Abstract
For and for positive integers and , we consider classes of harmonic functions , where and or and , and we prove that their convolution is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
1. Introduction
Let denote the class of functions that are analytic in the open unit disk and let be the subclass of consisting of functions with the normalization . Consider the family of complex-valued harmonic functions , where and are real harmonic in . Such functions can be expressed as , where and . By Lewy’s Theorem (see [1, 2] or [3]), a necessary and sufficient condition for the harmonic function to be locally one-to-one and sense-preserving in is that its Jacobian should be positive or equivalently if and only if in and the second complex dilatation of satisfies in . In the sequel, without loss of generality, we consider those locally one-to-one and sense-preserving harmonic functions that are normalized by and and have the representationThe Hadamard product or convolution of two power series and is given by . Similarly, the convolution of two harmonic functions and is given by .
A simply connected proper subdomain of the complex domain is said to be convex if the linear segment joining any two points of lies entirely in and 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 convex or close-to-convex in if is convex or close-to-convex there. For , a function is said to be in the class if , It can easily be verified that if . If for in , then is said to be convex of order in (e.g., see [3] or [4]). A function is simply called a convex function in .
Ruscheweyh and Sheil-Small [5] 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 can not be extended to the harmonic case, since the convolution of harmonic functions, unlike the analytic case, proved to be very challenging.
Recently, Ahuja and Jahangiri [6] proved the following theorem.
Theorem 1. Let the functions and be in the class in . If either , , or , , , then is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
The following question is asked in [6].
Question 2. Is Theorem 1 true for and if ?
In Theorem 3, we address Question 2. Moreover, in Theorem 4, we allow variations in the powers of for the dilatations of harmonic functions. Also note that the techniques presented here to prove our theorems are different from those used in [6].
Theorem 3. Let the functions and be so that is convex in . Set and , where . Then the convolution function is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
Theorem 4. Let the functions and be convex of order in . Set , , and , . Then the convolution function is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
In the following example, we demonstrate a case of close-to-convexity of convolutions of two harmonic functions.
Example 5. ConsiderFor , it is easy to verify that andAlso,Therefore, is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
The images of under , , and are shown in Figures 1, 2, and 3, respectively.
2. Preliminary Lemmas and Proofs
To prove our theorems, we shall need the following four lemmas. Lemmas 6 and 9 are according to Clunie and Sheil-Small [2], Lemma 7 is a well-established result by Robinson [7], and Lemma 8 is a celebrated result by Ruscheweyh and Sheil-Small [5].
Lemma 6. Let and be analytic in so that . If is close-to-convex analytic in for each , then is close-to-convex harmonic in .
Lemma 7. If and are analytic in , for all , , and if maps onto a region which is starlike with respect to origin, then for all .
A function analytic in is convex of order in if and only if is starlike of order in (e.g., see Duren [4]).
Lemma 8. Let and be analytic and starlike of order in . Then for each function analytic in , the convolution takes only values in the convex hull of .
Lemma 9. If and are analytic in , is convex in and if is locally univalent in , then the function is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
Proof of Theorem 3. For the convolution functionwe note that To satisfy the condition of Lemma 6, we will show that is close-to-convex analytic in for each . To do so, it suffices to show that there exists a function analytic and convex in so thatWe observe that Now letting yieldsThus by Lemma 6, is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
Proof of Theorem 4. We need to show that the convolution functionis locally univalent and sense-preserving in .
Without loss of generality, we consider only the two cases of and .
Case 1. If , then we have to show that in . We observe thatInequality (10) is equivalent toororSince is convex in , is starlike in . Therefore, by Lemma 7, inequality (13) yieldsororCase 2. If , then we haveWe note thatLetting , , and in Lemma 8 yieldsThereforeThis is exactly inequality (10). So a similar argument following inequality (10) will lead to the conclusion thatTherefore, for either of cases or ,is locally univalent and sense-preserving in . Thus, by Lemma 9,is locally one-to-one, sense-preserving, and close-to-convex harmonic in .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.