A Study of Spiral-Like Harmonic Functions Associated with Quantum Calculus

Abstract

This article introduces new subclasses of harmonic univalent functions associated with -difference operator. Modified -multiplier transformation is defined, and certain geometric properties such as the sufficient condition, distortion result, extreme points, and invariance of convex combination of the elements of the subclasses are discussed by employing the newly defined -operator. Also, various well-known results already proved in the literature are pointed out.

1. Introduction

A function is known to be real harmonic function in domain if and are continuous in and satisfies

Continuous function defined by is harmonic if both and are real harmonic in . We found that, in any simply connected domain , every harmonic function can be expressed by , where and are analytic in , and are called, respectively, the analytic and coanalytic parts of .

The class of complex-valued harmonic functions defined in the open unit disc and normalized by is denoted by . The function in the class has the following power series representation:

It is clear that when is identically zero, the class coincides with the class of normalized analytic functions in . Due to Lewy [1], a function is locally univalent and sense-preserving in if and only if

We indicate by the subclass of consisting of all sense-preserving univalent harmonic functions . Firstly, Clunie and Sheil-Small [2] discussed certain geometric properties of the class and its subclasses. Later on, several authors contributed in the study of subclasses of the class , for example, see [3–9]. The most prominent author Jahangiri [10] investigated various interesting properties of the class of starlike harmonic functions of order , , defined by

For the convenience, we present the notion of q-difference operator briefly. Jackson [11] introduced the q-difference operator and is defined by

for and with . We note that , where is the ordinary derivative of the function. It is clear that where for and . For some recent investigations involving -calculus, we may refer the interested reader to [12–17]. Recently, in [18], Shah and Noor introduced the -analogue of multiplier transformation by where , and . It is noted that for nonnegative integer and , the operator coincides with the Salagean -differential operator defined in [19]. Moreover, if in (8), then the multiplier transformation studied by the Cho and Kim in [20] is deduced. Nowadays, several subclasses of associated with operators and -operators were discussed by the prominent researchers, like [21–26]. In motivation of the above said literature, first, we modify the -multiplier transformation, and then we define certain new subclasses of . For given by (2), we define the modified -multiplier transformation of as where is given by (8) and

It is observed that, for , the modified -multiplier transformation defined by (9) turns out to be the multiplier transformation introduced in [18]. For , we define a new class as the following.

Definition 1. Let . Then if where , and .
Particularly, for , the class reduces to the class denoted by of functions that satisfies where and . Moreover, if then the class coincides with the class introduced by Jahangiri [10]. We further define , where denotes the subclass of consisting of functions of the type , where Now, by using modified -multiplier transformation given by (9), we define the following.

Definition 2. Let . Then if where , , , , and .

Also, we define , where denotes the subclass of consisting of functions given by (13). It is noted that, for , we have and . In particular, if we take and in above definitions, then we have well-known classes and introduced by Jahangiri [22].

The next section presents the main investigations such as the sufficient condition, distortion result, extreme points, and invariance of convex combination of the elements of the subclasses defined as above.

2. Main Results

Theorem 3. Let is given by (2) and satisfies where , , , , and . Then, .

Proof. We need to prove that if the coefficients of the harmonic function given by (2) satisfy the inequality (15), then it also satisfies (14). It is known that if and only if . So, it suffices to prove that or equivalently, From the left hand side, The above expression is nonnegative by (15). Hence, .
The harmonic function is where and show that the coefficient bound given by (15) is sharp. For different choices of parameters, we deduce certain results as follows. If in Theorem 3, then we have a following new result.

Corollary 4. Let a function given by (2) and satisfies where , , and . Then, .

If , then Corollary 4 reduces to a new result as follows:

Corollary 5. Let a function given by (2) and satisfies where and . Then, .

If we take and , then we have well-known result.

Corollary 6 (see [22]). Let a function given by (2) and satisfies where and . Then, .

When in Corollary 5, we get the sufficient condition for in proved by Jahangiri [10]. Moreover, for in Corollary 5, the sufficient condition for function in the class of starlike harmonic univalent mappings is obtained, see [4]. Now, we state and prove the necessary and sufficient conditions for the harmonic functions to be in as follows.

Theorem 7. Let given by (13). Then, if and only if where , , , , and .

Proof. The sufficient condition is obvious from the Theorem 3, because . We need to prove the necessary condition only; that is, if , then the coefficients of the function satisfy the inequality (23). Let . Then, by the definition of , we have where , , , , and . Equivalently, we can write (24) as Substituting in (25) and employing (8) along with (13), and also some computation yields For all values of in above required condition must hold. Selecting on the positive real axis where , we obtain The numerator in (27) is negative for sufficiently close to whenever the inequality (23) does not hold. Hence, there exists in for which the quotient in (27) is negative. This contradicts the required condition for , and so the proof is complete.

Next, we want to discuss the distortion bounds for the function , which yields a covering result for this class.

Theorem 8. If and , then with

Proof. Let . Taking absolute value of , we get where is given by (29). Hence, this is the required right hand inequality. Similarly, one can easily prove the required left hand inequality.

Letting and by making use of the left hand inequality of the above theorem, we obtain the following.

Corollary 9 (covering result). If , then where and .
In particular, we obtain the covering results for the newly defined classes and well-known classes of harmonic functions by choosing suitable choices of parameters.

Now, our task is to examine , the extreme points of closed convex hulls of .

Theorem 10. A function if and only if where , with , , and . Particularly, and are the extreme points of .

Proof. We assume function as given by (32) where and .
Equating (34) with (13), we get Now, Thus, by Theorem 7, . Conversely, let . We take with . We follow our required result by substituting the values of and from the above relations in (13).
Finally, we wish to show that the class is closed under the convex combination of its elements.

Theorem 11. The class is closed under the convex combination.

Proof. Let , , with Making use of Theorem 7, we have with .
Now, To prove our result, we use (40) and (41) Therefore, .

3. Conclusions

In this research, we have defined some new subclasses of harmonic univalent functions related to the -difference operator. Also, we have introduced and studied the modified -multiplier transformation. Several geometric properties such as sufficient condition, necessary conditions, distortion results, and invariance of classes under convex combination and extreme points are investigated. It is also noted that our investigations deduced various well-known results. In addition, this work can be extend for multivalent functions and -calculus.

Data Availability

No data were used to support this study.

Conflicts of Interest

There is no conflict of interest regarding the publication of this article.

Authors’ Contributions

All authors equally contributed to this manuscript and approved the final version.