Directional Convexity of Convolutions of Harmonic Functions

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.


Introduction
For  and V real harmonic in the open unit disk D fl { : || < 1}, the continuous complex-valued harmonic function  =  + V can be expressed as  = ℎ + , where ℎ and  are analytic in E. We call ℎ the analytic part and  the coanalytic part of the harmonic function  = ℎ + .By a result of Lewy [1] (see [2] or [3]), a necessary and sufficient condition for a harmonic function  = ℎ +  to be locally one-to-one and sense-preserving in D is that its Jacobian   = |ℎ  | 2 − |  | 2 is positive in D or equivalently, if and only if ℎ  () ̸ = 0 in D and the second complex dilatation  of  satisfies || = |  /ℎ  | < 1 in D. A simply connected domain D ⊂ C is said to be convex in the direction , 0 ≤  <  if every line parallel to the line through 0 and   either misses D, or is contained in D, or its intersection with D is either a line-segment or a ray.For the open unit disk D, an analytic or harmonic function  : D → C is said to be convex in the direction  if (D) 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 S H be the class of locally one-to-one and sensepreserving complex-valued harmonic univalent functions  = ℎ +  for which (0) =   (0) =   (0) − 1 = 0 and   (0) = ℎ(0) = ℎ  (0) − 1 = 0. We also let  1 *  2 = ℎ 1 * ℎ 2 + 1 *  2 be the convolution of two harmonic functions  1 = ℎ 1 +  1 and  2 = ℎ 2 +  2 , 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
As a follow-up to the above Lemma 1, Clunie and Sheil-Small [2] provided the following example.
Example .Since /(1 − ) is convex analytic in D, the harmonic function ℎ +  defined by is convex in the direction of imaginary axis.The second author in his doctoral dissertation [5] proved the following theorem.Since then a number of related articles were published and we refer the readers to three recently published articles ( [6][7][8]) and the citations therein.As an extension to the above Theorem 4, Liu et al. [7] proved the following theorem.
In the following Theorem 7 we improve the shear of the analytic map ℎ is locally univalent and sense-preserving in D, then  1 *  2 is convex in the direction of imaginary axis.

Preliminaries, Proof and Example
Making use of the fact that a function  is convex in the direction  if and only if the function  (/2−)  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 D. e function  maps D univalently onto a domain convex in the direction  if and only if there are numbers  and ], where 0 ≤  < 2 and 0 ≤ ] ≤  so that Proof of eorem .Adding the identities and Differentiating and International Journal of Mathematics and Mathematical Sciences 3 we obtain and 1 () may be written as If  is even, then If  is odd, then One can easily verify that is a positive real part function in D with real coefficients.So, by a result of Rogosinski [11] (or see Duren [12] page 56) we conclude that is typically real in D (also see Clunie and Sheil-Small [2] page 22).Therefore the integral function is also typically real in D (e.g., see Theorem 2 in Robertson [13] or Duren [12], page 247).Consequently, Similarly, for any positive integer So, for all positive integers of  and , we proved that Thus for ] =  =  = /2, it follows from Lemma 8 that the function  1 +  2 or the analytic convolution function (ℎ 1 * ℎ 2 )+( 1 *  2 ) is convex in the direction of the imaginary axis.This in conjunction with Lemma 1, for  = /2 prove that the harmonic convolution function ( 1 *  2 ) = (ℎ 1 * ℎ 2 )+ ( 1 *  2 ) ∈ S H 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 . For 𝑓
Then  1 *  2 is locally univalent, sensepreserving, 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 D, that is, || = |  /  | < 1 in D. Under the hypotheses of Example 9, a simple calculation reveals that International Journal of Mathematics and Mathematical Sciences It is easy to verify that |  (0)| = 0 < 1 = |  (0)|; therefore we shall take  in  0 , where  0 = { : 0 < || < 1}.Now for  ∈  0 In order to prove that || < 1 in  0 it suffices to show that The left hand side of the above inequality reduces to A result of Robinson [14] Similarly, Next we will show that the harmonic convolution function  1 *  2 =  +  is convex in the direction of imaginary axis.By Lemma 1, it suffices to show that + = ℎ 1 * ℎ 2 + 1 *  2 is convex in the direction /2.Equivalently, by Lemma 8, we need to show that We observe that  where  0 = 1 and   = 2/(2 − 1)(2 + 1);  = 1, 2, 3 . . . is a convex null sequence.Therefore, R{( 2 )} > 0, that is, 2 )[()+()]  ≥ 0 The images of || =  < 1 under  1 and  2 are shown in Figures 1 and 2, respectively.Figure 3 clearly demonstrates the directional convexity of the convolution  1 *  2 along the imaginary axis.
and expand the powers of  in the dilatation   () =      ;  = 1, 2 to  1 () = ± − 1   and  2 () = ± − 2   , 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  = 1, 2 and for positive integers  and  let   = ℎ  +   ∈ S H be the shear of the analytic map ℎ  ()+     () = (1/2) log((1+)/(1−)) with the dilatations