On convolutions of slanted half-plane mappings

The convolution of convex harmonic univalent functions in the unit disk, unlike analytic functions, may not be convex or even univalent. The main purpose of this work is to develop previous work involving the convolution of convex harmonic functions. Briefly, we obtain under which conditions the convolution of a right half-plane harmonic mapping having a dilatation and a slanted half-plane harmonic mapping with β having a dilatation and is univalent and convex in the direction . We also provide an example illustrating graphically with the help of Maple to illuminate the result.


Introduction
Consider the harmonic function k = v 1 + iv 2 defined in the unit disc E = {z ∈ C : |z| < 1} where v 1 (1) are analytic in E. The dilatation of a harmonic function k defined by (z) = t (z)/s (z) is analytic and satisfies | (z)| < 1 for all z ∈ E if and only if k is locally univalent (LU) and sense-preserving (SP) in E. Denote by SH 0 the class of all harmonic, SP and univalent mappings k = s + t in E, which are normalized by the conditions k(0) = k z (0) − 1 = k z (0) = 0. A simply connected subdomain of C is said to be close-to-convex if its complement in C can be written as the union of non-crossing half-lines. Further, let KH 0 and CH 0 be the subclass of SH 0 whose image domains are convex and closeto-convex domains, respectively. A domain M β is said to be convex in the direction β (β ∈ R) if and only if for every b ∈ C, the set M β ∩ {b + r e iβ : r ∈ R} is either connected or empty. In particular, the domain M 0 is said to be convex in the direction of the real axis. Denote by CH 0 β , the class of functions k = s + t ∈ SH 0 that map E onto the domain M β . It is clear that, CH 0 β ⊂ CH 0 . The shearing method of Clunie and Sheil-Small [1] enables us to construct new examples of harmonic univalent mappings with prescribed dilatations. In this method, they considered a harmonic function k = s + t which is LU in E (i.e. | (z)| < 1 for all z ∈ E) and they proved that the function K = s − e 2iβ t is analytic univalent mapping of E onto the domain M β if and only if k = s + t is a univalent mapping of E onto the domain M β . For basic details of harmonic univalent functions, see [1][2][3].
On the other hand, harmonic mappings and related functions have been used in the areas of description of the fluid flows, elasticity problems and approximation theory of plates, etc. Techniques in recent papers such as [4][5][6] can be used in further applications.
where Re means the real part. It is almost easy to obtain that such mapping can be expressed as (see [7, Lemma 1]) Denote by SH 0 β ⊂ KH 0 , the class of all β−SHP mappings. For β = 0, we get the class of right half-plane (RHP) mappings SH 0 0 ⊂ KH 0 . In particular, for K 0 = S 0 + T 0 ∈ SH 0 0 with 0 (z) = −z, we obtain Moreover, if k = s + t ∈ SH 0 be of the form (1) then There have been several publications involving the convolution of the harmonic functions that are convex in a particular direction such as [7][8][9][10][11]. Moreover, Dorff et al. [7] proved It is clear that, Theorem A implies for every k β ∈ SH 0 β , On the other hand, Dorff et al. [7] considered k ∈ SH 0 0 having a dilatation (z) = ρ+z 1+ρz , with ρ ∈ (−1, 1) and they proved that K 0 * k is a member of the class CH 0 0 . In [12], the authors collected some open problems on harmonic mappings.
One of the open problems given in reference [12, Problem 3.26(a)] that will be addressed in this study was first raised by Dorff et al. [7]. The problem in question is as follows: Determine other values of ρ ∈ E for the mapping K 0 * k to be a member of the class CH 0 0 . Furthermore, Li and Ponnusamy [13] proved Recently, Jiang et al. [14] studied the convolution of K 0 with a mapping K 1 ∈ SH 0 0 and they proved

Theorem C ([14, Theorem 1]):
In this paper, we deal mainly with the generalization of Theorem B such that considering of k β ∈ SH 0 β in a more general setting of the form which is an automorphism of the unit disk. We investigate conditions which guarantee the mapping K 0 * k β is a member of the class CH 0 −β . Our main result is Theorem 3.1 in Section 3, which improves Theorem B, and also Theorem C. Furthermore, an example, illustrating graphically with the help of Maple, is given to illuminate the main result.

Conclusion
In this study, we considered a RHP mapping K 0 = H 0 + G 0 having a form H 0 (z) + G 0 (z) = z 1−z with a dilatation −z and a β−SHP mapping k β = s β + t β having a form s β (z) + e −2iβ t β (z) = z 1−e iβ z with a dilatation e iμ ρ+z 1+ρz (ρ ∈ E and μ ∈ R). We derived the convolution of these mappings is a member of the class CH 0 −β provided that the conditions of (11)-(13) hold. We would like to emphasize an important innovation of our study as follows: We generalized Theorem B by adding a rotation parameter μ in the dilatation. In this context, as observed in Example 3.2, when the parameters are specially selected, Theorems B and C cannot be used, while our main result, Theorem 3.1 holds. In our further investigations we will consider some open problems related with the generalized RHP mapping defined by Muir [17,18] L ς (z) where ς > 0 and studied in recent papers such as [9,10]. In this regard, considering the mapping L ς instead of the mapping K 0 , the conditions will be determined for which the results of Theorems B, C, and 3.1 hold.

Disclosure statement
No potential conflict of interest was reported by the author(s).