Abstract
New sufficient conditions, concerned with the coefficients of harmonic functions in the open unit disk normalized by , for to be harmonic close-to-convex functions are discussed. Furthermore, several illustrative examples and the image domains of harmonic close-to-convex functions satisfying the obtained conditions are enumerated.
1. Introduction
For a continuous complex-valued function , we say that is harmonic in the open unit disk if both and are real harmonic in , that is, and satisfy the Laplace equations A complex-valued harmonic function in is given by where and are analytic in . We call and the analytic part and the coanalytic part of , respectively. A necessary and sufficient condition for to be locally univalent and sense preserving in is in (see [1] or [2]). Let denote the class of harmonic functions in with and . Thus, every normalized harmonic function can be written by where and , for convenience.
We next denote by the class of functions that are univalent and sense preserving in . Due to the sense-preserving property of , we see that . If , then reduces to the class consisting of normalized analytic univalent functions. Furthermore, for every function , the function is also a member of . Therefore, we consider the subclass of defined as Conversely, if , then for any .
We say that a domain is a close-to-convex domain if the complement of can be written as a union of nonintersecting half-lines (except that the origin of one half-line may lie on one of the other half-lines). Let , , and be the respective subclasses of , , and consisting of all functions , which map onto a certain close-to-convex domain.
Bshouty and Lyzzaik [3] have stated the following result.
Theorem 1.1. If satisfies for all , then .
A simple and interesting example is below.
Example 1.2. The function satisfies the conditions of Theorem 1.1, and therefore belongs to the class . We now show that is actually a close-to-convex domain. It follows that Setting for any , we see that Since we obtain that Also, noting that we know that which implies that Thus, maps onto the following close-to-convex domain as shown in Figure 1.
Remark 1.3. Let be the class of all functions satisfying the conditions of Theorem 1.1. Then, it was earlier conjectured by Mocanu [4, 5] that . Furthermore, we can immediately see that the function in Example 1.2 is a member of the class and it shows that is not necessarily starlike with respect to the origin in ( is starlike with respect to the origin in if and only if for all and ).
Remark 1.4. For the function given by
letting , we know that
which means that maps the unit circle onto a union of several concave curves (see [6, Theorem 2.1]).
Jahangiri and Silverman [7] have given the following coefficient inequality for to be in the class .
Theorem 1.5. If satisfies then .
Example 1.6. The function
belongs to the class and satisfies the condition of Theorem 1.5. Indeed, maps onto the following hypocycloid of six cusps (cf. [8] or [6]) as shown in Figure 2.
The object of this paper is to find some sufficient conditions for functions to be in the class . In order to establish our results, we have to recall here the following lemmas due to Clunie and Sheil-Small [1].
Lemma 1.7. If and are analytic in with and is close-to-convex for each , then is harmonic close-to-convex.
Lemma 1.8. If is locally univalent in and is convex for some , then is univalent close-to-convex.
We also need the following result due to Hayami et al. [9].
Lemma 1.9. If a function is analytic in and satisfies for some real numbers and , then is convex in .
2. Main Results
Our first result is contained in the following theorem.
Theorem 2.1. If satisfies the following condition for some real number , then .
Proof. Let be analytic in . If satisfies then it follows that This gives us that that is, . Then, it is sufficient to prove that for each by Lemma 1.7. From the assumption of the theorem, we obtain that This completes the proof of the theorem.
Example 2.2. The function
satisfies the condition of Theorem 2.1 with and belongs to the class . In particular, putting , we obtain Figure 3.
By making use of Lemma 1.8 with and applying Lemma 1.9, we readily obtain the next theorem.
Theorem 2.3. If is locally univalent in and satisfies for some real numbers and , then .
Putting in the above theorem, we arrive at the following result due to Jahangiri and Silverman [7].
Theorem 2.4. If is locally univalent in with then .
Furthermore, taking and in the theorem, we have the following corollary.
Corollary 2.5. If is locally univalent in and satisfies then .
Example 2.6. The function satisfies the conditions of Corollary 2.5 and belongs to the class as shown in Figure 4.
3. Appendix
A sequence of nonnegative real numbers is called a convex null sequence if as and for all .
The next lemma was obtained by Fejér [10].
Lemma 3.1. Let be a convex null sequence. Then, the function is analytic and satisfies in .
Applying the above lemma, we deduce the following theorem.
Theorem 3.2. For some and some convex null sequence with , the function belongs to the class .
Proof. Let us define by for each . Then, we know that By virtue of Lemmas 1.7 and 3.1, it follows that , that is, . Thus, we conclude that .
In the same manner, we also have the following theorem.
Theorem 3.3. For some and some convex null sequence with , the function belongs to the class .
Proof. Let us define by for each . Then, we know that Therefore, by the help of Lemmas 1.7 and 3.1, we obtain that , that is, , which implies that .
Remark 3.4. The sequence
is a convex null sequence because
Setting in Theorem 3.2 with the above sequence , we derive the following example.
Example 3.5. The function is in the class as shown in Figure 5.
Moreover, we know the following remark.
Remark 3.6. The sequence
is a convex null sequence because
Hence, letting in Theorem 3.3 with the sequence , we have the following example.
Example 3.7. The function is in the class as shown in Figure 6.
Dedication
This paper is dedicated to Professor Owa on the occasion of his retirement from Kinki University.
Acknowledgment
The author expresses his sincere thanks to the referees for their valuable suggestions and comments for improving this paper.