On Certain Classes of Biharmonic Mappings Defined by Convolution

and Applied Analysis 3 and satisfy the condition ∂ ∂θ ( arg f ( re )) Re { zh′ − zg ′ h g } > α 1.8 in D, where 0 ≤ α < 1. For two analytic functions f1 and f2, if f1 z ∞ ∑ j 1 ajz j , f2 z ∞ ∑ j 1 Ajz j , 1.9 then the convolution of f1 and f2 is defined by ( f1 ∗ f2 ) z f1 z ∗ f2 z ∞ ∑ j 1 ajAjz j . 1.10 By using the convolution, in 16 , Ali et al. introduced the class SH φ, σ, α of harmonic mappings in the form of 1.6 such that Re ⎧ ⎨ ⎩ z ( h ∗ φ′ z − σzg ∗ φ′ z ( h ∗ φ z σg ∗ φ z ⎫ ⎬ ⎭ > α 1.11 and the class SP 0 H φ, σ, α such that Re ⎧ ⎨ ⎩ ( 1 e )z ( h ∗ φ′ z − σzg ∗ φ′ z ( h ∗ φ z σg ∗ φ z − e ⎫ ⎬ ⎭ > α, 1.12 where σ ∈ R and α ∈ 0, 1 are constants, γ ∈ R and φ z z ∞n 2 φnz is analytic in D. Now we consider a class of biharmonic mappings, denoted by BH0 φk;σ, a, b , as follows: F ∈ BH0 D with the form 1.4 is said to be in BH0 φk;σ, a, b if and only if Re { a Φ z Ψ z − b } > 0, 1.13


Introduction
A four times continuously differentiable complex-valued function F u iv in a domain D ⊂ C is biharmonic if ΔF, the Laplacian of F, is harmonic in D. Note that ΔF is harmonic in D if F satisfies the biharmonic equation Δ ΔF 0 in D, where Δ represents the Laplacian operator It is known that, when D is simply connected, a mapping F is biharmonic if and only if F has the following representation: where G k are complex-valued harmonic mappings in D for k ∈ {1, 2} cf.1-6 .Also it is known that G k can be expressed as the form for k ∈ {1, 2}, where all h k and g k are analytic in D cf. 7, 8 .Biharmonic mappings arise in a lot of physical situations, particularly, in fluid dynamics and elasticity problems, and have many important applications in engineering and biology cf.9-11 .However, the investigation of biharmonic mappings in the context of geometric function theory is a recent one cf.1-6 .
In this paper, we consider the biharmonic mappings in D {z ∈ C : |z| < 1}.Let BH 0 D denote the set of all biharmonic mappings F in D with the following form: In 12 , Qiao and Wang proved that for each F ∈ BH 0 D , if the coefficients of F satisfy the following inequality: then F is sense preserving, univalent, and starlike in D see 12, Theorems 3.1 and 3.2 .
Let S H denote the set of all univalent harmonic mappings f in D, where In particular, we use S 0 H to denote the set of all mappings in S H with b 1 0. Obviously, S 0 H ⊂ BH 0 D .In 1984, Clunie and Sheil-Small 7 discussed the class S H and its geometric subclasses.Since then, there have been many related papers on S H and its subclasses see 13, 14 and the references therein .In 1999, Jahangiri 15 studied the class S * H α consisting of all mappings f h g such that h and g are of the form and satisfy the condition in D, where 0 ≤ α < 1.
For two analytic functions f 1 and f 2 , if A j z j , 1.9 then the convolution of f 1 and f 2 is defined by By using the convolution, in 16 , Ali et al. introduced the class S 0 H φ, σ, α of harmonic mappings in the form of 1.6 such that Re and the class SP 0 H φ, σ, α such that Re where σ ∈ R and α ∈ 0, 1 are constants, γ ∈ R and φ z z ∞ n 2 φ n z n is analytic in D. Now we consider a class of biharmonic mappings, denoted by BH 0 φ k ; σ, a, b , as follows: where where The object of this paper is to generalize the discussions in 16 to the setting of BH 0 φ k ; σ, a, b and TBH 0 φ k ; σ, a, b in a unified way.The organization of this paper is as follows.In Section 2, we get a convolution characterization for BH 0 φ k ; σ, a, b .As a corollary, we derive a sufficient coefficient condition for mappings in BH 0 D to belong to BH 0 φ k ; σ, a, b .The main results are Theorems 2.1 and 2.3.In Section 3, first, we get a coefficient characterization for TBH 0 φ k ; σ, a, b , and then find the extreme points of TBH 0 φ k ; σ, a, b .The corresponding results are Theorems 3.1 and 3.6.

A Convolution Characterization
We begin with a convolution characterization for BH 0 φ k ; σ, a, b .
for all z ∈ D \ {0} and all x ∈ C with |x| 1.

Abstract and Applied Analysis 5
Proof.By definition, a necessary and sufficient condition for a mapping F in BH 0 D to be in BH 0 φ k ; σ, a, b is given by 1.13 .Let Then G 0 1, and so the condition 1.13 is equivalent to for all z ∈ D \ {0} and all x ∈ C with |x| 1 and x / − 1. Obviously, 2.3 holds if and only if Straightforward computations show that As an application of Theorem 2.1, we derive a sufficient condition for mappings in BH 0 D to be in BH 0 φ k ; σ, a, b in terms of their coefficients.
Abstract and Applied Analysis here and in the following, z max max γ∈R {|x ye iγ |} x y, where z x ye iγ , x and y ∈ 0, ∞ are constants.
Proof.For F given by 1.4 , we see that

2.7
If F is the identity, obviously, L z |z|.
If F is not the identity, then

2.8
Hence the assumption implies that L z > 0 for all z ∈ D \ {0} and all x ∈ C with |x|

A Coefficient Characterization and Extreme Points
We start with a coefficient characterization for TBH 0 φ k ; σ, a, b .
Theorem 3.1.Let φ k z z ∞ j 2 φ k,j z j with φ k,j ≥ 0, and let F be of the form 1.15 .Then Proof.By similar arguments as in the proof of Theorem 2.3, we see that it suffices to prove the "only if" part.For F ∈ TBH 0 φ k ; σ, a, b , obviously, 1.13 is equivalent to

3.3
Letting z → 1 − through real values leads to the desired inequality.So the proof is complete.

3.4
The result is sharp with equality for mappings In particular, the extreme points of TBH 0 φ k ; σ, a, b are all mappings h kj and g kj listed in 1 , 3 , and 4 above.
Proof.It follows from the assumptions that x kj z j for k ∈ {1, 2} and all j ≥ 2. Then b k,j z j .

3.10
The proof of the theorem is complete.

σ 2 k 1 ∞ j 2 |z| 2 2 j
k−1 a − b σ 2 j a max b max φ k,j y kj z j , a max − b max a − b φ k,j • a − b j a max − b max φ k,so Theorem 3.1 implies that F ∈ TBH 0 φ k ; σ, a, b .Abstract and Applied Analysis 9 Conversely, assume F ∈ TBH 0 φ k ; σ, a, b , and let x 21 y 11 y 21 0, max − b max φ k,j a k,j a − b , y kj σ 2 j a max b max φ k,j b k,j a − b , 3.9 5 from which we see that 2.3 is true if and only if so is 2.1 .The proof is complete.If h 2 g 2 0, a 1 and b α, then Theorem 2.1 coincides with Theorem 2.1 in 16 , and if h 2 g 2 0, a 1 e iγ , and b α e iγ , then Theorem 2.1 coincides with Theorem 2.3 in 16 .
If h 2 g 2 0, a 1 and b α, then Theorem 2.3 coincides with Theorem 2.2 in 16 , and if h 2 g 2 0, a 1 e iγ and b α e iγ , then Theorem 2.3 coincides with Theorem 2.4 in 16 .
Let X be a topological vector space over the field of complex numbers, and let E be a subset of X.A point x ∈ E is called an extreme point of E if it has no representation of the form x ty 1 − t z 0 < t < 1 as a proper convex combination of two distinct points y and z in E cf. 17 .wherex21 y 11 y 21 0, all other x kj and y kj are nonnegative, and2 We now determine the extreme points of TBH 0 φ k ; σ, a, b .