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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 379130, 10 pages
On Certain Classes of Biharmonic Mappings Defined by Convolution
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Received 1 March 2012; Accepted 7 August 2012
Academic Editor: Saminathan Ponnusamy
Copyright © 2012 J. Chen and X. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce a class of complex-valued biharmonic mappings, denoted by , together with its subclass , and then generalize the discussions in Ali et al. (2010) to the setting of and in a unified way.
A four times continuously differentiable complex-valued function in a domain is biharmonic if , the Laplacian of , is harmonic in . Note that is harmonic in if satisfies the biharmonic equation in , where represents the Laplacian operator
It is known that, when is simply connected, a mapping is biharmonic if and only if has the following representation: where are complex-valued harmonic mappings in for (cf. [1####^~^~^~^~^~^####x2013;6]). Also it is known that can be expressed as the form for , where all and are analytic in (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####^~^~^~^~^~^####x2013;11]). However, the investigation of biharmonic mappings in the context of geometric function theory is a recent one (cf. [1####^~^~^~^~^~^####x2013;6]).
In this paper, we consider the biharmonic mappings in . Let denote the set of all biharmonic mappings in with the following form: with , , , and .
Let denote the set of all univalent harmonic mappings in , where with . In particular, we use to denote the set of all mappings in with . Obviously, .
In 1984, Clunie and Sheil-Small  discussed the class and its geometric subclasses. Since then, there have been many related papers on and its subclasses (see [13, 14] and the references therein). In 1999, Jahangiri  studied the class consisting of all mappings such that and are of the form and satisfy the condition in , where .
For two analytic functions and , if then the convolution of and is defined by
Now we consider a class of biharmonic mappings, denoted by , as follows: with the form (1.4) is said to be in if and only if where are analytic in for , is a constant, , , are constants with , , and . Here and in what follows, ####^~^~^~^~^~^####x201c;####^~^~^~^~^~^####x201d; always stands for ####^~^~^~^~^~^####x201c;####^~^~^~^~^~^####x201d;.
Obviously, if , and , then reduces to , and if , and , then reduces to .
Further, we use to denote the class consisting of all mappings in with the form where
The object of this paper is to generalize the discussions in  to the setting of and in a unified way. The organization of this paper is as follows. In Section 2, we get a convolution characterization for . As a corollary, we derive a sufficient coefficient condition for mappings in to belong to . The main results are Theorems 2.1 and 2.3. In Section 3, first, we get a coefficient characterization for , and then find the extreme points of . The corresponding results are Theorems 3.1 and 3.6.
2. A Convolution Characterization
We begin with a convolution characterization for .
Theorem 2.1. Let . Then if and only if for all and all with .
Proof. By definition, a necessary and sufficient condition for a mapping in to be in is given by (1.13). Let
Then , and so the condition (1.13) is equivalent to
for all and all with and . Obviously, (2.3) holds if and only if
Straightforward computations show that from which we see that (2.3) is true if and only if so is (2.1). The proof is complete.
As an application of Theorem 2.1, we derive a sufficient condition for mappings in to be in in terms of their coefficients.
Theorem 2.3. Let . Then if here and in the following, , where , and are constants.
Proof. For given by (1.4), we see that ####^~^~^~^~^~^####x2009;If is the identity, obviously, .####^~^~^~^~^~^####x2009;If is not the identity, then Hence the assumption implies that for all and all with . It follows from Theorem 2.1 that .
3. A Coefficient Characterization and Extreme Points
We start with a coefficient characterization for .
Theorem 3.1. Let with , and let be of the form (1.15). Then if and only if
Proof. By similar arguments as in the proof of Theorem 2.3, we see that it suffices to prove the ####^~^~^~^~^~^####x201c;only if####^~^~^~^~^~^####x201d; part. For , obviously, (1.13) is equivalent to in , where Letting through real values leads to the desired inequality. So the proof is complete.
It follows from Theorem 3.1 that we have the following.
Corollary 3.3. Let with and . If , then for , one has The result is sharp with equality for mappings
Corollary 3.4. Under the hypotheses of Corollary 3.3, one has that is closed under the convex combination.
Definition 3.5. Let be a topological vector space over the field of complex numbers, and let be a subset of . A point is called an extreme point of if it has no representation of the form () as a proper convex combination of two distinct points and in (cf. ).
We now determine the extreme points of .
Theorem 3.6. Let (1), (2), (3) for and all , (4) for and all . Under the hypotheses of Corollary 3.3, one has that if and only if it can be expressed as
where , all other and are nonnegative, and .
In particular, the extreme points of are all mappings and listed in , , and above.
Proof. It follows from the assumptions that
and so Theorem 3.1 implies that .
Conversely, assume , and let for and all . Then The proof of the theorem is complete.
The research was partly supported by NSFs of China (No. 11071063).
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