# Coefficient Inequalities for a Subclass of Symmetricq -Starlike Functions Involving Certain Conic Domains

Authors: Mohammad Faisal Khan; Nazar Khan; Serkan Araci; Shahid Khan; Bilal Khan
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we make use of a certain Ruscheweyh-typeq -differential operator to introduce and study a new subclass of q -starlike symmetric functions, which are associated with conic domains and the well-known celebrated Janowski functions in D. We then investigate many properties for the newly defined functions class, including for example coefficients inequalities, the Fekete–Szegö Problems, and a sufficient condition. There are also relevant connections between the results provided in this study and those in a number of other published articles on this subject.


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## Body

## 1. Introduction, and Preliminaries

LetℋD be a class of analytic functions where D is the open unit disk and is given by the following equation:(1)D=z:z∈ℂ,z<1,and let f∈A be those functions in the open unit disk D which are normalized by the following equation:(2)f0=0,f′0=1,thus, we have the following series form for f∈A(3)fz=z+∑n=2∞anzn,z∈D.Moreover, all normalized univalent functions inD are contained in the set A⊂S. For two given functions g1,g2∈A, we say that g1 is subordinate to g2, written symbolically as g1≺g2, if there exist a Schwarz function w, which is holomorphic in D with(4)w0=0,wz<1,so that(5)g1z=g2wzz∈D.Moreover, if the functiong2 is univalent in D, then the following equivalence hold true:(6)g1≺g2⇔g10=g20,g1D⊂g2D.LetP be the class of Carathedory function, an analytic function χ∈P if(7)χz=1+∑n=1∞χnzn,such that(8)χ0=1,ℜχz>0∀z∈D.Definition 1.
A given functionp is said to be in the class PX,Y if(9)pz≺1+Xz1+Yz−1≤Y<X≤1.
Janowski [1] investigated the class of functions PX,Y and found that pz∈PX,Y if and only if there exist a function χ∈P such that(10)pz=X+1χz−X−1Y+1χz−Y−1−1≤Y<X≤1.Definition 2.
A functionf of the form (3) be in the functions class S∗X,Y if and only if(11)zf′zfz=X+1χz−X−1Y+1χz−Y−1−1≤Y<X≤1.
Historically speaking, Kanas and Wiśniowska were the first (see [2], [3]), (see also [4]) who introduced and defined the class of λ-uniformly convex functions λ−UCV and λ-starlike functions λ−ST subject to the conic domain Ωλ, where(12)Ωλ=u+iv:u>λu−12+v2,u>0,λ≥0.
Moreover ifλ, is fixed and λ=0 then Ωλ denote the conic region bounded by the imaginary axis, if λ=1, we have a parabola, if 0<λ<1 this domain represents the right branch of hyperbola and for λ>1 an ellipse.
For these conic areas, the following functions serve as extremal functions(13)pλz=1+z1−z=1+2z+2z2+⋯,λ=0,1+2π2log1+z1−z2,λ=1,1+21−λ2sinh22πarccosλarctanpz,0≤λ<1,1+1λ2−1sinπ2Kκ∫0uz/κdt1−t21−κ2t2+1λ2−1,λ>1,where(14)uz=z−κ1−κz∀z∈D,and κ∈0,1 is chosen such that λ=coshπK′κ/4Kκ. Here, Kκ is Legendre’s complete elliptic integral of first kind and K′κ=K1−κ2, that is K′t is the complementary integral of Kt.
The functionpλz in [5] be given as follows:(15)pλz=1+ςλz+ς2λz2+ς3λz3+⋯,where(16)ςλ=8arccosκ2π21−κ20≤λ<1,8π2λ=1,π24κ2−1R2tt1+tλ>1,(17)ς2λ=ς1ςλ,(18)ς1=2/πarccosκ2+230≤λ<1,23λ=1,4R2tt2+6t+1−π224R2t1+ttλ>1,n!r!n−r!,and t∈0,1.
The following is defined by Noor et al. [6], who combine the ideas of Janowski functions and conic regions.Definition 3.
A functionp from the functions class P be in the functions class λ−PX,Y, if(19)pz≺X+1pλz−X−1Y+1pλz−Y−1,−1≤Y<X≤1,λ≥0,where pλz is defined by (13).
Geometrically for the domainΩλX,Y, we have the following(20)ΩλX,Y=w:ℜY−1w−X−1Y+1w−X+1>λY−1w−X−1Y+1w−X+1,or equivalently for w=u+iv, we have the following equation:(21)Y2−1u2+v2−2XY−1u+X2−12>λ−2Y+1u2+v2+2X+Y+2u−2X+12+4X−Y2v2.
The domainΩλX,Y denote the conic type regions (see [6]).Definition 4.
(see [6]). An analytic function f be in the class λ−STX,Y, if(22)zf′zfz∈λ0PX,Y∀z∈Dandλ≥0.
The angle2π/i at which domain D rotates around the origin maps D onto itself The domain D is thus said to as i-fold symmetric. In D a function f is said to be i-fold symmetric if for every z in D(23)fe2πI/iz=e2πI/ifz,i∈ℤ+.
Here,Si denote the i-fold symmetric functions and i symbolize the imaginary unit. For j=0,1,2,…,i−1;i=2,3,…;, the j,i-symmetric functions are an extension of the concept of even, odd, and i-symmetric functions. Several applications of the theory of j,i -symmetric functions may be found in [7]. For e=e2πI/i, the functions f:D⟶ℂ is known to be j,i -symmetric if(24)fejz=ejfz.
The set of allj,i-symmetric functions is denote by Sj,i . First of all, if j=0 and i=2, then Sj,i is called even symmetric functions. Secondly, if j=1 and i=2, then Sj,i is called odd symmetric functions. Thirdly, if j=1, then Sj,i is called i-symmetric functions.Theorem 1 (see [7]).
For mappingf:D⟶ℂ, there is just one series of j,i -symmetrical functions fj,iz exists that given as follows:(25)fz=∑j=0i−1fj,iz.From equation (25), we can get the following equation:(26)fj,iz=1i∑v=0i−1e−vjfevz=1i∑v=0i−1e−vj∑n=1∞anevzn,then(27)fj,iz=∑n=1∞ψnanzn,a1=1,ψ1=1,where f∈A and(28)e=e2πI/iℕ0=1,2,3,…,(29)ψn=1i∑v=0i−1en−jv=1,n=li+j0,n≠li+j.


## 2. Some Basic Concepts ofq-Calculus

The usage of the quantum (orq -) calculus in many diverse areas of mathematics and physics are quite significant. In the theory of Univalent functions, Srivastava [8] firstly apply the q-calculus in order to put the foundation of a new direction for other researchers. Motivated by [8] of Srivastava, many researchers have worked on this direction. For example, the convolution theory, enable us to investigate various properties of analytic functions. Due to the large range of applications of q-calculus and the importance of q-operators instead of regular operators, many researchers have explored q-calculus in depth, such as, Kanas and Reducanu [9], Muhammad and Sokol [10] and Noor et al. [11–15]. Also in [1–5, 9, 16–22], Ahmad et al. see also [21], have used the q-derivative operator to define a new subclass q-meromorphic starlike functions. They also developed some remarkable results for their defined classes of analytic functions. In addition, Srivastava [23] see also [8, 12–15, 23–26] recently published survey-cum-expository review paper this might be useful for researchers and scholars working on these subjects. For some recent and related study about q-series, we may refer the interested readers to see [27–29].Definition 5.
(see [30]). Let q∈0,1 and q-integer n, be defined as follows:(30)n,q=1−qn1−q=1+q+⋯+qn−1,0,q=0.Definition 6.
We define theq-shifted factorial as follows:(31)0,q!=1,n,q!=1,q2,q…n,q.
It could be seen that(32)limq⟶1−n,q=n,limq⟶1−n,q!=n!.
In generalt,q=1−qt/1−q.Definition 7 (see [20]).
For an analytic functionf, the q-deformation or q-generalization of derivative is defined by the following equation:(33)∂qfz=fz−fqz1−qzz∈D,(34)∂qfz=1+∑n=2∞n,qanzn−1,and(35)∂qzn=n,qzn−1.Definition 8.
The generalizedq-Pochhammer symbol is given by the following equation:(36)t,qn=t,qt+1,qt+2,q…t+n−1,q.and q-gamma function be given as follows:(37)Γqt+1=t,qΓqt,Γq1=1.Definition 9.
(see [31]). A function p∈λ-PqX,Y if and only if(38)pz≺pλ,qX,Y,z,where(39)pλ,qX,Y,z=X1+q+−q+3pλz−X1+q−−q+3Y1+q+−q+3pλz−Y1+q−−q+3,λ≥0.and pλz is defined by (13).
Geometrically, we have the following equation:(40)Ωλ,qX,Y=w=u+iv:ℜΨ>λΨ−1,where(41)Ψ=Y1+q−−q+3wz−X1+q−−q+3Y1+q+−q+3wz−X1+q+−q+3.
For more detail (see [31]).Definition 10.
(see [9]). For f∈A, the Rusceweyh q -differential operator be defined as follows:(42)Rqλ1fz=fz∗ℋq,λ1+1z,z∈D,λ1>−1,where(43)ℋq,λ1+1z=z+∑n=2∞Γqλ1+nn−1,q!Γq1+λ1zn=z+∑n=2∞λ1+1,qn−1n−1,q!zn=z+∑n=2∞φq,n−1zn,with(44)φq,n−1=λ1+1,qn−1n−1,q!.
By “∗” we mean convolution (or Hadamard product).
Moreover from (42), we have the following equation:(45)Rq0fz=fz,Rq1fz=z∂qfz,and(46)Rqmfz=z∂qmzm−1fzm,q!,m∈ℕ.
Making use of (42) and (43), the power series of Rqλ1fz is given by the following equation:(47)Rqλ1fz=z+∑n=2∞Γqλ1+nn−1,q!Γq1+λ1anzn=z+∑n=2∞λ1+1,qn−1n−1,q!anzn=z+∑n=2∞φq,n−1anzn.Similarly,(48)Rqλ1fj,iz=fj,iz∗ℋq,λ1+1z,z∈D,λ1>−1,=z+∑n=2∞λ1+1,qn−1n−1,q!ψnanzn,=z+∑n=2∞φq,n−1ψnanzn,a1=1,ψ1=1,φq,0=1,where φq,n−1 is given by (44) and ψn is given by (49).
Note that,(49)limq⟶1−ℋq,λ1+1z=z1−zλ1+1,(50)limq⟶1−Rqλ1fj,iz=fj,iz∗z1−zλ1+1,and(51)limq⟶1−Rqλ1fz=fz∗z1−zλ1+1.
Whenq⟶1−, Ruscheweyh q-differential operator reduce to Ruscheweyh differential operator [14]. Also, note that(52)z∂qℋq,λ1+1z=1+λ1,qqλ1ℋq,λ1+2z−λ1.qqλ1ℋq,λ1+1z,and(53)z∂qRqλ1fz=1+λ1,qqλ1Rqλ1+1fz−λ1,qqλ1Rqλ1fz.Definition 11.
A given functionf is said to be in the functions class λ-Vqj,iλ1,X,Y, λ≥0, −1≤Y<X≤1, if(54)ℜφX,Y,q,λ1>λφX,Y,q,λ1−1,or equivalently(55)z∂qRqλ1fzRqλ1fj,iz∈λ−PqX,Y,where(56)φX,Y,q,λ1=Y1+q−−q+3z∂qRqλ1fz/Rqλ1fj,iz−X1+q−−q+3Y1+q+−q+3z∂qRqλ1fz/Rqλ1fj,iz−X1+q+−q+3.
Each of the following special case of the above-defined functions classλ-Vqj,iλ1,X,Y is worthy of note.(i)
One can easily seen that(57)limq⟶1−λ−Vqj,i0,X,Y=λ−Vj,iX,Y,whereλ-Vj,iX,Y is the functions classes studied by Al-Sarari and Latha (see [18]).(ii)
If we set(58)q⟶1−,λ1=0,j=1.in Definition11, we have class λ-STX,Y,i introduced in [17].(iii)
If we put(59)q⟶1−,λ1=0,j=1=i.we get the functions classλ-STX,Y (see [30]).(iv)
We see that(60)limq⟶1−λ0Vq1,10,1,−1=λ0ST,where Kanas and Wisniowska [3] have studied the class λ-ST.(v)
If we set(61)q⟶1−,Y=1=i=j=λ1+1,X=1−2α.in Definition11, we get the class SDλ,α (see [24]).(vi)
It can be easily seen that(62)limq⟶1−00Vq1,10,X,Y=S∗X,Y,where the functions classS∗X,Y is studied in [1].


## 3. A Set of Lemmas

Lemma 1 (see [13]).
Letχz=1+∑n=1∞χnzn≺ℱz=1+∑n=1∞Cnzn. If ℱz is convex univalent in D, then(63)χn≤C1,n≥1.Lemma 2 (see [32]).
Ifχz=1+χ1z+χ2z2+⋯ is an analytic function with positive real part in D, then(64)χ2−vχ12≤−4v+2ifv<0,2if0≤v≤1,4v−2ifv>1.The equality holds for(65)χz=1+z1−z,whenv<0orv>1,or one of its relations. The equality holds for(66)χz=1+z21−z2,when0<v<1.or one of its relations. Ifv=0, the equality holds if and only if(67)χz=12+12λ1+z1−z+12−12λ1−z1+z,0≤λ≤1,or one of rotation of it. The above upper bound is sharp as well, when(68)χ2−vχ12+vχ12≤2,0<v≤12.and(69)χ2−vχ12+1−vχ12≤212<v≤1.Lemma 3 (see [32]).
If the functionχz given by (8) belongs to the class P of analytic functions and have positive real part in D, then(70)χn≤2,n∈ℕ.The equality holds for(71)fz=1+z1−z.Lemma 4 (see [32]).
If the functionχz given by (8) belongs to the class P of analytic functions and have positive real part in D, then(72)c2−μc12≤2max1,2μ−1,∀μ∈ℂ.Lemma 5.
Letλ∈0,∞ be fixed and(73)pλ,qX,Y,z=X1+q+−q+3pλz−X1+q−−q+3Y1+q+−q+3pλz−Y1+q−−q+3,then(74)pλ,qX,Y,z=1+Υ5qςλz+Υ5qς2λ−Υ6qςλ2z2+⋯,where Υ5q, Υ6q, ςλ and ς2λ given by (83), (84), (16), and (17).Proof.
From (39), we have the following equation:(75)pλ,qX,Y,z=X1+q+−q+3pλz−X1+q−−q+3Y1+q+−q+3pλz−Y1+q−−q+3=X1+q+−q+3pλz−X1+q−−q+3Y1+q+−q+3pλz−Y1+q−−q+3−1=X1+q−−q+3Y1+q−−q+31−X1+q+−q+3X1+q−−q+3pλz1+∑n=1∞Y1+q+−q+3Y1+q−−q+3pλzn=X1+q−−q+3Y1+q−−q+3+Υ1qpλz+Υ2qpλz2+Υ3qpλz3+⋯,where(76)Υ1q=X1+q−−q+3Y1+q+−q+3Y1+q−−q+32−X1+q+−q+3Y1+q−−q+3,Υ2q=X1+q−−q+3Y1+q+−q+32Y1+q−−q+33−X1+q+−q+3Y1+q−−q+3Y1+q−−q+32,and(77)Υ3q=X1+q−−q+3Y1+q+−q+33Y1+q−−q+34−X1+q+−q+3Y1+q−−q+32Y1+q−−q+33.
By using (15) and (75), we have the following equation:(78)pλ,qX,Y,z=∑n=1∞−2Y1+q+−q+3n−1Y1+q−−q+3n2nX1+q−Y1+qY1+q+−q+3n−1Y1+q−−q+3n+1+∑n=1∞ςλz+∑n=1∞2nX1+q−Y1+qY1+q+−q+3n−1Y1+q−−q+3n+1ς2λ+∑n=1∞Υ4qςλ2z2+⋯,where(79)Υ4q=2nY1+q+−q+3X1+q−Y1+q3−q+Y1+qnY1+q−3−qn+2.
The series(80)∑n=1∞−23−q+Y1+qn−1Y1+q−−q+3n,∑n=1∞2nX1+q−Y1+qY1+q+−q+3n−1Y1+q−−q+3n+1,and(81)∑n=1∞2nY1+q+−q+3X1+q−Y1+q3−q+Y1+qnY1+q−3−qn+2.
are convergent and convergent to 1,1+qX−Y/4 and Y1+q+−q+3X−Y1+q/8, respectively.
Therefore (78) becomes(82)pλ,qX,Y,z=1+Υ5qςλz+Υ5qς2λ−Υ6qςλ2z2+⋯,where(83)Υ5q=X−Y1+q4.and(84)Υ6q=3−q+Y1+qX−Y1+q8.
which is our required result.Remark 1.
If we setA=1, B=−1 and let q⟶1−, in Lemma 5, we will arrive it that of a result given by Sim et al. [25].Lemma 6.
Letχz=1+∑n=1∞χnzn∈λ-PqX,Y, then(85)χn≤Υ5qςλ=X−Y1+q4ςλ,n≥1.Proof.
By Definition9 and χz∈λ-PqX,Y if and only if(86)χz≺pλ,qX,Y,zλ≥0
pλ,qX,Y,z is given by (39).
By using (82) and (86), we have the following equation:(87)χz≺1+Υ5qςλz+Υ5qς2λ−Υ5qςλ2z2+⋯.
Now by using Lemma1 on (87), we have the following equation:(88)χn≤Υ5qςλ=X−Y1+q4ςλ,
which is our required result.


## 4. Main Results

Theorem 2.
An analytic functionf of the form (3) belongs to the class k-Vqj,iλ1,X,Y, if the following condition is satisfied:(89)∑n=2∞ℱ1+ℱ2φq,n−1an≤Y−X1+q,where(90)ℱ1=2k+13−qψn−n,q,ψn is given by (29) and(91)ℱ2=Yq+1−3−qn,q−Xq+1+3−qψn.Proof.
Suppose that (89) holds, then it is enough to show that(92)λφX,Y,q,λ1−1−ℜφX,Y,q,λ1−1<1.where φX,Y,q,λ1 is given by (56).
Now, we have the following equation:(93)1+λφX,Y,q,λ1−1=21+λ3−qRqλ1fj,iz−z∂qRqλ1fz3−q+Y1+qz∂qRqλ1fz−X1+q+−q+3Rqλ1fj,iz≤21+λ3−q∑n=2∞ψn−n,qφq,n−1anY−X1+q−∑n=2∞Λqφq,n−1an.where(94)Λq=3−q+Y1+qn,q−X1+q+−q+3ψn.
Expression in (93) is bounded above by 1 if(95)∑n=2∞ℱ1+ℱ2φq,n−1,an≤Y−X1+q,where ℱ1 and ℱ2 are defined by (90) and (91) respectively. Thus, we have completed the proof of our Theorem.
If in Theorem1 we put(96)q⟶1−λ1=0,i=1
then the result was reduced to the following known one.Corollary 1 (see [30]).
An analytic functionf of the form (3) is in the class λ-STX,Y, if it hold the condition(97)∑n=2∞21+λn−1+nY+1−X+1an≤Y−X.If in Theorem1, we set(98)λ1+1=j=1=i=X=Y+2.and letq⟶1−, then we get the following well-known consequence.Corollary 2 (see [2]).
An analytic functionf of the form (3) is in the class λ-ST, if it satisfies the condition(99)∑n=2∞n+λn−1an≤1.If we put(100)λ1+1=j=1=i=Y+2,X=1−2α0≤α<1,and letq⟶1−, in Theorem 2, then we have the following known result.Corollary 3 (see [24]).
An analyticf∈A and of the form (3) is in the class λ-SDλ,α, if it satisfies the condition(101)∑n=2∞n1+λ−λ+αan≤1−α0≤α<1,λ≥0.Theorem 3.
Iffz∈k-Vqj,iλ1,X,Y, then(102)an≤∏h=1n−1δkX−Y1+q−4h,q−ψhφq,h−1Y4h+1,q−ψh+1φq,hn≥2,where k≥0, −1≤Y<X≤1 and δk,ψn, φq,n−1 are defined by (16), (29) and (44).Proof.
Letfz∈λ-Vqj,iλ1,X,Y and(103)z∂qRqλ1fzRqλ1fj,iz=χz,
which implies that(104)z∂qRqλ1fz=Rqλ1fj,izχz.
By using (47) and (48) on (104), we have the following equation:(105)z+∑n=2∞n,qφq,n−1anzn=z+∑n=2∞φq,n−1ψnanzn1+∑n=1∞cnzn.
By making use of the well-known Cauchy Product formula on Right hand side of (105), we get the following equation:(106)∑n=2∞n,q−ψnφq,n−1anzn=∑n=2∞∑p=1n−1φq,n−1ψpcn−papzn.
On both sides of equation (106), comparing the coefficients of zn, we have the following equation:(107)n,q−ψnφq,n−1an=∑p=1n−1φq,n−1ψpcn−pap,a1=1,ψ1=1,φ0=1,which implies that(108)an≤1n,q−ψnφq,n−1∑p=1n−1φq,n−1ψpapcn−p.
By using Lemma6 together with (108), we have the following equation:(109)an≤X−Y1+qςλ4n,q−ψnφq,n−1∑p=1n−1φp−1ψpap.
We claim that(110)X−Y1+qςλ4n,q−ψnφq,n−1∑p=1n−1φq,p−1ψpap≤∏p=1n−1ςλX−Y1+q−4p,q−ψpφq,p−1Y4p+1,q−ψp+1φq,p.
We prove (110) by using the principle of the mathematical induction method.
Forn=2, from (109), we have the following equation:(111)a2≤ςλX−Y1+q42,q−ψ2φq,1.
Again from (110), we have the following equation:(112)a2≤ςλX−Y1+q42,q−ψ2φq,1.
Let the hypothesis be true forn=m, then from (109), we have the following equation:(113)am≤ςλX−Y1+q4m,q−ψmφq,m−1×∑p=1m−1φq,p−1ψpap.
From (110), we have the following equation:(114)am≤∏p=1m−1ςλX−Y1+q−4p,q−ψpφq,p−1Y4p+1,q−ψp+1φq,p.
Using modulus properties we get the following equation:(115)am≤∏p=1m−1ςλX−Y1+q+4p,q−ψpφq,p−14p+1,q−ψp+1φq,p.
By the induction hypothesis, we have the following equation:(116)ςλX−Y1+q4m,q−ψmφq,m−1∑p=1m−1φq,p−1ψpap≤∏p=1m−1ςλX−Y1+q+4p,q−ψpφq,p−14p+1,q−ψp+1φq,p.
Multiplying both sides of (116) by the following equation:(117)ςλX−Y1+q+4m,q−ψmφq,m−14m+1,q−ψm+1φq,m,
we have(118)ςλX−Y1+q+4m,q−ψmφq,m−14m+1,q−ψm+1φq,m∏p=1m−1ςλX−Y1+q+4p,q−ψpφq,p−14p+1,q−ψm+1φq,p≥ςλX−Y1+q+4m,q−ψmφq,m−14p+1,q−ψm+1φq,mςλX−Y1+q4m,q−ψmφq,m−1∑p=1m−1φq,p−1ψpap.
Furthermore,(119)∏p=1mςλX−Y1+q+4p,q−ψpφq,p−14p+1,q−ψp+1φq,p≥ςλX−Y1+q4m+1,q−ψm+1φq,mςλX−Y1+q4m,q−ψmφq,m−1∑p=1m−1φq,p−1ψpap+∑p=1m−1φq,p−1ψpap≥ςλX−Y1+q4m+1,q−ψm+1φq,mam+∑p=1m−1φq,p−1ψpap=ςλX−Y1+q4m+1,q−ψm+1φq,m∑p=1mφq,p−1ψpap.
That is,(120)ςλX−Y1+q4m+1,q−ψm+1φq,m∑p=1mφq,p−1ψpap≤∏p=1mςλX−Y1+q+4p,q−ψpφq,p−14p+1,q−ψp+1φq,p.
Hence inequality is true forn=m+1. Which completes the proof. □
If in Theorem2, we set(121)λ1=0,j=1=i,
and letq⟶1−, we get the following known result.Corollary 4 (see [30]).
An analytic functionf of the form )(3) belongs to the class λ-STX,Y, if it satisfies(122)an≤∏j=0n−2ςλX−Y−2jY2j+1.If we set(123)λ1=0,j=1=i=X=−Y,and letq⟶1−, in Theorem 2, we have the following result.Corollary 5 (see [3]).
An analytic functionf of the form (3) belongs to the class λ-ST, if it satisfies(124)an≤∏j=0n−2ςλ+jj+1.If we set(125)λ1=0,,j=1=i=X=−Yand letq⟶1−, also by taking λ=0, then ςλ=2, in Theorem 2, we have the following result.Corollary 6 (see [1]).
An analytic functionf of the form (3) is in the class S∗X,Y, if it satisfies(126)an≤∏j=0n−2X−Y−jYj+1,−1≤Y<X≤1.Theorem 4.
If the functionfz given by (3) belongs to the class k-Vqj,iλ1,X,Y, then(127)a2≤X−Y1+qδk23−q1+q−ψ2φq,1,and(128)a3≤X−Y1+q23−q1+q+q2−ψ3φq,2δ2k1−δk1−Y+Yq+3−q23−q+X−Y1+qδk2ψ223−q1+q−ψ2+δk.Proof.
Letχ∈P and defined by the following equation:(129)χz=1+wz1−wz=1+c1z+c2z2+⋯,where wz is a Schwarz function, such that w0=0 and wz<1. It follows seasily that(130)wz=χz−1χz+1=c12z+12c2−c124z2+12c3−12c1c2+c138z3….
Iffz∈λ-Vqj,iλ1,X,Y, thus from (55), we have the following equation:(131)z∂qRqλ1fzRqλ1fj,iz≺X1+q−−q+3pλz−X1+q−−q+3Y1+q+−q+3pλz−Y1+q−−q+3,where pλz is given in (15), thus from (131), we have the following equation:(132)z∂qRqλ1fzRqλ1fj,iz=X1+q−−q+3pλwz−X1+q−−q+3Y1+q+−q+3pλwz−Y1+q−−q+3.
From (130), we have the following equation:(133)pλwz=1+ςλwz+ς2λwz2+ς3λwz3+⋯=1+ςλc12z+12c2−c124z2+12c3−12c1c2+c138z3…+ς2λc12z+12c2−c124z2+12c3−12c1c2+c138z3…2+ς3λc12z+12c2−c124z2+12c3−12c1c2+c138z3…3+⋯.
After some simplification we have the following equation:(134)pλwz=1+ςλ2c1z+ς2λc124+12c2−c122ςλz2+ςλ8−ς2λ4+ς3λ8c13+ς2λ2−ςλ2c1c2+ςλ2c3z3+⋯.
Using (134) and (132), we have the following equation:(135)z∂qRqλ1fzRqλ1fj,iz=1+X−Y1+qςλ43−qc1z+X−Y1+q23−qς2λ4−ςλ4−Y+Yq+3−qςλ283−qc12+ςλ2c2z2+X−Y1+q2−q+3ςλ2c3+ς2λ2−ςλ2−Y+Yq+3−q43−qc1c2+ςλ8−ς2λ4+ς3λ8−Y+Yq+3−qςλς2λ83−q+Y+Yq+3−q83−qςλ2−Y+Yq+3−q2323−q2ςλ3c13z3+⋯.
From (47) and (48), we have the following equation:(136)z∂qRqλ1fzRqλ1fj,iz=1+1+q−ψ2φq,1a2z+1+q+q2−ψ3φq,2a3−1+q−ψ2φq,12ψ2a22z2+1+q1+q2−ψ4φq,3a4−1+q−ψ2ψ3+1+q+q2−ψ3φq,2ψ2φq,1a2a3+1+q−ψ2φq,13ψ22a23z3+⋯.
From (135) and (136), equating coefficients of z and z2 gives us the following equation:(137)a2=X−Y1+qςλ43−q1+q−ψ2φq,1c1,and(138)a3=X−Y1+q23−q1+q+q2−ψ3φq,2ς2λ4−ςλ4−Y+Yq+3−q83−q+X−Y1+qςλ2ψ283−q1+q−ψ2c12+ςλ2c2.
By using Lemma3, on (137) and (138) we obtain the result asserted by Theorem 3.Theorem 5.
Let−1≤Y<X≤1 and 0≤k<∞ be fixed and let fz∈k−Vqj,iλ1,X,Y, then for a complex number μ.(139)a3−μa22≤X1δk2−8X1δkμC12−X1Y1+2μ<X1Y12C12,2X1Y12C12≤μ≤X1δk+Y12C12,8X1δkμC12−X1Y1−2μ>X1δk+Y12C12,where v, X1,Y1 and C1 are given by (141), (142), (143), and (144), respectively.Proof.
Using (137) and (138), we have(140)a3−μa22=X1ςλ2c2−2X1ςλμC12−X1Y1c12,where(141)v=2X1ςλμC12−X1Y1,(142)X1=X−Y1+q2−q+31+q+q2−ψ3φq,2,(143)Y1=ς2λ4−ςλ4−Y+Yq+3−q83−q+X−Y1+qςλ2ψ283−q1+q−ψ2,and(144)C1=X−Y1+qςλ43−q1+q−ψ2φq,1.
Apply Lemma2 on (140) we have.Theorem 6.
Let0≤k<∞ and let fz∈k-Vqj,iλ1,X,Y of the form (3), then for a complex number μ(145)a3−μa22≤X1δk2max1,2v−1,∀μ∈ℂ.where v, X1,Y1,C1 given by (141), (142), (143), and (144).Proof.
From (140), we have the following equation:(146)a3−μa22=X1ςλ2c2−2X1ςλμC12−X1Y1c12,
Apply Lemma4 in conjunction with (146), we obtain the result asserted by Theorem 5.


## 5. Conclusion

We have successfully studied the uses of certain Ruscheweyh-typeq-differential operator to a new subclass of q-starlike symmetric functions, which involving both the conic domains and the well-known celebrated Janowski functions in the open unit disk D. We then investigated many properties for the newly defined functions class, including for example coefficients inequalities, the Fekete–Szegö Problems, and a sufficient condition. We have also emphasized certain known results of our key findings.The interested readers should be advised not to be misled to believe that the so-calledk-Gamma function provides a “generalization” of the classical (Euler’s) Gamma function. Similar remarks will apply also to all of the usages of the so-called k-Gamma function including (for example) the so-called k,s-extensions of the Riemann–Liouville and other operators of fractional integral and fractional derivatives.



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