Abstract

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

1. Introduction

A continuous complex-valued function is said to be harmonic in a simply connected domain if both and are real harmonic in . There is a close interrelation between analytic functions and harmonic functions. For example, for real harmonic functions and , there exist analytic functions and so that and . Then , where and are, respectively, the analytic functions and . In this case, the Jacobian of is given by . The mapping is orientation preserving and locally one-to-one in if and only if in . The function is said to be harmonic univalent in if the mapping is orientation preserving, harmonic, and one-to-one in . We call the analytic part and the coanalytic part of (see Clunie and Sheil-Small [1]).

Denote by the class of functions that are harmonic univalent and orientation preserving in the open unit disk for which . Then for , we may express the analytic functions and as Note that reduces to the class of normalized analytic univalent functions if the coanalytic part of its members is zero. For this class the function may be expressed as A function with and given by (1) is said to be harmonic starlike of order for , if The class of all harmonic starlike functions of order is denoted by and extensively studied by Jahangiri [2]. The cases and were studied by Silverman and Silvia [3] and Silverman [4]. Other related works of the class also appeared in [5–16].

Definition 1. Let where and are given by (1). Let and . Then if and only if

We note that for , the class reduces to the class . Further, if the coanalytic part is zero, the class reduces to the class of functions which satisfy the condition for some , , and . Observe that the classes and were introduced and studied by many authors and these include, for example, by Obradovic and Joshi [17], Padmanabhan [18], Li and Owa [19], Xu and Yang [20], Singh and Gupta [21], and Lashin [22]. We also note that for , the class was studied by Silverman [23].

2. Main Results

The first theorem of this section determines the sufficient coefficient condition for functions to belong to the class . The following lemma obtained by Jahangiri is needed.

Lemma 2 (see [2, Theorem 1]). Let with and of the form (1) and let where . Then is harmonic, orientation preserving, and univalent in , and .

Theorem 3. Let where and are of the form (1). If for some , and , then is harmonic, orientation preserving, and univalent in and .

Proof. Since and , , it follows from Lemma 2 that and hence is harmonic, orientation preserving, and univalent in . Now, we only need to show that if (7) holds then Using the fact that if and only if , it suffices to show that where Substituting for and in (9), we obtain by the given condition (7).
The harmonic function where shows that the coefficient bound given in (7) is sharp. The functions of the form (12) are in since

Remark 4. Setting in Theorem 3 yields the result obtained by Lashin [22, Theorem 2.1].

We denote by the class of functions whose coefficients satisfy the condition (7).

Theorem 5. Let and . Then .

Proof. For , it follows from (7) that Hence .

As a consequence of Theorem 5, the functions in are starlike harmonic in .

Corollary 6. For and ,.

3. Distortion Bounds and Extreme Points

In this section, we obtain the distortion bounds and extreme points for functions in the class .

Theorem 7. Let where and are of the form (1) and . Then for , we have where The result is sharp.

Proof. We shall prove the first inequality. Let . Then we have and so
The proof of the inequality (16) is similar to the proof of the inequality (15), thus we omit it.
The upper bound given for is sharp and the equality occurs for the function where . This completes the proof of Theorem 7.

Now, we determine the extreme points of the closed convex hull of the class denoted by .

Theorem 8. Let where and are given by (1). Then if and only if where In particular, the extreme points of the class are and , respectively.

Proof. For a function of the form (21), we have But Thus .
Conversely, suppose that . Set Then by the inequality (7), we have and . Define and note that . Thus we obtain . This completes the proof of the theorem.

4. Convolution and Convex Combinations

For two harmonic functions we define their convolution Using this definition, we show that the class is closed under convolution.

Theorem 9. For and , let . Then .

Proof. We note that and . For the convolution , we have Therefore .

We show that the class is closed under convex combination of its members.

Theorem 10. The class is closed under convex combination.

Proof. For , let where is given by Then by (7), we have For , , the convex combination of may be written as Then by (7), we have Therefore .

5. Neighborhood Results

Following the earlier investigations by Goodman [24], Ruscheweyh [25], Altintas et al. [26], and Porwal and Aouf [27], we define the -neighborhood of function by In particular, for the identity function , we immediately have

Theorem 11. , where

Proof. Let . Then, in view of (7), since and are increasing functions of , we have which yields On the other hand, we also find from (7) From (37) and (38), we obtain which is equivalent to

6. A Family of Class Preserving Integral Operator

In this section, we consider the closure property of the class under the Bernardi integral operator , which is defined by

Theorem 12. Let be in the class , where and are given by (1). Then defined by (41) also belongs to the class .

Proof. From the representation of , it follows that Now by (7). Thus .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to express his gratitude to Professor Dr. V. Ravichandran and the referees for their valuable comments which have essentially improved the presentation of this paper.