# Certain Families of Analytic Functions Characterized byp,q-Difference Operator

Authors: Rashid Murtaza; Adnan; Amna Aziz; Abdulaziz H. Alghtani; Ilyas Khan; Mulugeta Andualem
Journal: Journal of Mathematics
 (2022)

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## Abstract

The major motivation behind this work is to utilizep,q-calculus and inspected the generalized p,q-Ruscheweyh differential operator. At the present time, we utilized p,q-Ruscheweyh differential operator and examined some new subfamilies of S and study some essential properties, for example, inclusion and subordination properties.


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## Body

## 1. Introduction

Theq-calculus is the speculation of ordinary calculus where the limits has been supplanted by q. The chronicled background of q-calculus may be portrayed by its wide variety of use in quantum mechanics, logical number hypothesis, theta functions, Bernoulli and Euler polynomials, Sobolev spaces, and more in the analytic and univalent function theory. Jackson ought to be praised for the specific start of quantum calculus in his work in [1, 2], by presenting the q-derivative and integral, respectively. In q-calculus, we essentially center around q-analogue that arise normally, rather than in arbitrary q-analogues of known results. While zeroing in on advancements and improvement of q-calculus and its applications in explicit fields of numerical and actual science, we will moreover look at q-analogues of a part of the new results in Geometric Function Theory in unit disk. The early work of q-calculus in the field of Geometric Function Theory was finished by Ismail et al. [3], by summing up the class of star-like functions into the class of q-star-like functions.Assume thatf∈ℂ. Then,(1)fis normalized analytic function⟺f′0−1=0=f0and characterized as(2)fz=z+∑k=2∞akzk,z∈E=z∈ℂ:|z|<1.The symbolA stands for the family of all these type of functions. Also, by S⊂A, comprising of functions, which are univalent in E. By S∗α0≤α<1, we meant the subfamily of S consisting of star-like functions of order α. If α=0,thenS∗α=S∗.Let℘∈ℂ be analytic in E. Furthermore, we say that(3)℘∈P⟺℘0=1,Re℘z>0,z∈E,and presented as(4)℘z=1+∑k=1∞ckzk.Forf1 and f2 given by(5)f1z=z+∑k=2∞akzk,f2z=z+∑k=2∞bkzk,z∈E,the convolution is characterized as(6)f1∗f2z=z+∑k=2∞akbkzk=f2∗f1z,z∈E.Assumef and g be analytic in E. Then, f≺g if there exists ω analytic in E, along ω0=0 and |ωz|<1, in E such as(7)fz=gωz,z∈E.Also, it is observed by [4] that(8)f≺g⟺f0=g0andfE⊂gE.Kanas and Wisniowska [5] extended the parabolic region and characterized the region Υk,0≤k<∞ as(9)Υk=w=s+it∈ℂ:s>ks−12+t2,with 1∈Υk. Assume(10)∂Υk=w=s+it:s2=k2s−12+t2.Therefore,(11)k=0⟺Υ0=s>0,k=1⟺Υ1=t2<2s−1,k∈0,1⟺Υk=s>ks−12+t2.Identified with the regionΥk, the accompanying functions are extremal and ℘kE⊂Υk(12)℘kz=1+z1−z,k=0,1+21−k2sinh22πarccoskarctanhz,0<k<1,1+2π2log1+z1−z2,k=1.Two fascinating subfamilies ofS are k−UCV and k−ST which consist of k-uniformly convex and k-uniformly star-like functions, respectively, and characterized by [5, 6] for k≥0 as(13)k−UCV=g∈S:Re1+zg″zg′z>kzg″zg′z,z∈E,k−ST=g∈S:Rezg′zgz>kzg′zgz−1,z∈E.Note that(14)g∈k−UCV⟺zg′∈k−ST.Geometrically,g∈k−UCV means that the image of E under zg′z′/g′z is contained in Υk, where Υk is defined by (9).In recent years, researchers are using these conic domain defined by (9) very efficiently and obtained very excellent results some of them are [7, 8] and reference therein.Remark 1.
All through this study, we mean byℋ the family of functions h that are convex univalent in E along h0=1, except if in any case referenced.


## 2. (p,q)-Calculus

Expanding the concept ofq-calculus, p,q-number was freely considered by [9]. Let 0<q<p≤1; then, p,q-number is characterized for j∈ℕ by(15)jp,q=pj−qjp−q=pj−1+pj−2q+⋯+pqj−2+qj−1,which is the common generalization of q-number. Note that j1,1=j. Also, for ϱ∈ℕ, we obtain(16)ϱ+1p,q=qϱ+pϱp,q,ϱ+jp,q=qϱ+j−1+pjϱp,q.Furthermore,(17)ϱ+jp,q−pjϱp,q=pϱ+j−qϱ+jp−q−pjpϱ−qϱp−q=qϱpj−qjp−q=qϱjp,q.In [9], for f∈ℂ, p,q-derivative of is characterized as(18)dp,qfz=fqz−fpzq−pz,z≠0,f′0,z=0.Moreover,(19)fz=zk⟺dp,qfz=kp,qzk−1.Also, by utilizing (2), we attain(20)dp,qfz=1+∑k=2∞kp,qakzk−1.Note, ifp=1, at that point(21)d1,qfz=dqfz.Also,d1,1fz=f′z. It is notable [10] that(22)dp,qvzuz=vpzdp,quz+uqzdp,qvz,(23)ordp,qvzuz=upzdp,qvz+vqzdp,quz.Thep,q-integral of f presented in [10] is given by(24)1zp−q∫0zdp,qt=∑j=0∞qjpj+1fqjpj+1z,ifpq>1.Note that ifp=1, definition (24) reduces to the well-known Jackson integral defined in [2]. We refer to a notion of p,q-operators, that is, p,q-operator, that play imperative job in the hypothesis of hypergeometric series, operator theory, and quantum physics. The utilization of q-calculus was started by Jackson.As of late, Akbar et al. [11] started the possibility of symmetric p,q-calculus, presented symmetric p,q-derivative, and examined another class of star-like functions along intriguing results.


### 2.1.p,q−Generalized Pochhammer Symbol

The incestigation of star-like functions via (p,q)-calculus and various analytic functions classes were reported in [11, 12], respectively. The p,q−Gamma function is characterized by the accompanying relation:(25)Γp,qr=rp,qΓp,qr−1,Γp,q1=1.Expanding the concept ofq-generalized Pochhammer symbol, the p,q-shifted factorial (p,q-generalized symbol) is characterized as(26)rp,qk=1,k=0,rp,qr+1p,q⋯r+k−1p,q,k∈ℕ.Note(27)r1,qk=rqkandr1,1k=rk.Now, by generalizing the concept of [13], we introduced the p,q-Ruscheweyh differential operator as(28)ℜp,qϱfz=fz∗Fp,qϱ+1z,ϱ>−1,0<q<p≤1,z∈E,where(29)Fp,qϱz=z+∑k=2∞Γp,qk+ϱk−1p,q!Γp,q1+ϱzk,(30)=z+∑k=2∞ϱ+1p,qk−1k−1p,q!zk.


### 2.2. Special Cases

Ifp=1,q⟺1−, then(31)ℜ1,1ϱfz=fz∗F1,1ϱ+1z=ℜϱfz,see [14].Also, ifp=1, then(32)ℜ1,qϱfz=ℜqϱz=fz∗Fqϱ+1z,see [13].Therefore,(33)ℜp,qϱfz=z+∑k=2∞Γp,qk+ϱΓp,q1+ϱk−1p,q!akzk,z∈E.Usingp,q-derivative defined by (19), we observed that(34)dp,qℜp,qϱfz=1+∑k=2∞Γp,qk+ϱΓp,q1+ϱk−1p,q!kp,qakzk−1.Ifp=1,q⟺1−andϱ=0, then(35)d1,1ℜp,qϱfz=1+∑k=2∞kakzk−1=f′z,z∈E.Utilizing (35) and ϱ∈ℕ0=0,1,2,3,…, it can, without much of a stretch, be confirmed that(36)ϱ+1p,qℜp,qϱfz=qϱzdp,qℜp,qϱfz+pϱp,qℜp,qϱfz.By takingp=1, we obtain(37)ϱ+1qℜqϱfz=qϱzdqℜqϱfz+ϱqℜqϱfz,see [15].Also, ifp=1,q⟺1−, we observe(38)ϱ+1ℜϱfz=zℜϱfz′+ϱℜϱfz,see [14].Usingp,q-derivative defined by (19), we define the class k−STp,qϱ by the following.Definition 1.
Assume thatf∈A is defined by (2) and h∈ℋ is given as in Remark 1. Then, for 0<q<p≤1,(39)f∈k−STp,qϱ⟺zdp,qℜp,qϱfzℜp,qϱfz≺℘kz,k≥0,z∈E,where ℘k are defined by (12).
We can define the corresponding classk−UCVp,qϱ as(40)f∈k−UCVp,qϱ⟺zdp,qf∈k−STp,qϱ.
Equivalently,(41)ℜp,qϱf∈k−ST⟺f∈k−STp,qϱ.
Also,(42)f∈Sp,q∗ϱ⟺ℜp,qϱf∈S∗.


## 2.1.p,q−Generalized Pochhammer Symbol

The incestigation of star-like functions via (p,q)-calculus and various analytic functions classes were reported in [11, 12], respectively. The p,q−Gamma function is characterized by the accompanying relation:(25)Γp,qr=rp,qΓp,qr−1,Γp,q1=1.Expanding the concept ofq-generalized Pochhammer symbol, the p,q-shifted factorial (p,q-generalized symbol) is characterized as(26)rp,qk=1,k=0,rp,qr+1p,q⋯r+k−1p,q,k∈ℕ.Note(27)r1,qk=rqkandr1,1k=rk.Now, by generalizing the concept of [13], we introduced the p,q-Ruscheweyh differential operator as(28)ℜp,qϱfz=fz∗Fp,qϱ+1z,ϱ>−1,0<q<p≤1,z∈E,where(29)Fp,qϱz=z+∑k=2∞Γp,qk+ϱk−1p,q!Γp,q1+ϱzk,(30)=z+∑k=2∞ϱ+1p,qk−1k−1p,q!zk.


## 2.2. Special Cases

Ifp=1,q⟺1−, then(31)ℜ1,1ϱfz=fz∗F1,1ϱ+1z=ℜϱfz,see [14].Also, ifp=1, then(32)ℜ1,qϱfz=ℜqϱz=fz∗Fqϱ+1z,see [13].Therefore,(33)ℜp,qϱfz=z+∑k=2∞Γp,qk+ϱΓp,q1+ϱk−1p,q!akzk,z∈E.Usingp,q-derivative defined by (19), we observed that(34)dp,qℜp,qϱfz=1+∑k=2∞Γp,qk+ϱΓp,q1+ϱk−1p,q!kp,qakzk−1.Ifp=1,q⟺1−andϱ=0, then(35)d1,1ℜp,qϱfz=1+∑k=2∞kakzk−1=f′z,z∈E.Utilizing (35) and ϱ∈ℕ0=0,1,2,3,…, it can, without much of a stretch, be confirmed that(36)ϱ+1p,qℜp,qϱfz=qϱzdp,qℜp,qϱfz+pϱp,qℜp,qϱfz.By takingp=1, we obtain(37)ϱ+1qℜqϱfz=qϱzdqℜqϱfz+ϱqℜqϱfz,see [15].Also, ifp=1,q⟺1−, we observe(38)ϱ+1ℜϱfz=zℜϱfz′+ϱℜϱfz,see [14].Usingp,q-derivative defined by (19), we define the class k−STp,qϱ by the following.Definition 1.
Assume thatf∈A is defined by (2) and h∈ℋ is given as in Remark 1. Then, for 0<q<p≤1,(39)f∈k−STp,qϱ⟺zdp,qℜp,qϱfzℜp,qϱfz≺℘kz,k≥0,z∈E,where ℘k are defined by (12).
We can define the corresponding classk−UCVp,qϱ as(40)f∈k−UCVp,qϱ⟺zdp,qf∈k−STp,qϱ.
Equivalently,(41)ℜp,qϱf∈k−ST⟺f∈k−STp,qϱ.
Also,(42)f∈Sp,q∗ϱ⟺ℜp,qϱf∈S∗.


## 3. Main Results

To get our main results, the accompanying lemmas are helpful.Lemma 1.
Assume thats=s1+is2,t=t1+it2 and χ:D⊂ℂ×ℂ⟺ℂ along(i)
χs,tis continuous inD(ii)
χ1,0>0and1,0∈D(iii)
Reχis2,t1≤0,foris2,t1∈Dandt1≤−1/21+s22Suppose that℘z=1+c1z+c2z+... belongs to P along ℘,z℘′∈D∀z∈E. If Reχ℘z,z℘′z>0, then Re℘z>0; for details, see [16].The attendant lemma isp,q-analogue of the lemma presented by [17].Lemma 2.
Assume thath∈ℋ be defined as in Remark 1. Furthermore, let β0,γ0∈ℂ along β0≠0. If ℘∈P, fulfill Reβ0℘z+γ0>0 and(43)℘z+zdp,qfzβ0℘z+γ0≺hz⟺℘z≺hz,z∈E.The accompanying lemma isp,q-analogue of the lemma presented by [15] and will be required for later use.Lemma 3.
Assume thath∈ℋ and sz=1+s1z+s2z2⋯ be analytic in E. Furthermore, if(44)sqz+pγp,qzdp,qszγp,q≺hz,z∈E.Then, forγp,q≠0, we have(45)sqz≺tqz≺hz,where(46)tz=γp,qzγp,q∫0ztγp,q−1htdp,qt=p−qγp,q∑j=0∞qjγp,qpj+1γp,qhqjpj+1z,is the best super-ordinate of (46).Proof.
Assume thath∈ℋ and(47)sqz+pγp,qzdp,qszγp,q≺hz,z∈E.
To find the best dominant of (47), we have to find the solution of corresponding differential equation:(48)tqz+pγp,qzdp,qtzγp,q=hz,z∈E.
This means that, making use of product rules (22) or (23), we obtain(49)γp,qzγ−1hz=pzγp,qdp,qtz+γp,qzγp,q−1tqz=pzγp,qdp,qtz+tqzdp,qzγp,q=dp,qzγp,qtz.
Consequently,(50)tz=γp,qzγp,q∫0ztγp,q−1htdp,qt,which is best dominant of (46). By using (24), we can right(51)tz=γp,qzγp,qzp−q∑j=0∞qjpj+1qjpj+1zγp,q−1hqjpj+1zγp,qp−q∑j=0∞qjγp,qpj+1γp,qhqjpj+1z,which is required.
If we takep=1 in Lemma 3, we obtain the well-known result from [15].Theorem 1.
Assume thatk≥0,0<q<p≤1,(52)k−UCVp,qϱ⊂k−STp,qϱ,z∈E.Proof.
Assumef∈k−UCVp,qϱ and consider(53)zdp,qℜp,qϱfzℜp,qϱfz=ℏz,ℏ∈P.
Presently utilizing (36), we obtain(54)ℏz=ϱ+1p,q/qϱℜp,qϱ+1fz−pϱp,q/qϱℜp,qϱfzℜp,qϱfz=ϱ+1p,q/qϱℜp,qϱ+1fzℜp,qϱfz−pϱp,qqϱ.
Differentiating (53), p,q-logarithmically, we have(55)ℏz+zdp,qℏzℏz=dp,qzdp,qℜp,qϱfzdp,qℜp,qϱfz≺℘kz.
Therefore,(56)ℏz+zdp,qℏzℏz≺℘kz,z∈E.
Along these lines by utilizing Lemma2, we have ℏz≺℘kz, and thus,(57)zdp,qℜp,qϱfzℜp,qϱfz≺℘k⟺f∈k−STp,qϱwhich is required.
Note that, by taking specific values to the parameters in Theorem1, we can get some notable results.
Forϱ=0,p=1,andq⟺1−, we obtain accompanying crucial result, see [6].Corollary 1.
Assume thatk≥0 and ∀z∈E,(58)k−UCV⊆k−ST.Corollary 2.
Assuming thatk=0 and ∀z∈E, we obtain C⊆S∗.Theorem 2.
Suppose that, fork≥0,andϱ∈ℕ0,(59)k−STp,qϱ+1⊆k−STp,qϱ⊆⋯⊆k−ST⊆S∗η0⊆S∗⊆S.Equivalently, forη0=k/k+1,(60)Sp,q∗ϱ+1,η0⊆Sp,q∗ϱ,η1⊆⋯⊆Sp,q∗ϱ−m−1,ηm⊆Sp,q∗ϱ−m,ηm+1,where ϱ>m∈ℕ0 and(61)ηm+1=2qϱη0−2pϱp,q−mp,q−qϱ−2qϱη0−2pϱp,q−mp,q−qϱ2+82η0pϱp,q−mp,q+qϱ4.Proof.
Firstly, consider(62)k−STp,qϱ+1⊆k−STp,qϱ,or(63)Sp,q∗ϱ+1,η0⊆Sp,q∗ϱ,η1,where(64)η1=2qϱη0−2pϱp,q−qϱ−2qϱη0−2pϱp,q−qϱ2+82η0pϱp,q+qϱ4.
Letf∈Sp,q∗ϱ+1,η0. Furthermore, let(65)zdp,qℜp,qϱfzℜp,qϱfz=1−η1p1z+η1,p1∈P.
Using (37), we obtain(66)ϱ+1p,q/qϱℜp,qϱ+1fzℜp,qϱfz−pϱp,qqϱ=1−η1p1z+η1.
This implies that(67)ϱ+1p,qℜp,qϱ+1fzℜp,qϱ=qϱ1−η1p1z+qϱη1+pϱp,q.
Differentiatingp,q−logarithmically, we attain(68)zdp,qℜp,qϱ+1fzℜp,qϱ+1fz=1−η1p1z+η1+1−η1zdp,qp1zqϱ1−η1p1z+qϱη1+pϱp,q.
Since(69)f∈k−STp,qϱ+1⊆Sp,q∗ϱ+1,η0,η0=kk+1.
This means that(70)Re1−η1p1z+η1+qϱ1−η1zdp,qp1zqϱ1−η1p1z+qϱη1+pϱp,q>η0,z∈E.
Now, we construct a functionalΨs,ν with s=s1+is2=p1z and ν=ν1+iν2=zdp,qp1z as(71)Ψs,ν=1−η1s+η1−η0+qϱ1−η1νqϱ1−η1s+qϱη1+pϱp,q.
It very well may be seen that, for the functionalΨs,ν,(i)
Ψs,νis continuous inD=ℂ−qϱη1+pϱp,q/qϱη1−1×ℂ(ii)
1,0∈D=ℂ−qϱη1+pϱp,q/qϱη1−1×ℂandReΨ1,0=1−η0>0
Now, consider(72)ReΨis,v1−Re1−ηis2+η1−η0+qϱ1−η1v1qϱ1−η1is2+qϱη1+pϱp,q=η1−η0+qϱ1−η1v1q2ϱ1−η12s22+qϱη1+pϱp,q2≤η1−η0−qϱ1−η121+s22+qϱη1+pϱp,q2q2ϱ1−η12s22+qϱη1+pϱp,q2,because ν1≤−1+s22/2. Therefore, after some simplifications, we obtain(73)ReΨis2,ν1=A+Bs222C,where(74)A=2η1−η0qϱη1+pϱp,q2−qϱ1−η1qϱη1+pϱp,q,B=2q2ϱη1−η01−η12−qϱ1−η1qϱη1+pϱp,q,C=2q2ϱ1−η12s22+qϱη1+pϱp,q2.
The R.H.S of (76) is negative if A≤0,B≤0. By applying Lemma 1, we finally attain(75)Sp,q∗ϱ,η0⊆Sp,q∗ϱ−1,η1,and A≤0 yields(76)η1=2qϱη0−2pϱp,q−qϱ−2qϱη0−2pϱp,q−qϱ2+82η0pϱp,q+qϱ4,where η0=k/k+1.
Similarly, proceeding as before, one can prove(77)Sp,q∗ϱ,η1⊆Sp,q∗ϱ−1,η2,where(78)η2=2qϱη0−2pϱp,q−1p,q−qϱ−2qϱη0−2pϱp,q−1p,q−qϱ2+82η0pϱp,q−1p,q+qϱ4.
Continuing this process, we obtain(79)Sp,q∗ϱ−m−1,ηm⊆Sp,q∗ϱ−m,ηm+1,where(80)ηm+1=2qϱη0−2pϱp,q−mp,q−qϱ−2qϱη0−2pϱp,q−mp,q−qϱ2+82η0pϱp,q−mp,q+qϱ4.
By using the principal of mathematical induction for post-quantum calculus, one can prove that formula (83) is true for all m∈ℕ0 and the proof is complete.
Note by allocating special values for parameters involved in Theorem2, we gain some notable results. Assuming p=1andq⟺1−, in Theorem 2, we acquire notable results from [14].Corollary 3.
Assume thatp=1andq⟺1−, which implies(81)k−STϱ+1⊆k−STϱEquivalently,(82)S∗ϱ+1,η0⊆S∗ϱ,η1where(83)η1=2η0−2ϱ−1−2η0−2ϱ−12+82η0ϱ+14,η0=kk+1.In this portion, we generalize some subordination results presented in [15].Theorem 3.
Letα>0,ϱ>0,0<q<p≤1,−1≤D<C≤1. If f∈A presented by (2) fulfills(84)1−αℜp,qϱfzz+αℜp,qϱ+1fzz≺1+Cz1+Dz,then, for η≥1, we obtain(85)Retz1/η>ϱ+1p,qαqϱpγp,q∫01sϱ+1p,q/αqϱpγp,q−11−Cs1−Dsdp,qs1/η.Proof.
Let(86)gz=ℛp,qϱfzz,g∈P.
Differentiating this equationp,q−logarithmically, we obtain(87)zdp,qgzgz=zdp,qℜp,qϱfzℜp,qϱfz−1.
Making use of (37), we can obtain(88)zdp,qgzgz=ϱ+1p,qqϱℜp,qϱ+1fzℜp,qϱfz−pϱp,qqϱ−1.
This implies from (16) that(89)zdp,qgzgz=ϱ+1p,qqϱℜp,qϱ+1fz−ℜp,qϱfzℜp,qϱfzqϱzdp,qgzϱ+1p,q=ℜp,qϱ+1fz−ℜp,qϱfzz.
Equivalently,(90)gz+αqϱzdp,qgzϱ+1p,q=1−αℜp,qϱfzz+αℜp,qϱ+1fzz≺1+Cz1+Dz,z∈E.
Therefore,(91)gz+αqϱzdp,qgzϱ+1p,q≺1+Cz1+Dz,z∈Eor(92)gz+pγp,qzdp,qgzϱ+1p,q/αqϱpγp,q≺1+Cz1+Dz,z∈E.
Using Lemma3, we obtain(93)gz≺tz=ϱ+1p,qαqϱpγp,qz−ϱ+1p,q/αqϱpγp,q∫0zsϱ+1p,q/αqϱpγp,q−11+Cwz1+Dwzdp,qs.
Therefore,(94)Retz>ϱ+1p,qαqϱpγp,q∫01sϱ+1p,q/αqϱpγp,q−11−Cs1−Dsdp,qs.
Presently, utilizing the realityRew1/η<Rew1/η for 0<Rew and 1≤η, we get (85), which is required.
Note that, by offering specific values to parameters in Theorem3, we attain certain fascinating results from [15].Corollary 4.
Assumes thatα>0,ϱ>0,−1≤D<C≤1,andp=1. Furthermore, if f∈A is presented as in (2), which satisfies(95)1−αℜqϱfzz+αℜqϱ+1fzz≺1+Cz1+Dz,Then, for η≥1, one can have(96)Reℜqϱfzz1/η>ϱ+1qαqϱ∫01sϱ+1q/αqϱ−11−Cs1−Dsdqs1/η.Corollary 5.
Assume thatα>1,ϱ>1,C=1−2β,0≤β<1,D=1 and p=1. Furthermore, if f∈A is presented as in (2), which satisfies(97)1−αℜqϱfzz+αℜqϱ+1fzz≺1+1−2βz1−z,then, for η≥1, we obtain(98)Reℜqϱfzz1/η>2β−1sϱ+1q/αqϱ+21−βϱ+1qαqϱ∫01sϱ+1q/αqϱ−11+sdqs1/η.


## 4. Conclusion

The essential explanation behind this investigation work is to usep,q−calculus and inspected the generalized p,q-Ruscheweyh differential operator. At the present time, we utilized p,q-Ruscheweyh differential operator and examined some new subfamilies of S and study some essential properties, for example, inclusion and subordination properties. Our results sum up some charming known consequences. The way is open for researchers to investigate all the more right currently related areas.



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*Source: 7255866-2022-02-14.xml*