Abstract

The purpose of the present paper is to obtain some inclusion relation between various subclasses of harmonic univalent functions by applying certain convolution operators associated with Wright’s generalized hypergeometric functions.

1. Introduction

A continuous complex-valued function defined in a simply connected domain is said to be harmonic in if both and are real harmonic in . In any simply connected domain , we can write , where and are analytic in . In 1984, Clunie and Sheil-Small [1] introduced a class of complex-valued harmonic maps which are univalent and sense-preserving in the open unit disk . The function can be represented by , where are analytic in the open unit disk They also proved that the function is locally univalent and sense-preserving in , if and only if For more basic studies, one may refer to Duren [2] and Ahuja [3]. It is worthy to note that if in (1), then the class reduces to the familiar class of analytic functions. For this class, may be expressed as of the form

Further, we suppose subclass of consisting of function of the form (1) with Now, we let , , and denote the subclasses of of harmonic functions which are, respectively, convex, starlike, and close-to-convex in . Also, let be the class of sense-preserving, typically real harmonic functions in For a detailed study of these classes, one may refer to [1, 2].

A function of the form (1) is said to be in the class , if it satisfy the condition

Similarly, a function of the form (1) is said to be in the class , if it satisfy the condition where and .

Now, we define the subclass of consisting of functions , so that and are of the form

Define and , where consists of the functions in . The classes , , , and were studied, respectively, by Ahuja and Jahangiri [4] and Rosy et al. [5].

Let , and , for and with Wright’s generalized hypergeometric functions [6] is defined by which is analytic for suitable bounded values of (see also [7, 8]). The generalized Mittag-Leffler, Bessel-Maitland, and generalized hypergeometric functions are some of the important special cases of Wright’s generalized hypergeometric functions, and for their details, one may refer to [8].

For with and with , we define Wright’s generalized hypergeometric functions: with

We consider a harmonic univalent function where and and are given by

From (12), we have for

For some fixed value of and for we denote provided that

Making use of (13) and (15), we have provided that (16) holds true.

The convolution of two functions of the form (1) and of the form is given by

Now, we introduce a convolution operator as where and given by (1) and (10), respectively. Hence

The application of the special functions on the geometric function theory always attracts researchers with various kinds of special functions, for example, hypergeometric functions [9–11], confluent hypergeometric functions [12], generalized hypergeometric functions [6, 13], Bessel functions [14], generalized Bessel functions [15–17], Wright functions [18–21], Fox-Wright functions [6, 22], and Mittag-Leffler functions [23] that have rich applications in analytic and harmonic univalent functions. By using special functions, some researchers introduce operators, for example, Carlson-Shaffer operator [24], Hohlov operator [25], and Dziok-Srivastava operator [26, 27], and obtain interesting results. Motivated with the work of [20], we obtain some inclusion relation between the classes , , , , and by applying the convolution operator

2. Main Results

In order to establish our main results, we shall require the following lemmas.

Lemma 1 [1]. If , where and are given by (5) with , then

Lemma 2 [1]. Let , where and are given by (1) with . Then

Lemma 3 [5]. Let be given by (5). If and then is a sense-preserving Goodman-Rønning-type harmonic univalent function in and

Remark 4. In [5], it is also shown that given by (5) is in the family , if and only if the coefficient condition (24) holds. Moreover, if , then

Theorem 5. Let and , and if the inequality holds, then

Proof. Let , where and are given by (1) with . We have to prove that , where is defined by (21). To prove , in view of Lemma 3, it is sufficient to prove that , where By using Lemma 1, by the given hypothesis. This completes the proof of Theorem 5.

The result is sharp for the function

Theorem 6. Let and , and if the inequality holds, then and

Proof. Let , where and are given by (1) with ; we need to prove that , where is defined by (21). In view of Lemma 3, it is sufficient to prove that , where is given by (27). Now using Lemma 2, we have by the given hypothesis. Thus, the proof of Theorem 6 is established.

The result is sharp for the function where

In our next theorem, we establish connections between and .

Theorem 7. Let and , and if the inequality holds, then

Proof. Let be given by (1). We have to prove that , where Now, using Remark 4, we have by the given hypothesis. This completes the proof of Theorem 7.

The result is sharp for the function where

3. Some Consequences of the Main Results

If we let and in (10), then reduces to a harmonic univalent function involving the following generalized Mittag-Leffler functions as where

With these specializations, the convolution operator reduces to the operator , which is defined as

For these specific values of and , Theorems 5–7 yield the following results.

Corollary 8. If the inequality holds, then .

Corollary 9. If the inequality holds, then and .

Corollary 10. If the inequality holds, then .

Remark 11. If we put , , , and , then and results of Theorems 5–7 reduce to corresponding results of Maharana and Sahoo [28].

Remark 12. If we put , , , and , then and results of Theorems 5–7 reduce to corresponding results of Porwal and Dixit [11].

Data Availability

No data is required.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally worked on the results, and they read and approved the final manuscript.