Abstract

We introduced a new subclass of univalent harmonic functions defined by the shear construction in the present paper. First, we showed that the convolutions of two special subclass harmonic mappings are convex in the horizontal direction. Secondly, we proved a necessary and sufficient condition for the above subclass of harmonic mappings to be convex in the horizontal direction. We also presented some basic examples of univalent harmonic functions explaining the behavior of the image domains.

1. Introduction

Let be a continuous complex-valued harmonic mapping in the open unit disc , where and are real-valued harmonic functions in . Such functions can be expressed as ; here is known as the analytic part and the coanalytic part of , respectively. The Jacobian of the mapping is given by . A necessary and sufficient condition (see [1] or [2]) for to be locally univalent and sense-preserving in is that , or equivalently, there exists an analytic complex dilatation of such that

We denote by the class of harmonic, sense-preserving, and univalent mappings in , normalized by the conditions and . Thus, a harmonic mapping in the class can be expressed as , where

Let be the subclass of whose members satisfy the additional condition . Let and be the subclasses of whose image domains are convex and close to convex, respectively.

A domain is said to be convex in the direction , , if every line parallel to the line joining and has a connected intersection with . In particular, a domain is said to be convex in horizontal direction (CHD) if its intersection with each horizontal line is connected (or empty). In this paper, a function is called a mapping if maps onto a domain. We denote by the class of all mappings. Clearly, .

The shear construction is essential to the present work as it allows one to study harmonic functions through their related analytic functions (see [3]); the shear construction produces a univalent harmonic function that maps to a region, that is, . This construction relies on the following theorem due to Clunie and Sheil-Small.

Theorem A (see [1]). A harmonic function locally univalent in is a univalent mapping of onto a domain convex in the horizontal direction if and only if is a conformal univalent mapping of onto a domain convex in the horizontal direction.

For two harmonic functions; and , their convolution is denoted by and defined as follows:

One can find the recent results involving harmonic convolutions in [48]. In [9, 10], explicit descriptions are given for half-plane and strip mappings. Specifically, the collection of the mappings that map onto the right half-plane, . Such mappings satisfy the condition

In [4], the following result was derived.

Theorem B. Let , be the right half-plane mappings. If is locally univalent and sense-preserving, then .

Let be the canonical right half-plane mapping with the dilatation ; then

Recently, Dorff et al. [5] obtained some results involving convolutions of with right half-plane mappings and vertical strip mappings. They proved the following.

Theorem C. Let with and . If , then .

Theorem D. Let with , where and . If , then .

In this paper, we consider the harmonic mapping satisfies the condition and when , applying the shear construction, we have Hence If is analytic in and , then

We have known that . The image of under the harmonic mapping is shown in Figure 1.

Writing , we can express as

The generalization is done in the following way. For a univalent analytic function with , define Clearly, .

Obviously, Hence So .

If is analytic in and , then we have

In [5], authors showed that Theorems C and D do not hold for . In the present paper, we construct a new subclass of harmonic mappings defined by (11). In Section 2, we show that convolutions of with (where and dilatation ) are in the subclass for and for all . In Section 3, we apply the transformation to a normalized analytic univalent function and show that the sufficient and necessary condition for is that is convex. Furthermore, we present some basic examples of harmonic mappings satisfying the conditions of the theorems and illustrate them graphically with the help of the Mathematica software. With these examples we explain the behavior of the image domains.

2. The Convolution of

To prove our main results, we need several lemmas.

Lemma 1 (see [11]). Let be an analytic function in with and , and let where . If then is convex in the horizontal direction.

Lemma 2. Let be a mapping defined by (12) and with and dilatation . Then , the dilatation of , is given by

Proof. Since and , then . We immediately get From (14), we have

Lemma 3. Let with . If is locally univalent, then .

Proof. Recall that and Hence Thus Next, we will show that is convex in the horizontal direction.
Letting , we havewhere ,   and , . So is convex in the horizontal direction by Lemma 1, and then is convex in the horizontal direction. Finally, since we assumed that is locally univalent, we apply Theorem A to get the desired result.

Now we turn to the distribution theory of the roots of polynomial for the unit disc. Given a polynomial of degree : with complex coefficients, the parallel algorithm for finding zeros of polynomial (24) inside the unit disc is worth studying. Let it is easy to verify that the zeros of (24) and (25) are inverse points with respect to the unit disc . To derive the main judging theorem, we need the following lemma.

Lemma 4 (see [12]). If all zeros of (24) are inside the unit disc , then .

By Lemma 4, we can obtain the following result.

Lemma 5. If  , then not all zeros of (24) are inside the unit disc .

Next, we construct a function sequence where where . If , then by Lemma 5 we know that not all roots of are inside the unit disc ; we would not need to construct the function . If , then use (28) to construct (26). The following lemma is a necessary and sufficient condition of all zeros of (24) lying inside the unit disc.

Lemma 6 (see [12]). A necessary and sufficient condition that all zeros of (24) inside the unit disc is , where and are given by (27).

The main result of this section is the following.

Theorem 7. Let be a mapping defined by (12) and with and dilatation . Then for .

Proof. In view of Lemma 3, it suffices to show that the dilatation of satisfies , for all . Setting into (17), we getwhere and .
Obviously, if is a zero of , the is a zero of . Hence, if are the zeros of , we can write
Now for ,   maps onto . So in order to prove our result, we will show that all zeros of named lie inside the unit disc for . In the following we divide our proof in two cases.
Case  1 (when ). In this case, substituting into (29), we see that Case  2 (when ). From (31), it is enough to show that all zeros of (30) are inside the unit disc for . Since , we can apply (28) to ; thus we have where and .
Since for , by using (28) on again, we get where and .
Since ; we have Continuing in this manner we derive that where and then . Hence, let , we have If , , then ; by Lemma 6, thus, lie inside unit disc.
If , then . As is a zero of , therefore we can write It suffices to show that zeros of lie inside unit disc. Since whenever , by applying (28) on , we obtain analogously that all zeros of lie inside unit disc.
To sum up, we showed that all zeros of lie inside or on the unit circle for . The proof of our theorem is now completed.

Next we give an example showing given values of , , and in Theorem 7.

Example 8. In Theorem 7, if we take , , then and , and we get . Let ; then and . From (7) and (9), we have
The images of concentric circles inside under the harmonic mapping and concentric circles under the convolution map are shown in Figures 1 and 2, respectively.

3. Harmonic Univalent Mappings Convex in One Direction

For the local univalence of defined by (11), we have the following.

Lemma 9. The function defined by (11) is locally univalent if and only if is convex.

Proof. Write . By (1), is locally univalent if and only if for all , and will be locally univalent if and only if or equivalently, since and , It can be easily seen that the inequality above is equivalent to which is the analytic condition for convexity. Thus, is locally univalent if and only if is convex.

We immediately have the following result.

Theorem 10. The function is defined by (11) if and only if is convex.

Proof. Again write . Since is convex, particularly, is convex in the horizontal direction; by Lemma 9 and Theorem A, the proof is complete.

The following example satisfies the condition of Theorem 10.

Example 11. Consider the univalent analytic function ; obviously, , and So, is convex analytic in . Therefore, in view of Theorem 10, the function . However Now, we let ; then The images of under and are shown in Figures 3 and 4, respectively. Obviously, we see that the image of under is convex in the horizontal direction.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors completed the paper together. They also read and approved the final paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grants 11301008 and 11226088, the Key Courses Construction of Honghe University underGrant ZDKC1003 and the Scientific Research Fund of Yunnan Province under Grant 2010ZC150.