Abstract
In this paper, we establish some results concerning the convolutions of harmonic mappings convex in the horizontal direction with harmonic vertical strip mappings. Furthermore, we provide examples illustrated graphically with the help of Maple to illuminate the results.
1. Introduction
For real-valued harmonic functions and in the open unit disk the complex-valued continuous function is said to be harmonic and can be expressed as where and are analytic in . Let be the class of harmonic mappings normalized by and have the following power series representations:
We call the analytic part and the coanalytic part of , respectively. The Jacobian of is given by . Lewy’s theorem [1] implies that is locally univalent and sense-preserving if and only if in . The condition is equivalent to that dilatation satisfying for all (see [2, 3]).
We denote by the class of all harmonic, sense-preserving, and univalent mappings in , which are normalized by the condition and . Let be the subset of all in which Further, let (resp., ) be the subset of (resp., ) whose images are convex and close-to-convex domains. A domain is said to be convex in the horizontal direction (CHD) if the intersection of with each horizontal line is connected (or empty). A function is said to be a CHD mapping if maps onto a CHD domain. Let be the subset of which consist of CHD mappings. The following basic theorem of Clunie and Sheil-Small [2] is known as shear construction that constructs harmonic mappings with prescribed dilatations onto a domain convex in one direction.
Theorem 1 (see [2]). A locally univalent harmonic mapping in is a univalent mapping of onto a domain convex in a direction if and only if is a conformal univalent mapping of onto a domain convex in the direction of
Let be the convolution of two harmonic functions and where the operator is convolution (or Hadamard product) of two power series.
There are several research papers in recent years which investigate the convolution of harmonic univalent functions. In particular, Dorff [4] and Dorff et al. [5] studied the convolution of harmonic univalent mappings in the right half-plane. For some recent investigations involving convolution of harmonic mappings, we refer the reader to [6–13].
Let sheared by with the dilatation where . Using shear construction of Clunie and Sheil-Small [2], we have
It is clear that by setting in (2) and (3), we obtain which satisfy the conditions and , studied by Liu and Li [8]. Wang et al. [14] also studied convolutions of this mapping. Note that is a CHD mapping.
Recently, Liu and Li [8] introduced the following generalized harmonic univalent mappings: where and . Obviously, . If , then
Also, maps onto the domain which is a CHD domain. Very recently, Yasar and Ozdemir [15] studied convolutions of these generalized harmonic mappings.
Let with where .
In this paper, we investigate the conditions under which the convolutions of harmonic mappings , , and with prescribed dilatations are univalent and CHD provided that the convolutions are locally univalent and sense-preserving. Furthermore, we provide two examples illustrated graphically with the help of Maple to illuminate our results.
2. Preliminary Results
Lemma 2 (see [16]). Let be an analytic fuction in with and and let where . If then is convex in the horizontal direction.
Lemma 3 (see [17]). Let and be analytic in with If is convex and is starlike, then for each function analytic in and satisying we have
Lemma 4 ([18], Cohn’s rule). Given a polynomial of degree , let Denote by and the number of zeros of inside the unit circle and on it, respectively. If , then is of degree with and the number of zeros of inside the unit circle and on it, respectively.
Lemma 5. Let be defined by (4) and be defined by (6) with dilatation . Then the dilatation of is given by
Proof. Since and then We immediately get From (4), we have ☐
Lemma 6. Letbe defined by (4) andbe defined by (6). Ifis locally univalent and sense-preserving, thenis univalent and convex in the horizontal direction.
Proof. Let Thus, By Theorem 1, we need to prove that is convex in the horizontal direction. Since we have where satisfies the condition Thus, we have Now, we consider where satisfies the condition Using the fact that and is convex, by Lemma 3, we have ☐
Finally, using Lemma 2, we obtain that is convex in the horizontal direction.
Lemma 7. Let be given by (6) with dilatation and be a mapping defined by (2) and (3). Then the dilatation of is given by
Proof. From (2) and (3), we have Using (14) and (15), then we obtain the dilatation of as follows: ☐
Lemma 8 ([14], Lemma 2.4). Let be a mapping defined by (2), (3) and be defined by (6). If is locally univalent and sense-preserving, then is univalent and convex in the horizontal direction.
Lemma 9 ([19], Gauss-Lucas theorem). Let be a nonconstant polynomial with complex coefficients. Then, the zeros of the derivative are contained in the convex hull of the set of the zeros of
Lemma 10. Letbe a complex polynomial of degree , where , and Then, all zeros of lie in the closed unit disk
Proof. Note that , where It is obvious that the roots of lie on the unit circle. Also, which are the roots of lie on the unit circle as well. Hence, the result follows from Lemma 9.☐
3. Main Results
Theorem 11. Let be a mapping given by (4) and be given by (6) with the dilatation Then is univalent and convex in the horizontal direction.
Proof. By Lemma 6, we need to prove that the dilatation of satisfies for all . Substituting in (13), we yield where If we substitute into (30), then , and it is clear that for all Now, we need to show that for Obviously, if is a zero , then is zero of Then, we may write Using Lemma 4, we only need to show that all zeros of (31) lie in the closed unit disk for Since for , thus we have By Lemma 10, we know that all zeros of lie inside the closed disk. Then, by Cohn’s rule, given by (31) has all its zeros in the closed unit disk. The proof is complete.☐
Theorem 12. Let be a mapping given by (2) and be a mapping given by (6) with the dilatation Then, is univalent and convex in the horizontal direction for
Proof. By Lemma 8, we need to prove that is locally univalent and sense-preserving, i.e., the dilatation of satisfies for all . Substituting in (25),
where
If we substitute into (36), we yield
Hence,
Next, we will show that for all If is a zero of then is zero of ; hence,
By Lemma 4, we need to show that all zeros of lie inside or on the unit disk for Since
from (12), we have
where
Because for it follows that both and have the same zeros. By Lemma 10, we know that all zeros of lie inside the closed unit disk. Then, by Cohn’s rule, we know that all zeros lie inside or on the boundary of the unit disk. The proof is completed.☐
Theorem 13. Let be a harmonic mapping with and dilatation . Let be a mapping defined by (6) with and dilatation Then the mapping is univalent and convex in the horizontal direction.
Proof. Since is a mapping defined by (6) with we have Therefore, we know that Substituting into (44) yields Setting in the above equation, we get , and hence, for all .☐
Example 14. Supposebe given by ((6)). If we setandthen by shear construction of Clunie and Sheil-Small [2], we haveRecall that, if , then So, we have
Now, in view of Theorem 11, if we set the parameter , then is univalent and CHD. Also, if we choose then is not guaranteed to be univalent. The images of under and are shown in Figures 1–4.
Example 15. Suppose be given by (6). If we set and , then calculations lead to If , then So, we have
Now, if we set the parameter , in view of Theorem 11, is univalent and CHD. If we choose then is not guaranteed to be univalent (see Figures 5–8).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest.