Convolutions of Harmonic Functions with Certain Dilatations

Abstract

The convolution of harmonic functions, unlike the analytic case, proved to be very challenging. In this paper, we introduce dilatation conditions that guarantee the convolution of two harmonic functions to be locally one-to-one, sense-preserving, and close-to-convex harmonic in the unit disk.

1. Introduction

Let denote the class of functions that are analytic in the open unit disc and let be the subclass of consisting of functions with the normalization . We let denote the class of functions so that

Consider the family of complex-valued harmonic functions defined in , where and are real harmonic in . Such functions can be expressed as , where and . Clunie and Sheil-Small in their remarkable paper [1] explored the functions of the form that are locally one-to-one, sense-preserving, and harmonic in . By Lewy’s Theorem (see [2] or [1]), a necessary and sufficient condition for the harmonic function to be locally one-to-one and sense-preserving in is that its Jacobian is positive or equivalently, if and only if in and the second complex dilatation of satisfies in .

In an interesting article, Bshouty and Lyzzaik [3] proved the following.

Theorem 1. Let be a harmonic mapping of , with , that satisfies and for all . Then is a univalent close-to-convex mapping.

A simply connected proper subdomain of is said to be close-to-convex if its complement in is the union of closed half-lines with pairwise disjoint interiors. Consequently, a univalent analytic or harmonic function is said to be close-to-convex if is close-to-convex (e.g., see Clunie and Sheil-Small [1] or Bshouty and Lyzzaik [3]).

Ruscheweyh and Sheil-Small in a striking article [4] proved that the Hadamard product or convolution of two analytic convex functions is also convex analytic and that the convolution of an analytic convex function and an analytic close-to-convex function is close-to-convex analytic in the unit disk . Ironically, these results could not be extended to the harmonic case, since the convolution of harmonic functions, unlike the analytic case, proved to be very challenging. The purpose of the present paper is to introduce dilatation conditions that guarantee the convolution of two harmonic functions to be locally one-to-one, sense-preserving, and close-to-convex harmonic in the unit disk . In other words, we extend Theorem 1 to the convolution of two harmonic functions and with certain dilatations, where .

The operator stands for the convolution or Hadamard product of two power series and given by . Similarly, the convolution of two harmonic functions and is given by .

In regard to the convolution of harmonic univalent functions, Clunie and Sheil-Small [1] proved the following.

Theorem 2. If and if is convex harmonic in , then their convolution is close-to-convex harmonic in .

A mapping is called convex harmonic if is a convex domain.

The convexity condition for the function in Theorem 2 cannot be compromised as it is demonstrated in the following.

Example 3. Set and consider the starlike analytic function ; in . Letting in Theorem 2, we observe that the harmonic convolution is not even univalent in .

In an attempt to investigate the possibilities of improving the required convexity condition for , the authors in [5] proved the following.

Theorem 4. Let and . Also let be a Schwarz function. Then the convolution function is close-to-convex harmonic in .

Theorem 4 for and yields a theorem given by Bshouty et al. ([6], Theorem ). From what is said above, especially Example 3, one wonders if there are other conditions that guarantee the close-to-convexity of the convolution of two harmonic functions. In the following theorem, we find such conditions.

Theorem 5. Let and so that , where is given by inequality (1). If one of the following conditions hold(i); and ,(ii); and , where , then the convolution function is locally one-to-one, sense-preserving, and close-to-convex harmonic in .

Since the convolution of two convex analytic functions is also convex (see Ruscheweyh and Sheil-Small [4]), an obvious consequence of the above theorem would be as follows.

Corollary 6. Let and and set and Then the convolution function is locally one-to-one, sense-preserving, and close-to-convex harmonic in .

2. Preliminary Lemmas and Proof of Theorem 5

To prove our Theorem 5, we shall need the following three lemmas, the first of which is a celebrated result by Clunie and Sheil-Small [1] and the second one is given by Kaplan [7]. The third lemma which is on subordination is a modification of a result given by Miller and Mocanu (e.g., see [8] Lemma or [9]). For functions and , where , we write (i.e., is subordinate to ) if there exists an analytic function with and so that in .

Lemma 7. (i) If and are analytic in so that and if is close-to-convex analytic in for each , then the function is close-to-convex harmonic in .
(ii) If and are analytic in so that and if is locally univalent in , then the function is close-to-convex harmonic in .

Lemma 8. A necessary and sufficient condition for the analytic function to be close-to-convex is that is nonvanishing in and

Lemma 9. If and is analytic in , then implies .

Proof of Theorem 5.  
Proof of Part (i). The convolution function is locally univalent and sense-preserving since Obviously ; therefore, in view of Lemma 7, it suffices to prove that for is close-to-convex analytic in .
We note that and .
We also observe that is nonvanishing in since . Therefore, Now, by Lemma 8 and inequality (5) for and , it suffices to show thatFor , one may verify (also see Bshouty and Lyzzaik [3] p. 770) that For , replacing by and letting yield where is the Poisson Kernal. It then follows that On the other hand, since , we obtain Therefore, in view of the required condition (9), we get Proof of Part (ii). In view of Lemma 7, it suffices to show that the convolution function is locally univalent and sense-preserving in . In other words, we need to show that Using the Hadamard product properties of power series, we have Therefore, On the other hand, since , is starlike or Thus, in view of Lemma 9, or

Remark 10. It is left as an open problem whether Theorem 5(i) can be extended to the case and if .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.