Abstract
Harmonic functions can be constructed using two analytic functions acting as their analytic and coanalytic parts but the prediction of the behavior of convolution of harmonic functions, unlike the convolution of analytic functions, proved to be challenging. In this paper we use the shear construction of harmonic mappings and introduce dilatation conditions that guarantee the convolution of two harmonic functions to be harmonic and convex in the direction of imaginary axis.
1. Introduction
For and real harmonic in the open unit disk , the continuous complex-valued harmonic function can be expressed as , where and are analytic in . We call the analytic part and the coanalytic part of the harmonic function . By a result of Lewy [1] (see [2] or [3]), a necessary and sufficient condition for a harmonic function to be locally one-to-one and sense-preserving in is that its Jacobian is positive in or equivalently, if and only if in and the second complex dilatation of satisfies in . A simply connected domain is said to be convex in the direction , if every line parallel to the line through 0 and either misses , or is contained in , or its intersection with is either a line-segment or a ray. For the open unit disk , an analytic or harmonic function is said to be convex in the direction if is convex in the direction there. We note that if a mapping is convex in every direction, then it is simply called a convex mapping.
We let be the class of locally one-to-one and sense-preserving complex-valued harmonic univalent functions for which and . We also let be the convolution of two harmonic functions and , where the operator stands for the Hadamard product or convolution of two Taylor power series. Even though the harmonic functions can be constructed using two analytic functions acting as their analytic and coanalytic parts, the prediction of the behavior of convolution of harmonic functions, unlike the convolution of analytic functions, proved to be challenging. In a striking result (see the following Lemma 1), Clunie and Sheil-Small [2] introduced a method of constructing harmonic mappings known as the shear construction that produces harmonic functions with a specific dilatation onto a domain convex in one direction.
Lemma 1. A harmonic function locally univalent in is a univalent mapping of onto a domain convex in the direction if and only if is a conformal univalent mapping of onto a domain convex in the direction .
As a follow-up to the above Lemma 1, Clunie and Sheil-Small [2] provided the following example.
Example 2. Since is convex analytic in , the harmonic function defined by is convex in the direction of imaginary axis.
Along the same line as the above Example 2, Dorff [4] proved the following result.
Theorem 3. Let and with . If is locally univalent and sense-preserving, then and is convex in the direction of real axis.
The second author in his doctoral dissertation [5] proved the following theorem.
Theorem 4. For consider the harmonic function sheared by with the dilatation . If is the harmonic right half plane mapping given by with the dilatation ; , , then the convolution and is convex in the horizontal direction for .
Since then a number of related articles were published and we refer the readers to three recently published articles ([6–8]) and the citations therein. As an extension to the above Theorem 4, Liu et al. [7] proved the following theorem.
Theorem 5. Let be as given in Theorem 4 and let be a convex harmonic mapping so that maps onto the symmetric vertical strip domain with the dilatation , , and . Then is univalent and convex in the horizontal direction for .
We remark that for the function given in Theorems 4 and 5 reduce to the harmonic function given in Example 2, where . Recently, Dorff et al. [9] presented the following Theorem 6 on the directional convexity for the convolution of harmonic functions for which
Theorem 6. For let , , and . If is locally univalent and sense-preserving in , then is convex in the direction of real axis.
In the following Theorem 7 we improve the shear of the analytic map to the general case , , and expand the powers of in the dilatation to and , where and are arbitrary positive integers. The arguments presented here to prove our Theorem 7 and Example 9 are new and have not yet been used in any of the preceding related articles.
Theorem 7. For and for positive integers and let be the shear of the analytic map with the dilatations and , where ; . If is locally univalent and sense-preserving in , then is convex in the direction of imaginary axis.
2. Preliminaries, Proof and Example
Making use of the fact that a function is convex in the direction if and only if the function is convex in the direction of imaginary axis, in the following we state a lemma that is a variation of a result due to Royster and Ziegler [10].
Lemma 8. Let be a nonconstant analytic function in . The function maps univalently onto a domain convex in the direction if and only if there are numbers and , where and so that
Proof of Theorem 7. Adding the identities and we get Substituting for yields Differentiating and we obtain and Since may be written as If is even, then If is odd, then One can easily verify that is a positive real part function in with real coefficients. So, by a result of Rogosinski [11] (or see Duren [12] page 56) we conclude that is typically real in (also see Clunie and Sheil-Small [2] page 22). Therefore the integral function is also typically real in (e.g., see Theorem 2 in Robertson [13] or Duren [12], page 247). Consequently, is of positive real part in with real coefficients. The argument for odd would be similar since is a positive real part function in with real coefficients.
Therefore, for any positive integer we have
Similarly, for any positive integer we have
So, for all positive integers of and , we proved that
Thus for , it follows from Lemma 8 that the function or the analytic convolution function is convex in the direction of the imaginary axis. This in conjunction with Lemma 1, for prove that the harmonic convolution function and is convex in the direction of imaginary axis.
To demonstrate the beauty of Theorem 7, we give an example of two harmonic functions that satisfy the dilatation stated in Theorem 7 and we then prove that their convolution is locally one-to-one, sense-preserving, and convex in the direction of imaginary axis.
Example 9. For let with the dilatation and for let with the dilatation . Then is locally univalent, sense-preserving, and convex in the direction of imaginary axis.
First we will show that the harmonic convolution function is locally one-to-one and sense-preserving in , that is, in Under the hypotheses of Example 9, a simple calculation reveals that It is easy to verify that ; therefore we shall take in , where . Now for In order to prove that in it suffices to show that for all orThe left hand side of the above inequality reduces to A result of Robinson [14] states that if and are analytic in so that in , then if is star-like in . Applying this fact to the functions and given in Example 9, we obtain since and is star-like in .
On the other hand, observe that Similarly, Thus in and hence for all .
Next we will show that the harmonic convolution function is convex in the direction of imaginary axis. By Lemma 1, it suffices to show that is convex in the direction . Equivalently, by Lemma 8, we need to show thatWe observe that
The Taylor Expansion of yieldsBy a result of Fejér [15] (or see Goodman [16], Chapter 7), where and is a convex null sequence.
Therefore, , that is,
The images of under and are shown in Figures 1 and 2, respectively. Figure 3 clearly demonstrates the directional convexity of the convolution along the imaginary axis.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.