Landau’s theorem for certain biharmonic mappings

Abstract

In this paper, we show the existence of Landau and Bloch constants for biharmonic mappings of the form $L(F)$. Here L represents the linear complex operator $L=z\frac{\mathrm{\partial }}{\mathrm{\partial }z}-\overline{z}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}$ defined on the class of complex-valued $C^1$ functions in the plane, and F belongs to the class of biharmonic mappings of the form $F(z)=|z{|}^{2}G(z)+K(z)(|z|<1)$, where G and K are harmonic.

Introduction

A four times continuously differentiable complex-valued function $F=u+\mathit{iv}$ in a domain $D\subset \mathbb{C}$ is biharmonic if and only if the Laplacian of F is harmonic. Note that $\mathrm{\Delta }F$ is harmonic in D if F satisfies the biharmonic equation $\mathrm{\Delta }(\mathrm{\Delta }F)=0$, where $\mathrm{\Delta }$ represents the Laplacian operator

$$\mathrm{\Delta }=4\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }z\mathrm{\partial }\overline{z}}\text{.}$$

Biharmonic functions arise in a lot of physical situations, particularly in fluid dynamics and elasticity problems, and have many important applications in engineering and biology (see [10], [12], [13] for the details).

It has been shown that a mapping F is biharmonic in a simply connected domain D if and only if F has the following representation:

$$F=|z{|}^{2}G+K\text{,}$$

where G and K are complex-valued harmonic functions in D (see [2], [3]). Also, it is known that (see [5], [7]) G and K can be expressed as

$$G=g_1+\overline{g_2}\mathrm{and}K=k_1+\overline{k_2}\text{,}$$

where $g_1$, $g_2$, $k_1$ and $k_2$ are analytic in D. The Jacobian $J_f$ of f is defined by

$$J_f=|f_z{|}^2-|f_{\overline{z}}{|}^2\text{.}$$

We now introduce

$${\lambda }_f=||f_z|-|f_{\overline{z}}||\mathrm{and}{\Lambda }_f=|f_z|+|f_{\overline{z}}|\text{.}$$

Then $J_f={\lambda }_f{\Lambda }_f$ if $J_f⩾0$.

In [3], the authors considered the following differential operator L defined on the class of complex-valued $C^1$ functions:

$$L=z\frac{\mathrm{\partial }}{\mathrm{\partial }z}-\overline{z}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\text{.}$$

Clearly, L is a complex linear operator and satisfies the usual product rule:

$$L(\mathit{af}+\mathit{bg})=\mathit{aL}(f)+\mathit{bL}(g)\mathrm{and}L(\mathit{fg})=\mathit{fL}(g)+\mathit{gL}(f)\text{,}$$

where $a\text{,}b$ are complex constants, f and g are $C^1$ 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 [3].

Throughout the paper, we use the notation ${\mathbb{D}}_r=\{z\in \mathbb{C}:|z|<r\}$, $\mathbb{D}={\mathbb{D}}_1$, and we now recall one of the main results from [3].

Theorem A [3, Corollary 1(3)]

Let F be a univalent biharmonic function in $\mathbb{D}$. If F is convex and $L(F)$ is univalent, then $L(F)$ is starlike.

At this place, it is also important to recall that (see [3, Corollary 1(3)]) the operator $L(F)$ for biharmonic functions behaves much like ${\mathit{zf}}^{\prime }(z)$ for analytic functions, for example in the sense that for F univalent and biharmonic, F is starlike if and only if $\mathrm{Re}(L(F)(z)/F(z))⩾0$. A similar characterization has also been obtained by the authors for convex (biharmonic) functions.

The classical theorem of Landau shows that there exists a $\rho =\rho (M)>0$ such that every function f, analytic in the unit disk $|z|<1$ with $f(0)=f^{\prime }(0)-1=0$ and $|f(z)|<M$ in $|z|<1$, is univalent in the disk ${\mathbb{D}}_{\rho }$ and in addition, the range $f({\mathbb{D}}_{\rho })$ contains a disk of radius $M{\rho }^2$. Recently, many authors considered Landau’s theorem for planar harmonic mappings (see, for example, [4], [6], [8], [14]) and biharmonic mappings (see [1]). From Theorem A, we see that it is significant to consider Landau’s theorem for biharmonic mappings of the form $L(F)$, where F belongs to the class of biharmonic mappings. The main aim of this paper is to consider this problem. Our main results are the following.

Theorem 1.1

Let $F=|z{|}^{2}G+K$ be a biharmonic mapping in $\mathbb{D}$ such that $F(0)=K(0)=0$, $J_K(0)=1$, where G and K are harmonic in $\mathbb{D}$. Assume that both $|G|$ and $|K|$ are bounded by M. Then there is a constant $\rho$ $(0<\rho <1)$ such that $L(F)$ is univalent in ${\mathbb{D}}_{\rho }$, where $\rho$ satisfies the following equation:

$$\frac{\pi }{4M}-\frac{6M{\rho }^2}{(1-\rho {)}^2}-\frac{4M{\rho }^3}{(1-\rho {)}^3}-\frac{16M}{{\pi }^2}{m}_1\arctan \rho -\frac{4M\rho }{(1-\rho {)}^3}=0\text{,}$$

where $m_1\approx 6.059$ is the minimum value of the function

$$\frac{2-x^2+\frac{4}{\pi }\arctan x}{x(1-x^2)}$$

for $0<x<1$. The minimum is attained at $x\approx 0.588$. Moreover, the range $L(F)({\mathbb{D}}_{\rho })$ contains a schlicht disk ${\mathbb{D}}_{R_1}$, where

$$R_1=\rho \left[\frac{\pi }{4M}-\frac{2M{\rho }^2}{(1-\rho {)}^2}-\frac{16M}{{\pi }^2}{m}_1\arctan \rho \right]\text{.}$$

It is important to remark that the bounds of G and K are not preserved under the differential operator L. In our next result we deal with the case $K=0$. Then the result differs from Theorem 1.1, because for $K=0$, the Jacobian $J_K(0)=0$. Therefore, we need to assume that $J_G(0)=1$ instead.

Theorem 1.2

Let $F=|z{|}^{2}G$ be a biharmonic mapping in $\mathbb{D}$ such that $G(0)=0$, $J_G(0)=1$ and $|G(z)|<M$, where G is harmonic in $\mathbb{D}$. Then there is a constant $\rho$ $(0<\rho <1)$ such that $L(F)$ is univalent in ${\mathbb{D}}_{\rho }$, where $\rho$ satisfies the following equation:

$$\frac{\pi }{4M}-\frac{48M}{{\pi }^2}{m}_1\arctan \rho -\frac{2M\rho }{(1-\rho {)}^3}=0\text{,}$$

where $m_1$ is defined as in Theorem 1.1. Moreover, $L(F)({\mathbb{D}}_{\rho })$ contains a schlicht disk ${\mathbb{D}}_{R_2}$ with

$$R_2={\rho }^3\left[\frac{\pi }{4M}-\frac{16M}{{\pi }^2}{m}_1\arctan \rho \right]\text{.}$$

Denote the $\rho$ values in Theorem 1.1, Theorem 1.2 by ${\rho }_1$ and ${\rho }_2$, respectively. In the left half of Table 1, we indicate the precise values of these ${\rho }_1$’s and the corresponding values of $R_1$’s for various choices of M. These values are obtained using Mathematica.

Remark 1.3

Theorem 1.1, Theorem 1.2 are not sharp.

Also, it could be of some interest to see some numerical examples to compare the situation for biharmonic F obtained in [1] and for $L(F)$ obtained in the present article. Thus, for a computational comparison with the recent results from [1], we recall their results in Theorem B, Theorem C to present the corresponding numerical values in Table 1, Table 2. The numerical values given in these two tables are self-explanatory.

Theorem B

Let $F=|z{|}^{2}G+K$ be a biharmonic mapping of the unit disk $\mathbb{D}$, where G and K are harmonic in $\mathbb{D}$ such that $F(0)=K(0)=0$, $J_K(0)=1$, and both $|G|$ and $|K|$ are bounded by M. Then there is a constant $0<{\rho }_3<1$ so that F is univalent for $|z|<{\rho }_3$. In specific ${\rho }_3$ satisfies

$$\frac{\pi }{4M}-2{\rho }_{3}M-2M\left(\frac{{\rho }_3^2}{(1-{\rho }_3{)}^2}+\frac{1}{(1-{\rho }_3{)}^2}-1\right)=0$$

and $F({\mathbb{D}}_{{\rho }_3})$ contains a schlicht disk ${\mathbb{D}}_{R_3}$, where

$$R_3=\frac{\pi {\rho }_3}{4M}-\frac{2M({\rho }_3^3+{\rho }_3^2)}{1-{\rho }_3}\text{.}$$

This result is not sharp.

Theorem C

Let G be harmonic in $\mathbb{D}$ such that $G(0)=0$, $J_G(0)=1$ and $|G(z)|<M$. Then there is a constant $0<{\rho }_4<1$ so that $F=|z{|}^{2}G$ is univalent in the disk $|z|<{\rho }_4$. ${\rho }_4$ is the solution of the equation

$$\frac{\pi }{4M}-\frac{4{\rho }_{4}M}{1-{\rho }_4}-2M\left(\frac{1}{(1-{\rho }_4{)}^2}-1\right)=0$$

and $F({\mathbb{D}}_{{\rho }_4})$ contains a disk ${\mathbb{D}}_{R_4}$ with

$$R_4=\frac{\pi {\rho }_4^3}{4M}-\frac{2M{\rho }_4^4}{1-{\rho }_4}\text{.}$$

The result is not sharp.

In Section 2, some elementary results will be recalled. The proofs of the main theorems will be presented in Sections 3 Proof for the case, 4 Proof for the case.