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

This article introduces new subclasses of harmonic univalent functions associated with q -di ﬀ erence operator. Modi ﬁ ed q -multiplier transformation is de ﬁ ned, and certain geometric properties such as the su ﬃ cient condition, distortion result, extreme points, and invariance of convex combination of the elements of the subclasses are discussed by employing the newly de ﬁ ned q -operator. Also, various well-known results already proved in the literature are pointed out.


Introduction
A function θ : ℝ ⟶ ℝ is known to be real harmonic function in domain D if θ xx and θ yy are continuous in D and satisfies θ xx x, y ð Þ+ θ yy x, y ð Þ= 0: ð1Þ Continuous function h : Ωð⊂ℂÞ ⟶ ℂ defined by hðzÞ = θ 1 ðx, yÞ + iθ 2 ðx, yÞ is harmonic if both θ 1 ðx, yÞ and θ 2 ðx, yÞ are real harmonic in Ω. We found that, in any simply connected domain Ω, every harmonic function hðzÞ can be expressed by hðzÞ = h 1 ðzÞ + h 2 ðzÞ, where h 1 and h 2 are analytic in Ω, and are called, respectively, the analytic and coanalytic parts of h.
The class of complex-valued harmonic functions h = h 1 + h 2 defined in the open unit disc U = fz : jzj < 1g and normalized by h 1 ð0Þ = h 2 ð0Þ = h 1 ′ ð0Þ − 1 = 0 is denoted by H . The function in the class H has the following power series representation: It is clear that when h 2 ðzÞ is identically zero, the class H coincides with the class A of normalized analytic functions in U. Due to Lewy [1], a function h ∈ H is locally univalent and sense-preserving in U if and only if We indicate by S H the subclass of H consisting of all sense-preserving univalent harmonic functions h. Firstly, Clunie and Sheil-Small [2] discussed certain geometric properties of the class S H and its subclasses. Later on, several authors contributed in the study of subclasses of the class S H , for example, see [3][4][5][6][7][8][9]. The most prominent author Jahangiri [10] investigated various interesting properties of the class S * H ðςÞ of starlike harmonic functions of order ς, ð0 ≤ ς < 1Þ, 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 q ∈ ð0, 1Þ and h 1 ∈ A with h 1 ðzÞ = z + ∑ ∞ n=2 a n z n . We note that lim q⟶1 − ∂ q h 1 ðzÞ = h 1 ′ðzÞ, where h 1 ′ðzÞ is the ordinary derivative of the function. It is clear that where for n ∈ ℕ = f1, 2, 3, ::g and z ∈ U. For some recent investigations involving q-calculus, we may refer the interested reader to [12][13][14][15][16][17]. Recently, in [18], Shah and Noor introduced the q-analogue of multiplier transformation I s q,τ : A ⟶ A by where h 1 ∈ A, s ∈ ℝ and τ > −1. It is noted that for nonnegative integer s and τ = 0, the operator I s q,τ coincides with the Salagean q-differential operator defined in [19]. Moreover, if q ⟶ 1 − in (8), then the multiplier transformation studied by the Cho and Kim in [20] is deduced. Nowadays, several subclasses of S H associated with operators and q-operators were discussed by the prominent researchers, like [21][22][23][24][25][26]. In motivation of the above said literature, first, we modify the q-multiplier transformation, and then we define certain new subclasses of S H . For h = h 1 + h 2 given by (2), we define the modified q-multiplier transformation of h as where I s q,τ h 1 ðzÞ is given by (8) and It is observed that, for h 2 = 0, the modified q-multiplier transformation defined by (9) turns out to be the q − multiplier transformation introduced in [18]. For h = h 1 + h 2 ∈ S H , we define a new class HST q ðζ, ςÞ as the following.
Also, we define HST s,τ q ðζ, ςÞ = HST s,τ q ðζ, ςÞ ∩ S H , where S H denotes the subclass of S H consisting of functions given by (13). It is noted that, for s = τ = 0, we have HST s,τ q ðζ, ςÞ = HST q ðζ, ςÞ and HST s,τ q ðζ, ςÞ = HST q ðζ, ςÞ. In particular, if we take ζ = τ = 0 and s = m ∈ ℕ in above definitions, then we have well-known classes H m q ðζ, ςÞ and H m q ðζ, ςÞ 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.

Journal of Function Spaces
If we take ζ = τ = 0 and s = m ∈ ℕ, then we have wellknown result.
When ζ = 0 in Corollary 5, we get the sufficient condition for f in S * H ðςÞ proved by Jahangiri [10]. Moreover, for ζ = ς = 0 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 h = h 1 + h 2 to be in HST s,τ q ðζ, ςÞ as follows.
where s ∈ ℝ, τ > −1, ς ∈ ½0, 1Þ, jζj < π/2, and q ∈ ð0, 1Þ. Equivalently, we can write (24) Substituting h s = h + h 2,s in (25) and employing (8) along with (13), and also some computation yields For all values of z in U above required condition must hold. Selecting z on the positive real axis where 0 ≤ z = r < 1, we obtain The numerator in (27) is negative for r sufficiently close to 1 whenever the inequality (23) does not hold. Hence, there exists z 0 = r 0 in ð0, 1Þ for which the quotient in (27) is negative. This contradicts the required condition for h s ∈ HST s,τ q ðζ, ςÞ, and so the proof is complete.
Next, we want to discuss the distortion bounds for the function h ∈ HST s,τ q ðζ, ςÞ, which yields a covering result for this class.

Theorem 8. If h ∈ HST
s,τ q ðζ, ςÞ and jzj = r < 1, then Proof. Let h ∈ HST s,τ q ðζ, ςÞ. Taking absolute value of h, we get Journal of Function Spaces where T is given by (29). Hence, this is the required right hand inequality. Similarly, one can easily prove the required left hand inequality.
Letting r ⟶ 1 and by making use of the left hand inequality of the above theorem, we obtain the following.
Corollary 9 (covering result). If h ∈ HST s,τ q ðζ, ςÞ, then where L =f½2 + τ q − ½1 + τ q ς cos ζg½2 + τ s q and M = ½1 + τ s+1 q . 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.
Equating (34) with (13), we get a n j j = ν n R n and b n j j = Ω n R * n : Now, with ∑ ∞ n=1 ðν n + Ω n Þ = 1. We follow our required result by substituting the values of ja n j and jb n j from the above relations in (13).
Finally, we wish to show that the class HST s,τ q ðζ, ςÞ is closed under the convex combination of its elements.

Journal of Function Spaces
Now, To prove our result, we use (40) and (41) Therefore, ∑ ∞ i=1 u i h i s ∈ HST s,τ q ðζ, ςÞ.

Conclusions
In this research, we have defined some new subclasses of harmonic univalent functions related to the q-difference operator. Also, we have introduced and studied the modified q-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 ðp, qÞ-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.