Abstract
In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.
1. Motivation and Preliminaries
A complex-valued function defined on the unit disk is called harmonic mapping if and are real-valued harmonic functions. In addition, since is a simply connected domain, has a unique representation , where and are analytic and co-analytic parts of , respectively. It is known from Lewy [1] that the mapping defined on is locally univalent and sense-preserving if and only if
The class of all univalent, harmonic sense-preserving mappings defined in is denoted by . Moreover, let be the class of all functions which is normalized by . Therefore, each member of has the following representation:for all .
The class of such type of functions given in (2) with is a subclass of and is denoted by . The dilatation of belonging to is the function given by and satisfies for all . Further, denote by (or ) all (or ) which are mapping onto convex regions. For comprehensive and fundamental knowledge on planar harmonic mappings, see Duren [2]. The subclass of denoted by consists of all univalent harmonic functions which maps onto domain convex in the direction of the real axis.
Section 2 demonstrates that the convolution of two special subclasses of univalent harmonic mappings is convex in the horizontal direction. The convolution of any two arbitrary harmonic functionsis defined by
The convolution of harmonic functions is different from the convolution of univalent functions (for more details, one can refer to [3]). Indeed, the convolution of two harmonic functions does not preserve the convexity and the convolution of any two univalent harmonic functions need not be univalent. These facts generated significant interest in the analysis of harmonic convolutions of univalent harmonic functions, and several articles on this subject recently appeared in the literature [3–11]. Specifically, the collection of the mappings that map onto the right half-plane and have the form with for all . Kumar et al. [12] proposed a class of right half-plane harmonic mappings, , satisfying
Using the shear construction of Clunie and Sheil-Small [13], it follows that
Letting in (5), the mapping with and , which is called the standard right half-plane mapping, is obtained. In [9], the following result was obtained.
Theorem 1 (see [9]). Let satisfy the condition and dilatation . If , then.
Recently, Liu and Ponnusamy [8] generalized Theorem 1 in the case as follows.
Theorem 2 (see [8]). Letsatisfy the conditionwith dilatation, where, andwith dilatation. Then, .
Liu and Ponnusamy [8] also proposed the following problem.
Problem 1. Let with and dilatation function , where , and let with dilatation . Then, .
Ali et al. [14] solved Problem 1 for . In [8], the authors conjectured that is not locally univalent for . In Section 2, new subclass of harmonic mappings is obtained. It is also shown that the necessary and sufficient condition for , if and only if is convex. The function satisfies the condition and dilatation with the mapping belonging to subclass for and for all are also explored.
The results proposed by Hengartner and Schober [15] and Pommenrenke [16] are very useful in checking the convexity of an analytic function in the direction of the imaginary axis as well as the convexity in the direction of the real axis, respectively.
Lemma 1 (see [15]). Supposeis a nonconstant analytic function of. Then,if and only if(1)is univalent in.(2)is convex in the direction of imaginary axis.(3)There exist sequencesconverging toandconverging to, such that
Lemma 2 (see [16]). Supposeis a nonconstant analytic function ofsatisfying the conditionand, and supposewhere . Ifthen is convex in the direction of the real axis.
Recall that the region is said to be convex in the direction , , if every line parallel to the line through 0 and has either connected or empty intersection with , and if is convex in every direction, then it is called convex. If (or ), then is convex in the direction of the real axis “CHD” or (is convex in the direction of the imaginary axis “CID”). Clunie and Sheil [13] proposed an integrated way to construct a univalent harmonic mapping convex in a given direction.
Theorem 3 (see [13]). Let. A locally univalent harmonic functioninis a univalent mapping ofonto a domain convex in the direction ofif and only ifis a univalent mapping ofonto a domain convex in the direction of.
The study of the geometric properties of the convex combination of the univalent mappings is another important topic in the geometric function theory. Dorff and Rolf [17] developed a necessary condition for convex combination to be univalent maps onto a domain convex in the direction of the imaginary axis.
Theorem 4 (see [17]). Letandbe two univalent harmonic mappings defined on, withandhaving the same second complex dilatation and satisfying conditions in (4); then,is univalent and is convex in the direction of the imaginary axis.
It is known that even if and are convex analytic functions, the convex combination of and may not be a univalent function (see [18], and for more recent studies of linear combinations of harmonic mappings, see [17, 19–23]). Moreover, Kumar et al. [20] studied the convexity of linear combination of harmonic mappings, which are shears of the analytic mapping and , for . Beig et al. [23] studied and found necessary conditions for the convex combination of the right half-plane mappings, the vertical strip mapping, their rotations, and some other harmonic mappings to be univalent and convex in a particular direction. In Section 3, the convex combination of mappings of the family of sense-preserving and locally univalent harmonic mappings , by shearing the function whereare studied. Further, the appropriate conditions for the convex combination in the vertical direction between univalent harmonic mappings to be univalent and convex in the vertical direction are also established.
2. Convolutions of Subclasses of Univalent Harmonic Right Half-Plane Mappings
In this section, we consider the harmonic mapping with with the second complex dilatation , . Hence, we can solve for and as follows:and thus
Taking , given in (14) can be rewritten as
For an analytic univalent function normalized by , definefor all . It is clear that . Therefore,
If is an analytic function with , thenand
Lemma 3. Letsatisfy the conditionwith second complex dilatationsuch thatandbe family of harmonic mappings given in (17). Then, , the second complex dilatation of, is given by
Proof. Since thenNow, from (18) and (19), it follows thatSince , thenFrom , it follows thatand thus
Lemma 4. Letwith. Ifis locally univalent, thenis in.
Proof. Since thenThus,Let . Consequently, we havewhere and . Thus, from Lemma 3, we deduce that is convex in the direction of the real axis. Theorem 3 implies that .
Lemma 5. The mappinggiven in (16) is locally univalent if and only if the function is convex.
Proof. The mapping is given in (16). Since is locally univalent if and only if ,and henceThus,It can be easily shown that the above inequality is identical toHence, is locally univalent if is convex function.
Theorem 5. The mappingis in the class, if and only ifis convex.
Proof. The mapping is given in (16), and where is convex. This implies that is convex in the direction of the real axis. Consequently, from Lemma 5 and Theorem 3, we deduce that in .
In order to prove the next result, we will employ the distribution theory of the polynomial function roots of the disc unit. Given a polynomial function of degree :with complex coefficients, the parallel algorithm for discovering polynomial zeros for (35) inside is worth exploring. LetThe inverse zeros for (35) and (36) can be easily checked with respect to . We need the following lemma to prove the main theorem.
Lemma 6 (Cohn’s rule) (see [24]). Letbe a polynomial function as given in (35) of degreeand let be as given in (36). The number of zeros of in or on the unit circle is denoted by and , respectively. If, thenis a polynomial function of degreewithand, whereandare the number of zeros of the polynomialin or on the unite circle, respectively.
Theorem 6. Letbe mapping given in (17) and let with and dilatation ,. Then, for and for all n belonging to .
Proof. According Lemma 4, it suffices to show that the mapping has dilatation such that for all . Setting in (20), we obtainwhereIf , is zero of the polynomial function , then will be zero of the polynomial function . Hence, we can rewrite asIn order to show that in , it suffices to prove that for , or equivalently, all the zeros for of the polynomial function lie in or on the unit circle .
Note thatso Cohn’s rule can be applied on . Consider the polynomial function given byIt is easy to verify that Cohn’s rule is applicable to also. So, let a polynomial function given byThen, the zeros of polynomial function lie on or in . We deduce that all zeros of lie inside or on , and this implies that all the zeros of lie inside or on .
Corollary 1. Letbe mapping given in (17) and letwithand dilatation,. Then, for.
The following examples illustrate Theorems 5 and 6.
Example 1. In Theorem 6, if , , and , thenIt follows from (17) thatand . Further,Figures 1–4, respectively, display representations of concentrated circles within under the harmonic mapping and concentrated circles under the convolution map . We take various values of .
Example 2. Consider an analytic univalent function , where and . Since and ,This implies that the function maps into convex region. In view of Theorem 1, the mapping is in for . NowFigures 5–7 display images of under for various values of , respectively. The image of under is convex in the direction of the real axis.
3. Convex Combination of a Family of Univalent Harmonic Mappings
Let , wherebe normalized, sense-preserving, and locally univalent harmonic mapping defined on . First of all, we have to show that belongs to class and maps onto region convex in the direction of the imaginary axis. Letfor all , , and we will prove that
Since , for all in , andit follows that
Note that and for each ,
Hence, by the minimum principle for harmonic functions, we obtainfor all , and this yields the result.
In addition, by Lemma 1, we can deduce that the analytic function is univalent and maps onto domain convex in the direction of the imaginary axis. Further, Theorem 3 yields that the harmonic mapping belongs to the class and is convex in the direction of imaginary axis. Figure 8 gives an illustration of images of under for several values of .
(a)
(b)
(c)
(d)
Let us begin by presenting our own dilatation .
Theorem 7. Let , for , be two normalized harmonic mapping satisfying , , and dilatation with in . Then, the second complex dilatation of the mapping , , is defined to be
Proof. Since , it follows that the dilatation of the mapping is given byMaking use of the mapping and , for , we obtainReplacing the above expressions with and in (57), it follows thatwhich yields the results in (56) through some computation.
The next result clarifies that the necessary and sufficient condition for the convex combination of and is in the class and convex in the direction of the imaginary axis if it is locally univalent and sense-preserving.
Theorem 8. Let , for , be two normalized harmonic mapping satisfying , , and dilatation with in . Then, the mapping ,with dilatation as given in (56), is in and maps into region convex in the direction of the imaginary axis, providedis locally univalent and sense-preserving.
Proof. DefineLetwhere . Since the mapping satisfies (49), it follows from (7) thaton for all . Hence, according to Lemma 1, one can deduce that the mapping is univalently convex in the direction of the imaginary axis. Also, If is sense-preserving locally univalent mapping in , then in view of Theorem 3, is univalent analytic mapping of onto a region convex in the direction of imaginary axis.
Theorem 9. For, defineto be normalized sense-preserving harmonic mapping such that,with. Letbe the dilatation of(in). If, then the mapping,, is inand is convex in the direction of the imaginary axis.
Proof. From Theorem 8, it will be sufficient if we show that the dilatation of given in (56) satisfies the condition in . Suppose that in (56); then,From the proof of Theorem 3 in [21], it is clear that . Hence, we deduce that the mapping is locally univalent and sense-preserving.
Next, we will consider one of the harmonic mappings involved in the linear combination induced by shearing analytic mapping where .
Theorem 10. Define , satisfying , , with dilatation lies in and let , satisfying , , with dilatation lies in , be two normalized harmonic mapping. Then, the mapping,is univalent and convex in the direction of the imaginary axis ifis locally univalent and sense-preserving.
Proof. Since and , it follows thatwhich implies thatHence,from (55). Sincefor all , we can easily see thatIn view of Theorem 3, we deduce that the mapping is univalent and maps onto domain convex in the direction of the imaginary axis provided is locally univalent and sense-preserving on .
It is known that for , the mapping is called vertical strip mapping if it satisfiesThus, Theorem 10 can be expressed in terms of vertical strip mappings.
Corollary 2. Let , where with dilatation lies in and , where , with dilatation lies in , to be two normalized harmonic mapping. Then, the mapping,, is univalent and convex in the direction of the imaginary axis ifis locally univalent and sense-preserving.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first and second authors were supported by the Fundamental Research Grant Scheme (Ministry of Education Malaysia (MOE), Acct No. 203.PMATHS.6711939).