Abstract

In this paper, we define a new class of Sakaguchi type-meromorphic harmonic functions in the Janowski domain that are starlike with respect to symmetric point. Furthermore, we investigate some important geometric properties like sufficiency criteria, distortion bound, extreme point theorem, convex combination, and weighted means.

1. Introduction and Definitions

One of the contemporary developments in Mathematics is the solicitations of harmonic analysis in other fields. Like various other fields, it has immensely influenced and nurtured the branch of geometric function theory. Jahangiri et al. [1] defined and studied a subclass of harmonic and univalent functions. Another example of such work would be an article of Porwal and Dixit [2], who used a certain convolution operator involving hypergeometric functions to define a class of univalent functions. As a consequence, many mathematicians generalized many ideas of this field and various important results with the help of some operators; the work of Porwal et al. [3], Porwal et al. [4], and Porwal and Dixit [5] are worth mentioning here. Recently, some subclasses of harmonic functions were investigated by Arif et al. [6] and Khan et al. [7]. To start with, we give preliminaries which will be useful in understanding the concepts of this research.

A real-valued function is said to be harmonic in a domain if it has a continuous second partial derivative and satisfy the Laplace’s equation

A continuous complex-valued function is said to be harmonic in a complex domain if both its real and imaginary parts are real harmonic in . In any simply connected domain , one can write where and are analytic in . The class of such functions is denoted by . The condition is necessary and sufficient for to be locally univalent and sense preserving in , see [8]. There are different papers on univalent harmonic functions defined in unit disc for details, see [9–14]. For in the punctured open unit disc and let denote the class of functions which are harmonic in where is analytic in and has a simple pole at the origin with residue 1, while is analytic in . The class was studied in [15–17]. Furthermore, denoted by , a subclass of , consisting a functions of the form which are harmonic univalent in punctured unit disc

For functions given by (2) and given by we recall the Hadamard product (or convolution) of and by

In terms of the Hadamard product (or convolution), we choose as a fixed function in such that exists for any and for various choices of , we get different linear operators which have been studied in the recent past.

Recently, Khan et al. [18] introduced and studied a class of meromorphic starlike functions with respect to symmetric point in circular domain i.e.,

Motivated from the above discussion on harmonic functions and class of meromorphic starlike functions with respect to symmetric point, we introduced the class of meromorphic harmonic univalent functions as:

Let Then, the function is in the class if it satisfies the condition where the symbol represent well-known subordination and

or equivalently

Furthermore, we denote subclass of consisting of harmonic meromorphic functions of the form (3).

2. Main Results

Theorem 1. Let be of the form (2) and satisfies the condition with then is harmonic univalent sense-preserving in and

Proof. For we obtain where we have used (10) and this shows that the function is univalent.

Now to show is sense-preserving harmonic mapping in , consider This shows that is sense-preserving.

Now, to show that from (9), it is enough to show that

For this, consider

Hence, complete the proof.

Example 2. The meromorphic univalent function such that we have

Thus, and above coefficient bound given in (10) is sharp for this function.

Theorem 3. Let and of the form (3), then if it satisfies the conditionwith

Proof. The proof is similar to Theorem 1, so omitted.

Theorem 4. Let and of the form (3), Then,

Proof. Consider

Similarly, proceeding as above we get

Hence, this completes the result.

Theorem 5. Let and of the form (3) then if and only if where where , and

Proof. Let

Thus, hence, Conversely, let . Set

Therefore, can be written as

Theorem 6. The class is closed under convex combination.

Proof. For let be of the form

Then form (10), we get

For , the convex combination of is

Using (10), we have thus prove our desired results.

Theorem 7. Let for be of the form (29), then, their weighted mean is also in the class where is defined below

Proof. From (33), one may easily write

To show that it is enough to show that

Now consider

Hence,

Data Availability

Data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

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

Acknowledgments

The third author is supported by UKM Grant: GUP-2019-032.