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

Authors: Shujaat Ali Shah; Luminita-Ioana Cotirla; Adriana Catas; Calin Dubau; Gabriel Cheregi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This article introduces new subclasses of harmonic univalent functions associated withq-difference operator. Modified q-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 q-operator. Also, various well-known results already proved in the literature are pointed out.


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## Body

## 1. Introduction

A functionθ:ℝ⟶ℝ is known to be real harmonic function in domain D if θxx and θyy are continuous in D and satisfies
(1)θxxx,y+θyyx,y=0.Continuous functionh:Ω⊂ℂ⟶ℂ defined by hz=θ1x,y+iθ2x,y is harmonic if both θ1x,y and θ2x,y are real harmonic in Ω. We found that, in any simply connected domain Ω, every harmonic function hz can be expressed by hz=h1z+h2¯z, where h1 and h2 are analytic in Ω, and are called, respectively, the analytic and coanalytic parts of h.The class of complex-valued harmonic functionsh=h1+h2¯ defined in the open unit disc U=z:z<1 and normalized by h10=h20=h1′0−1=0 is denoted by H. The function in the class H has the following power series representation:
(2)hz=z+∑n=2∞anzn+∑n=2∞bnzn.¯It is clear that whenh2z 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
(3)h1′z>h2′z,forz∈U.We indicate bySH 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 SH and its subclasses. Later on, several authors contributed in the study of subclasses of the class SH, for example, see [3–9]. The most prominent author Jahangiri [10] investigated various interesting properties of the class SH∗ς of starlike harmonic functions of order ς, 0≤ς<1, defined by
(4)Rzh′zhz=Rzh1′z−zh2′z¯h1z+h2z¯≥ς.For the convenience, we present the notion ofq-difference operator briefly. Jackson [11] introduced the q-difference operator and is defined by
(5)∂qh1z=h1z−h1qz1−qz;q≠1,z≠0,forq∈0,1 and h1∈A with h1z=z+∑n=2∞anzn. We note that limq⟶1−∂qh1z=h1′z, where h1′z is the ordinary derivative of the function. It is clear that
(6)∂q∑n=1∞anzn=∑n=1∞nqzn−1,where
(7)nq=1−qn1−q=1+q+q2+⋯,for n∈ℕ=1,2,3,.. and z∈U. For some recent investigations involving q-calculus, we may refer the interested reader to [12–17]. Recently, in [18], Shah and Noor introduced the q-analogue of multiplier transformation Iq,τs:A⟶A by
(8)Iq,τsh1z=z+∑n=2∞n+τq1+τqsanzn,where h1∈A, s∈ℝ and τ>−1. It is noted that for nonnegative integer s and τ=0, the operator Iq,τs 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 SH associated with operators and q-operators were discussed by the prominent researchers, like [21–26]. In motivation of the above said literature, first, we modify the q-multiplier transformation, and then we define certain new subclasses of SH. For h=h1+h2¯ given by (2), we define the modified q-multiplier transformation of h as
(9)Iq,τshz=Iq,τsh1z+−1sIq,τsh2z¯,where Iq,τsh1z is given by (8) and
(10)Iq,τsh2z=∑n=1∞n+τq1+τqsbnzn.It is observed that, forh2=0, the modified q-multiplier transformation defined by (9) turns out to be the q−multiplier transformation introduced in [18]. For h=h1+h2¯∈SH, we define a new class HSTqζ,ς as the following.Definition 1.
Leth∈SH. Then h∈HSTqζ,ς if
(11)R1+eiζz∂qhzhz−1≥ςcosζ,where ς∈0,1, ζ<π/2 and q∈0,1.
Particularly, forq⟶1−, the class HSTqζ,ς reduces to the class denoted by SH∗ζ,ς of functions h∈SH that satisfies
(12)R1+eiζzh′zhz−1≥ςcosζ,where ς∈0,1 and ζ<π/2. Moreover, if ζ=0, then the class SH∗ζ,ς coincides with the class SH∗ς introduced by Jahangiri [10]. We further define HST¯qζ,ς=HSTqζ,ς∩SH¯, where SH¯ denotes the subclass of SH consisting of functions of the type hsz=h1z+h2,sz¯, where
(13)h1z=z−∑n=2∞anznandh2,sz=−1s∑n=2∞bnzn.
Now, by using modifiedq-multiplier transformation given by (9), we define the following.Definition 2.
Leth∈SH. Then h∈HSTqs,τζ,ς if
(14)R1+eiζIq,τs+1hzIq,τshz−1≥ςcosζ,where s∈ℝ, τ>−1, ς∈0,1, ζ<π/2, and q∈0,1.Also, we defineHST¯qs,τζ,ς=HSTqs,τζ,ς∩SH¯, where SH¯ denotes the subclass of SH consisting of functions given by (13). It is noted that, for s=τ=0, we have HSTqs,τζ,ς=HSTqζ,ς and HST¯qs,τζ,ς=HST¯qζ,ς. In particular, if we take ζ=τ=0 and s=m∈ℕ in above definitions, then we have well-known classes Hqmζ,ς and H¯qmζ,ς 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.
Leth=h1+h2¯∈SH is given by (2) and satisfies
(15)∑n=1∞n+τq1+τq−ςcosζan+n+τq1+τq+ςcosζbn×n+τq1+τqs≤21−ςcosζ,where s∈ℝ, τ>−1, ς∈0,1, ζ<π/2, and q∈0,1. Then, h∈HSTqs,τζ,ς.Proof.
We need to prove that if the coefficients of the harmonic functionh=h1+h2¯∈SH given by (2) satisfy the inequality (15), then it also satisfies (14). It is known that Rw≥ξ if and only if 1−ξ+w≥1+ξ−w. So, it suffices to prove that
(16)1−ςcosζ+1+eiζIq,τs+1hzIq,τshz−1≥1+ςcosζ−1−eiζIq,τs+1hzIq,τshz−1,or equivalently,
(17)2−ςcosζIq,τshz+eiζIq,τs+1hz−Iq,τshz−ςcosζIq,τshz−eiζIq,τs+1hz−Iq,τshz≥0.
From the left hand side,(18)2−ςcosζIq,τshz+eiζIq,τs+1hz−Iq,τshz−ςcosζIq,τshz−eiζIq,τs+1hz−Iq,τshz=2−ςcosζIq,τsh1z+−1sIq,τsh2z¯−eiζIq,τsh1z+−1sIq,τsh2z¯−Iq,τs+1h1z+−1sIq,τs+1h2z¯−ςcosζIq,τsh1z+−1sIq,τsh2z¯+eiζIq,τsh1z+−1sIq,τsh2z¯−Iq,τs+1h1z+−1sIq,τs+1h2z¯≥2−ςcosζz+2−ςcosζ×∑n=2∞n+τq1+τqsanzn+−1s∑n=1∞n+τq1+τqsbnzn¯−eiζ∑n=2∞n+τq1+τqsanzn+−1s∑n=1∞n+τq1+τqsbnzn¯−∑n=2∞n+τq1+τqs+1anzn+−1s∑n=1∞n+τq1+τqs+1bnzn¯−ςcosζz+ςcosζ×∑n=2∞n+τq1+τqsanzn+−1s∑n=1∞n+τq1+τqsbnzn¯+eiζ∑n=2∞n+τq1+τqsanzn+−1s∑n=1∞n+τq1+τqsbnzn¯−∑n=2∞n+τq1+τqs+1anzn+−1s∑n=1∞n+τq1+τqs+1bnzn¯≥2−ςcosζz−∑n=2∞2−ςcosζ+n+τq1+τq−1n+τq1+τqsanzn−∑n=1∞ςcosζ+n+τq1+τq−1n+τq1+τqsbnzn−∑n=2∞n+τq1+τq−1−ςcosζn+τq1+τqsanzn−∑n=2∞n+τq1+τq+ςcosζ+1n+τq1+τqsbnzn≥21−ςcosζ×1−11−ςcosζ∑n=2∞n+τq1+τq−ςcosζan+∑n=1∞n+τq1+τq+ςcosζbnn+τq1+τqs.
The above expression is nonnegative by (15). Hence, h∈HSTqs,τζ,ς.
The harmonic function is(19)hz=z+∑n=2∞1−ςcosζ1+τqn+τq−1+τqςcosζψntnzn+∑n=1∞1−ςcosζ1+τqn+τq+1+τqςcosζψnυ¯nz¯n,where ψn=n+τq/1+τqs and ∑n=2∞tn+∑n=1∞υn=1 show that the coefficient bound given by (15) is sharp. For different choices of parameters, we deduce certain results as follows. If s=τ=0 in Theorem 3, then we have a following new result.Corollary 4.
Let a functionhz=h1z+h2z¯∈SH given by (2) and satisfies
(20)∑n=1∞nq−ςcosζ1−ςcosζan+nq+ςcosζ1−ςcosζbn≤2,where ς∈0,1, ζ<π/2, and q∈0,1. Then, h∈HSTqζ,ς.Ifq⟶1−, then Corollary 4 reduces to a new result as follows:Corollary 5.
Let a functionh=h1+h2¯∈SH given by (2) and satisfies
(21)∑n=1∞n−ςcosζ1−ςcosζan+n+ςcosζ1−ςcosζbn≤2,where ς∈0,1 and ζ<π/2. Then, h∈SH∗ζ,ς.If we takeζ=τ=0 and s=m∈ℕ, then we have well-known result.Corollary 6 (see [22]).
Let a functionh=h1+h2¯∈SH given by (2) and satisfies
(22)∑n=2∞nqmnq−ςan+∑n=1∞nqmnq+ςbn≤1−ς,where ς∈0,1 and q∈0,1. Then, f∈Hqmζ,ς.Whenζ=0 in Corollary 5, we get the sufficient condition for f in SH∗ς 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=h1+h2¯ to be in HST¯qs,τζ,ς as follows.Theorem 7.
Leths=h+h2,s¯∈SH given by (13). Then, hs∈HST¯qs,τζ,ς if and only if
(23)∑n=1∞n+τq1+τq−ςcosζan+n+τq1+τq+ςcosζbnn+τq1+τqs≤21−ςcosζ,where s∈ℝ, τ>−1, ς∈0,1, ζ<π/2, and q∈0,1.Proof.
The sufficient condition is obvious from the Theorem3, because HST¯qs,τζ,ς⊂HSTqs,τζ,ς. We need to prove the necessary condition only; that is, if hs∈HST¯qs,τζ,ς, then the coefficients of the function hs=h+h2,s¯ satisfy the inequality (23). Let hs∈HST¯qs,τζ,ς. Then, by the definition of HST¯qs,τζ,ς, we have
(24)R1+eiζIq,τs+1hszIq,τshsz−1−ςcosζ≥0,where s∈ℝ, τ>−1, ς∈0,1, ζ<π/2, and q∈0,1. Equivalently, we can write (24) as
(25)R1−ςcosζIq,τshsz+eiζIq,τs+1hsz−Iq,τshszIq,τshsz≥0.
Substitutinghs=h+h2,s¯ in (25) and employing (8) along with (13), and also some computation yields
(26)R1−ςcosζz−∑n=2∞n+τq/1+τq−ςcosζn+τq/1+τqsanzn−∑n=1∞n+τq/1+τq+ςcosζn+τq/1+τqsbnznz−∑n=2∞n+τq/1+τqsanzn−∑n=1∞n+τq/1+τqsbnzn≥0.
For all values ofz in U above required condition must hold. Selecting z on the positive real axis where 0≤z=r<1, we obtain
(27)1−ςcosζ−∑n=2∞n+τq/1+τq−ςcosζn+τq/1+τqsanrn−1−∑n=1∞n+τq/1+τq+ςcosζn+τq/1+τqsbnrn−1z−∑n=2∞n+τq/1+τqsanrn−1−∑n=1∞n+τq/1+τqsbnrn−1≥0.
The numerator in (27) is negative for r sufficiently close to 1 whenever the inequality (23) does not hold. Hence, there exists z0=r0 in 0,1 for which the quotient in (27) is negative. This contradicts the required condition for hs∈HST¯qs,τζ,ς, and so the proof is complete.Next, we want to discuss the distortion bounds for the functionh∈HST¯qs,τζ,ς, which yields a covering result for this class.Theorem 8.
Ifh∈HST¯qs,τζ,ς and z=r<1, then
(28)1−b1r−Tr2≤hz≤1+b1r+Tr2,with
(29)T=1+τqs+12+τqs1−ςcosζ2+τq−1+τqςcosζ−1+ςcosζ2+τq−1+τqςcosζb1.Proof.
Leth∈HST¯qs,τζ,ς. Taking absolute value of h, we get
(30)hz≤1+b1r+∑n=2∞an+bnr2≤1+b1r+1−ςcosζ1+τqs+12+τq−1+τqςcosζ2+τqs×∑n=2∞2+τq−1+τqςcosζ2+τqs1−ςcosζ1+τqs+1an+bnr2≤1+b1r+1−ςcosζ1+τqs+12+τq−1+τqςcosζ2+τqs∑n=2∞n+τq−1+τqςcosζ1−ςcosζ1+τqn+τq1+τqsan+bnr2≤1+b1r+1−ςcosζ1+τqs+12+τq−1+τqςcosζ2+τqs1−1−ςcosζ1−ςcosζb1r2,by2.2≤1+b1r+Tr2,where T is given by (29). Hence, this is the required right hand inequality. Similarly, one can easily prove the required left hand inequality.Lettingr⟶1 and by making use of the left hand inequality of the above theorem, we obtain the following.Corollary 9 (covering result).
Ifh∈HST¯qs,τζ,ς, then
(31)w:w<L−M1−ςcosζL−L−M1+ςcosζLb1⊂fE,where L=2+τq−1+τqςcosζ2+τqs and M=1+τqs+1.
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 examineclcoHST¯qs,τζ,ς, the extreme points of closed convex hulls of HST¯qs,τζ,ς.Theorem 10.
A functionhs∈HST¯qs,τζ,ς if and only if
(32)hsz=∑n=1∞νnhnz+ωngsnz,where h1z=z,
(33)hnz=z−1−ςcosζ1+τqn+τq−1+τqςcosζψnzn;n=2,3,⋯,gsnz=z+−1s1−ςcosζ1+τqn+τq+1+τqςcosζψnz¯n;n=1,2,3,⋯,with ∑n=1∞νn+Ωn=1, νn,Ωn≥0, and ψn=n+τq/1+τqs. Particularly, hn and gsn are the extreme points of HST¯qs,τζ,ς.Proof.
We assume functionhs as given by (32)
(34)hsz=∑n=1∞νnhnz+Ωngsnz=∑n=1∞νn+Ωnz−∑n=2∞νnRnzn+−1s∑n=1∞ΩnRn∗z¯n,where Rn=1−ςcosζ1+τq/n+τq−1+τqςcosζψn and Rn∗=1−ςcosζ1+τq/n+τq+1+τqςcosζψn.
Equating (34) with (13), we get
(35)an=νnRnandbn=ΩnRn∗.
Now,(36)∑n=1∞n+τq−1+τqςcosζψn1−ςcosζ1+τqan+n+τq−1+τqςcosζψn1−ςcosζ1+τqbn=1−X1+∑n=1∞νn+Ωn=2−X1≤2.
Thus, by Theorem7, hs∈HST¯qs,τζ,ς. Conversely, let hs∈HST¯qs,τζ,ς. We take
(37)νn=n+τq−1+τqςcosζψn1−ςcosζ1+τqan;n=2,3,⋯,(38)Ωn=n+τq+1+τqςcosζψn1−ςcosζ1+τqbn;n=1,2,⋯,with ∑n=1∞νn+Ωn=1. We follow our required result by substituting the values of an and bn from the above relations in (13).
Finally, we wish to show that the classHST¯qs,τζ,ς is closed under the convex combination of its elements.Theorem 11.
The classHST¯qs,τζ,ς is closed under the convex combination.Proof.
Lethis∈HST¯qs,τζ,ς, i=1,2,⋯, with
(39)his=z−∑n=2∞ai,nzn+−1s∑n=2∞bi,nz¯n.
Making use of Theorem7, we have
(40)∑n=2∞n+τq−1+τqςcosζψn1−ςcosζ1+τqai,n+∑n=1∞n+τq−1+τqςcosζψn1−ςcosζ1+τqbi,n≤1,with ψn=n+τq/1+τqs.
Now,(41)∑i=1∞ùihis=z−∑n=2∞∑i=1∞ùiai,nzn+−1s∑n=1∞∑i=1∞ùibi,nz¯n.
To prove our result, we use (40) and (41)
(42)∑n=2∞n+τq−1+τqςcosζψn1−ςcosζ1+τq∑i=1∞ùiai,n+∑n=1∞n+τq−1+τqςcosζψn1−ςcosζ1+τq∑i=1∞ùibi,n≤∑i=1∞ùi=1.
Therefore,∑i=1∞ùihis∈HST¯qs,τζ,ς.


## 3. Conclusions

In this research, we have defined some new subclasses of harmonic univalent functions related to theq-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 p,q-calculus.



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