=h(0)=JF(0)-1=0, and both g(z) and h(z) are bounded by M. Then L(F) is univalent in the disk 𝔻r1, where r1 is the minimum root of the equation
(39)λ0(M)-12Mr2π(1-r2)-8Mr3π(1-r)3-λ0(M)M4-1·[2r4r2+r4+1(1-r2)5/2+rr4-3r2+4(1-r2)3/2]=0,
and L(F)(𝔻r1) contains a schlicht disk 𝔻σ1, where
(40)σ1=r1[r1r14-3r12+4(1-r12)3/2λ0(M)-λ0(M)M4-1hhhhhh·r1r14-3r12+4(1-r12)3/2-4Mr12π(1-r1)2].In order to show Corollary12 improves Theorem A, we use Mathematica to compute the approximate values for various choices of M as in Table 1.Table 1
The values ofr1,σ1 are in Corollary 12. The values of ρ1,R1 are in Theorem A.
M
=
1
M
=
2
M
=
3
M
=
4
M
=
5
ρ
1
0.0527621
0.0139445
0.00626165
0.00353488
0.0022661
r
1
0.357671
0.0593158
0.0269865
0.015355
0.00988556
R
1
0.013793
0.00164514
0.00048245
0.00020277
0.00010364
σ
1
0.216467
0.0119479
0.00357231
0.00151701
0.00077955Remark 13.
From Table1 we can see, for the same M,
(41)r1>ρ1,σ1>R1.
Finally we improve Theorems B as follows.Theorem 14.
LetF(z)=|z|2g(z) be a biharmonic mapping in 𝔻 such that g(0)=0, Jg(0)=1 and |g(z)|<M, where M⩾1 and g(z) is harmonic in 𝔻. Then L(F) is univalent in the disk 𝔻r2, where r2 is the minimum positive root in (0,1) of the following equation:
(42)1-M4-1·[3rr4-3r2+4(1-r2)3/2+2r4r2+r4+1(1-r2)5/2]=0,
and L(F)(𝔻r2) contains a schlicht disk 𝔻σ2 with
(43)σ2=r23λ0(M)[1-M4-1·r2r24-3r22+4(1-r22)3/2],
where λ0(M) is defined by (16).Proof.
Let(44)g(z)=g1(z)+g2¯(z)=∑n=1∞‍anzn+∑n=1∞‍bn¯z¯n
LetH(z):=L(F)=|z|2L(g); then we have
(45)Hz=2|z|2gz-z¯2gz¯+z|z|2gzz,Hz¯=-2|z|2gz¯+z2gz-z¯|z|2gz¯z¯.
Forz1≠z2 in 𝔻r(0<r<r2), by Lemmas 4, 5, 8, and 9, we get
(46)|H(z1)-H(z2)|⩾|z1-z2|(∫01‍|tz1+(1-t)z2|2dt)×[λg(0)-3∑n=2∞‍n(|an|+|bn|)rn-1hhhhhhh-∑n=2∞‍n(n-1)(|an|+|bn|)rn-1]⩾|z1-z2||z1|3+|z2|33(|z1|+|z2|)×[λg(0)-3λg(0)M4-1·rr4-3r2+4(1-r2)3/2hhhhhhh-M4-1·λg(0)·2r4r2+r4+1(1-r2)5/2]⩾|z1-z2||z1|3+|z2|33(|z1|+|z2|)λ0(M)×[1-3M4-1·rr4-3r2+4(1-r2)3/2hhhhhhh-M4-1·2r4r2+r4+1(1-r2)5/2]>0,
which implies H(z1)≠H(z2).
For anyz such that z∈∂𝔻r2, by Lemmas 4 and 5, we obtain
(47)|H(z)|=|L(|z|2g)|⩾||z|2(zgz(0)-z¯gz¯(0))|-||z|2(z(gz-gz(0))-z¯(gz¯-gz¯(0)))|⩾r23[λg(0)-∑n=2∞‍n(|an|+|bn|)r2n-1]⩾r23λ0(M)[1-M4-1·r2r24-3r22+4(1-r22)3/2]=σ2.
This completes the proof of Theorem 14.In order to show Theorem14 improves Theorem B, we use Mathematica to compute the approximate values for various choices of M as in Table 2.Table 2
The values ofr2,σ2 are in Theorem 14. The values of ρ2,R2 are in Theorem B.
M
=
2
M
=
3
M
=
4
M
=
5
ρ
2
0.00623234
0.00277176
0.00155948
0.00099817
r
2
0.032209
0.0139701
0.00782686
0.00500376
R
2
6.54254
×
1
0
-
8
3.83564
×
1
0
-
9
5.12297
×
1
0
-
10
1.07466
×
1
0
-
10
σ
2
9.84416
×
1
0
-
6
5.35363
×
1
0
-
7
7.06092
×
1
0
-
8
1.47596
×
1
0
-
8Remark 15.
From Table2 we can see, for the same M,
(48)r2>ρ2,σ2>R2.



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