# Certain New Class of Harmonic Functions Involving Quantum Calculus

Authors: Mohammad Faisal Khan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Here in this paper, we are using the concepts ofq-calculus operator theory associated with harmonic functions and define the q-Noor integral operator for harmonic functions f∈H0. We investigate a new class SH0m,q,α of harmonic functions f∈H0. In this class, we prove a necessary and sufficient convolution condition for the functions f∈H0 and also we proved that this sufficient coefficient condition is sense preserving and univalent in the class SH0m,q,α. It is proved that this coefficient condition is necessary for the functions in its subclass TSH0m,q,α. By using this necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of q-starlikeness and q-convexity of order α in the class TSH0m,q,α. Also we obtained extreme points for the functions in the class TSH0m,q,α.


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## Body

## 1. Introduction and Definitions

A complex-valued functionf=u+iv is said to be harmonic in in open unit disc U=z∈ℂ:z<1 if both u and v are real valued harmonic functions in U. Also the complex-valued harmonic function f=u+iv can also be expressed as f=h+g¯, where h and g are analytic in U. In particular, h is called analytic part, and g is called coanalytic part of f. The Jacobian of the function f=u+iv is given by
(1)Jfz=h′z2−g′z2.It is known (see [1]) that every harmonic function f=h+g¯ to be locally univalent and sense preserving in U if and only if Jfz>0 in U which is equivalent to uz=g′z/h′z in U such that
(2)uz<1,forallz∈U.For detail (see [2]). Let H indicates the class of harmonic functions in U. Also let H0 denoted by the family of harmonic functions f=h+g¯∈H which have the series expansion of the form:
(3)fz=z+∑n=2∞anzn+∑n=1∞bnzn¯,z∈U,where h and g are analytic functions with the following series expansion:
(4)hz=z+∑n=2∞anzn,gz=∑n=1∞bnzn,b1<1,z∈U.The series defined in (3) and (4) are convergent in the open unit disc U. Also let S represents all functions (say f∈S) which are univalent analytic in U and satisfy the condition
(5)f0=f′0−1=0.Further, letSH denotes the class of all harmonic functions f=h+g¯∈H0 which are sense preserving and univalent in U. The class SH reduces to the class S if coanalytic part of f is zero.Clunie and Small [3] and Small [4] studied the class SH along with some of their subfamilies. Particularly, they explored and studied the families of starlike harmonic and convex harmonic functions in U, which are given as follows:
(6)SH∗=f∈SH:DHfzfz≺1+z1−z,z∈U,SHc=f∈SH:DHfz∈SH∗z∈U,where
(7)DHfz=zh′z−zg′z¯.In [5], Dziok introduced a new family SH∗L,M,L,M∈ℂ, and L≠M of Janowski harmonic functions and defined by
(8)SH∗L,M=f∈SH:DHfzfz≺1+Lz1+Mz,z∈U,where DHfz is given by (5). We can see that
(9)SH∗1,−1=SH∗.The convolution of two functionsh,g∈A, is defined by
(10)h∗gz=∑n=1∞anbnzn,where
(11)hz=∑n=1∞anzn,gz=∑n=1∞bnzn.Similarly, the convolution of two harmonic functionsf=h+g¯ and f1=h1+g1¯ is defined by
(12)f∗f1z=h∗h1z+g∗g1z¯.The functionh subordinate to a function g and write hz≺gz,z∈U, if there exists a complex-valued function v which map U into itself such that v0=0 and hz=gvz. In particular, if g is univalent in U, then we have the following equivalence:
(13)hz≺gz,z∈U⇔h0=g0,hU⊂gU.In the nineteen century, several mathematician has been usingq-calculus operator theory in various area of science, such that fractional calculus, q-difference equation, optimal control, q-integral equations, and geometric function theory (GFT). In 1908, Jackson [6] introduced the q-derivative and q-integral operator and discussed some of their applications. In the year 1990, Ismail et al. [7] gave the idea of q-extension of class of q-starlike functions by implementing the q-calculus theory. Kanas and Raducanu [8] used q-calculus operator theory and introduced the q-Ruscheweyh differential operator for analytic functions. Zhang et. al [9] introduced a generalized conic domain Ωk,α,q by using the basic concepts of q-calculus and studied new subclass of q-starlike functions. Arif et al. defined q-Noor integral operator [10] by using the concept of convolution and used it to investigated some new subclasses of analytic functions. Further, in article [11], Khan et al. discussed some applications of q-derivative operator for multivalent functions, while coefficient estimates for a certain family of analytic functions involving a q-derivative operator were discussed by Raza et al. [12]. Recently, Srivastava et. al published few articles in which they implemented basic concepts of q-calculus operator theory and studied class of q-starlike functions from different aspects (see [13–16]). Additionally, a recently published article by Srivastava [17] is very suitable for researchers to work on this topic. For more recently, Khan et al. [18, 19] used the concepts of q-calculus operator theory to define some new subclasses of analytic functions. Also for more detail, we may refer to [20–25].For,q∈0,1, the q-derivative operator ∂q of f is defined as follows:
(14)∂qfz=fz−fqz1−qz,z≠0,0<q<1,=1+∑n=2∞nqanzn−1.Making use of (3) and (14), and for n∈ℕ, we have
(15)∂qfz=∂qhz+∂qgz¯,=1+∑n=2∞nqanzn−1+∑n=2∞nqbnzn−1¯,nq=1−q−11−qn.For more detail (see [26, 27]).Definition 1 (see [10, 28]).
Theq-Noor integral operator for the analytic function h is defined by
(16)Iqmhz=hz∗Tq,m+1z−1,z∈U,m>−1,where
(17)Tq,m+1z−1=z+∑n=2∞nq!Γq1+mΓqn+mzn,Tq,m+1z−1∗Tq,m+1z=z∂qhz.
Thus, we have(18)Iqmhz=z+∑n=2∞nq!Γq1+mΓqn+manzn=z+∑n=2∞nq!m+1n−1anzn=z+∑n=2∞Ψnqanzn,where
(19)Ψnq=nq!m+1n−1.
Clearly,(20)Iq0hz=z∂qhz=z1−z1−qz,Iq1hz=z1−z.Remark 2.
Whenq⟶1−, then q-Noor integral operator reduces to Noor integral operator (see [29]).
First of all Jahangiri [30] applied certain q-calculus operators to complex harmonic functions and obtained some useful results, while Porwal and Gupta discussed some application of q-calculus to harmonic univalent functions in [31]. Recently Arif et al. [27] introduced some new families of harmonic functions associated with the symmetric circular region. For some more recent investigation about harmonic univalent functions, we may refer to [32, 33]. By taking the motivation from the article Arif et al. [27], we define the q-Noor integral operator for the harmonic function f=h+g¯.Definition 3.
Let theq-Noor integral operator Iqm of order m>−1, for the harmonic function f=h+g¯ be defined as
(21)Iqmfz=Iqmhz+−1mIqmgz¯,where hz and gz is given by (4).In this paper, by using the concepts ofq-calculus operator theory and q-Noor integral operator for harmonic functions f, we define a new class SH0m,q,α of harmonic functions f∈H0. In this class, we prove a necessary and sufficient convolution condition for the functions f∈H0 and prove that this sufficient coefficient condition is sense preserving and univalent in the class SH0m,q,α. It is proved that this coefficient condition is necessary for the functions in its subclass TSH0m,q,α. By using this necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of q-starlikeness and q-convexity of order α and extreme points for the functions in the class TSH0m,q,α. This research work will motivate future research to work in the area of q-calculus operators together with harmonic functions.Definition 4.
LetSH0m,q,α be the family of harmonic functions f∈H0 that satisfy the subordination condition
(22)Iqm+1fzIqmfz≺1+Qz1+Tz,q∈0,1,0≤α<1,z∈U,where
(23)Q=α1+q−1,T=q.
Inequalities (22) is equivalent to the condition
(24)Iqm+1fz−IqmfzTIqm+1fz−QIqmfz<1.
We denote byTSH0m,q,α a subclass of harmonic functions f=h+g¯∈SH0m,q,α, where for m, functions h and g are of the form:
(25)hz=z−∑n=2∞anzn,gz=−1m∑n=2∞bnzn,z∈U.


## 2. Main Results

Theorem 5.
Letf∈H0. Then the function f∈SH0m,q,α if and only if
(26)Iqmfz∗ϕz,γ≠0,γ∈ℂ,γ=1,z∈U\0,where
(27)ϕz,γ=T−Qζz−1+Tζqz21−z1−qz−2z+Q+Tζz−1+Tζqz21−z1−qz¯.Proof.
Letf=h+g¯∈H0 be of the form (3). Then the function f∈SH0m,q,α if and only if (22) holds or equivalently
(28)Iqm+1fzIqmfz≠1+Qζ1+Tζ,ζ∈ℂ,ζ=1,z∈E0,which by (21) is given by
(29)1+TζIqmIqhz+−1m+1IqmIqgz¯−1+QζIqmhz+−1mIqmgz¯≠0.
On using (20), the condition (29) may also be given by
(30)Iqmhz∗1+Tζz1−z−1+Qζz1−z1−qz−−1mIqmgz¯∗1+Tζz¯1−z¯+1+Qζz¯1−z¯1−qz¯≠0.
Which on using the convolution∗ between two harmonic functions, we get
(31)Iqmfz∗ϕz,γ≠0,where the harmonic function ϕz,γ is given by (27).Theorem 6.
Letf=h+g¯∈H0 be of the form (3) and q∈0,1,0≤α<1. If
(32)∑n=2∞Lnan+Mnbn≤T−Q,where
(33)Ln=ΨnqmΨnq1+T−1+Q,(34)Mn=ΨnqmΨnq1+T+1+Q,where Ψnq is given by (19), then.
(i)
the functionf is locally univalent and sense-preserving as q⟶1−(ii)
the functionf∈SH0m,q,α
Equality occurs for the function(35)fz=z+∑n=2∞T−QLnγnzn+∑n=2∞T−QMnβnzn¯,∑n=2∞γn+βn=1.Proof.
For part (i), it is clear that the theorem is true for the functionfz≡z. Let f=h+g¯ and assume that there exist n≥2 such that an≠0 or bn≠0. Since Ψnq>1, we observe from (33) and (34) that Ln≥Mn>ΨnqT−Q, by which the condition (32) implies the condition
(36)∑n=2∞Ψnqan+bn<1,∂qhz−∂qgz≥1−∑n=2∞Ψnqanzn−1−∑n=2∞Ψnqbnzn−1,>1−z∑n=2∞Ψnqan+an≥1−z>0,in D which implies as q⟶1− that h′z>g′z in D that is a function f is locally univalent and sense-preserving in D.
For part (i), to prove thatf∈SH0m,q,α, we only need to show that f satisfy the condition (24). Consider for f=h+g¯ and for z=r0<r<1, we can write (24) as
(37)Iqm+1fz−Iqmfz−TIqm+1fz−QIqmfz=∑n=2∞ΨnqmΨnq−1anzn−−1m∑n=2∞ΨnqmΨnq+1bnzn¯−T−Qz+∑n=2∞ΨnqmTΨnq−Qanzn−−1m∑n=2∞ΨnqmTΨnq+Qbnzn¯≤∑n=2∞ΨnqmΨnq−1anrn−−1m∑n=2∞ΨnqmΨnq+1bnrn¯+T−Qr+∑n=2∞ΨnqmTΨnq−Qanrn+∑n=2∞ΨnqmTΨnq+Qbnrn¯<∑n=2∞Lnan+Mnbnrn−T−Q≤∑n=2∞Lnan+Mnbnrn−T−Q≤0,
if the condition (32) holds. This proves the condition (24). This completes the proof of Theorem 6.Theorem 7.
Letf=h+g¯∈H0 where h and g are given by (25). Then f∈TSH0m,q,α if and only if the condition (32) holds that is
(38)∑n=2∞Lnan+Mnbn≤T−Q,where Ln and Mn are given by (33) and (34).Proof.
If part is proved in Theorem6. To prove only if part, let f∈TSH0m,q,α. Then by the class condition (22), we have from (24) that for any z∈E.
(39)AT,Qanzn+BT,Qbnzn¯T−Qz−CT,Qanzn−DT,Qbnzn¯<1,where
(40)AT,Q=∑n=2∞ΨnqmΨnq−1,BT,Q=∑n=2∞ΨnqmΨnq+1,CT,Q≪∑n=2∞ΨnqmTΨnq−Q,DT,Q≪∑n=2∞ΨnqmTΨnq+Q.
Forz=r0≤r<1, we obtain
(41)AT,Qanrn−1+BT,Qbnrn−1¯T−Q−CT,Qanrn−1−DT,Qbnrn−1¯<1,which proves for Ln and Mn defined by (33) and (34) that
(42)∑n=2∞Lnan+Mnbnrn−1<T−Q.
Letσn be the sequence of partial sums of the series
(43)∑n=2∞Lnan+Mnbn.
Thenσn is a nondecreasing sequence, and by (42), it is bounded above. Thus, as r⟶1−, it is convergent and
(44)∑n=2∞Lnan+Mnbn=limn⟶∞σn≤T−Q.
This gives the condition (32).Remark 8.
Theorem7 gives a necessary and sufficient condition for the functions f=h+g¯∈H0, where h and g are given by (25) to be q-starlike and q-convex of order α in E if we put m=0 and m=1, respectively, in (38) and are given by
(45)∑n=2∞Ψnq−αan+Ψnq+αbn≤1−α,(46)∑n=2∞ΨnqΨnq−αan+Ψnq+αbn≤1−α.Theorem 9.
The classTSH0m,q,α is a convex and compact subclass of the class of functions f=h+g¯∈H0, where h and g are given by (25).Proof.
Leti=1,2,fi∈TSH0m,q,α; and let this m is of the form
(47)fiz=z−∑n=2∞ai,nzn+−1m∑n=2∞bi,nz¯n,z∈U.
Then for0≤ρ≤1(48)Fz=ρf1z+1−ρf2z=z−∑n=2∞ρa1,n+1−ρa2,nzn+−1m∑n=2∞ρb1,n+1−ρb2,nz¯n.and by Theorem 7, we get Ln and Mn given by (33) and (34) that
(49)∑n=2∞Lnρa1,n+1−ρa2,n+Mnρb1,n+1−ρb2,n=ρ∑n=2∞Lna1,n+Mnb1,n+1−ρ∑n=2∞Lna2,n+Mnb2,n≤ρT−Q+1−ρT−Q=T−Q.
Therefore,F∈TSH0m,q,α. Hence, the class TSH0m,q,α is convex.
On the other hand, if we considerfi∈TSH0m,q,α,i∈N=1,2,3⋯ of the form (47), and then by Theorem 7, we get for Ln and Mn defined by (33) and (34).
(50)∑n=2∞Lnai,n+Mnbi,n≤T−Q.
Hence, forz≤r0<r<1(51)fiz≤r+∑n=2∞ai,n+bi,nrn≤T−Q≤r+∑n=2∞Lnai,n+Mnbi,nrnΨ2qmΨ2q1+T−1+Q<r+T−QΨ2qmΨ2q1+T−1+Qr2.
Similarly, we get forz≤r0<r<1,(52)fiz>r−T−QΨ2qmΨ2q1+T−1+Qr2.
Therefore, classTSH0m,q,α is locally uniformly bounded.
If we assume thatfi⟶f, then we conclude that ai,n⟶an and bi,n⟶bn as i⟶∞ for any n=2,3⋯. Hence, from (50), we get
(53)∑n=2∞Lnan+Mnbn≤T−Q,which proves that f∈TSH0m,q,α. Therefore, the class TSH0m,q,α is closed. This proves that class TSH0m,q,α is compact.Corollary 10.
Letf∈TSH0m,q,α. Then for z=rr<1(54)r−T−QΨ2qmΨ2q1+T−1+Qr2,<fz<r+T−QΨ2qmΨ2q1+T−1+Qr2.
Furthermore,(55)w∈ℂ:w<1−T−QΨ2qmΨ2q1+T−1+Q⊂fU.Remark 11.
The minimum of all values of the radiusr∈0,1 for functions f∈TSH0m,q,α such that
(56)frzr∈Hq∗αis called the radius of q-starlikeness of order α and is denoted by rHq∗αTSH0m,q,α.
Now in next theorem, we obtain the radius ofq-starlikeness of order α for functions f∈TSH0m,q,α.Theorem 12.
Let0≤α<1, and Ln and Mn are defined by (33) and (34). Then
(57)rHq∗αTSH0m,q,α=infn≥21−αT−QminLnΨnq−α,MnΨnq+α1/n−1,where Ψnq defined by (19).Proof.
Letf=h+g¯∈TSH0m,q,α, then by Theorem 7, we have
(58)∑n=2∞Lnan+Mnbn≤T−Q,where Ln and Mn are defined, respectively, by (33) and (34). Let r0 be the radius of q-starlikeness of order α. Then fr0z/r0∈Hq∗α if and only if from (45) that
(59)∑n=2∞Ψnq−αan+Ψnq+αbnr0k−1≤1−α,which is true if
(60)Ψnq−α1−αr0k−1≤LnT−Q,n=2,3⋯,Ψnq+α1−αr0k−1≤MnT−Q,n=2,3⋯.
Or if(61)r0≤1−αT−QminLnΨnq−α,MnΨnq+α1/n−1.It follows that the radiusrHq∗αTSH0m,q,α in (57).Similarly, we may find the radius ofq-convexity of order α for functions f=h+g¯∈TSH0m,q,α, which is as below:Theorem 13.
Let0≤α<1, and Ln and Mn are defined by (33) and (34). Then
(62)rHqcαTSH0m,q,α=infn≥21−αT−QΨnqminLnΨnq−α,MnΨnq+α1/n−1,where Ψnq is defined by (19).Theorem 14.
f=h+g¯∈TSH0m,q,α be of the form (25). Then if and only if
(63)fz=∑n=1∞xnhnz+yngnz,where
(64)h1z=z,hnz=z−T−QLnzn,g1z=z,gnz=z−T−QMnz¯n,forn=2,3,⋯,xn,yn≥0,x1=1−∑n=2∞xn−∑n=2∞yn.
In particular the pointshn and gn are called the extreme points of the closed convex hull of the class TSH0m,q,α, denoted by clco TSH0m,q,α.Proof.
Letf be given by (63). Then from (64), it is of the form
(65)fz=z−∑n=2∞xnT−QLnzn+−1m∑n=2∞ynT−QMnz¯n,which by Theorem 7 proves that f∈TSH0m,q,α. Since for this function
(66)∑n=2∞LnxnT−QLn+MnynT−QMn=T−Q∑n=2∞xn+yn=T−Q1−x1−y1≤T−Q.
Conversely, letf=h+g¯∈TSH0m,q,α and set
(67)xn=LnT−Qan,xn=MnT−Qbn.
Then on using (64), we obtain
(68)fz=z−∑n=2∞anzn+−1m∑n=2∞bnz¯n=z−∑n=2∞xnT−QLnzn+−1m∑n=2∞ynT−QMnz¯n=z−∑n=2∞xnz−hnz+∑n=2∞yngnz−z=1−∑n=2∞xn+ynz+∑n=2∞xnhnz+yngnz,which is of the form (63). This proofs Theorem 14.Corollary 15.
Letf∈TSH0m,q,α be of the form (25). Then
(69)an≤T−QLn,bn≤T−QMn,n=2,3,4⋯,where Ln and Mn are defined, respectively, by (33) and (34). Equality in the inequalities (69) occurs for the extremal functions hnz and gnz given in (64) for n=1,2,3.


## 3. Conclusion

In this paper, we defined a new classSH0m,q,α of harmonic functions f∈H0 associated with newly defined q-Noor integral operator for harmonic functions f. In this class, we proved necessary and sufficient convolution condition for the functions f∈H0. We proved that, sufficient coefficient condition for the functions f∈H0 to be sense preserving and univalent and also this coefficient condition is necessary for subclass TSH0m,q,α. By using the necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of q-starlikeness and q-convexity of order α in the class TSH0m,q,α. Also we investigated extreme points for the functions f∈TSH0m,q,α.



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