# Onq-Convex Functions Defined by the q-Ruscheweyh Derivative Operator in Conic Regions

Authors: Mehwish Jabeen; Sarfraz Nawaz Malik; Shahid Mahmood; S. M. Jawwad Riaz; Md. Shajib Ali
Journal: Journal of Mathematics
 (2022)

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## Abstract

The core objective of this article is to introduce and investigate a new classβ−UCVqλA,B of convex functions associated with the conic domain defined by the Ruscheweyh q-differential operator. Many interesting properties such as sufficiency criteria, coefficient bounds, partial sums, and radius of convexity of order α for the functions of the said class are investigated here.


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## Body

## 1. Introduction

Quantum calculus has emerged as one of the most vibrant areas of research in recent years. Researchers have discussed and found its applications in numerous dimensions, such as hypergeometric series, complex analysis, and applied physics. It has developed techniques to be used inq-calculus, time scales, partitions, and continued fractions. Jackson, for the first time, in the beginning of the 20th century, introduced quantum calculus, where he developed and standardized it. For more details about quantum calculus, see [1–14]. To make a good pace and understanding of the results presented in this article, we are going to give below some primary definitions and relevant details of quantum calculus. Suppose ℱ represents the class of holomorphic functions of type(1)yz=z+∑n=2∞bnzn.in open unit disk E=z:z∈ℂandz<1 and normalized by the conditions y′0=1 and y0=0. Moreover, S represents the class of all functions in ℱ which are univalent in E; see [15].A domainD is starlike with respect to a point z0∈D if all possible lines which are confined by two points, connecting z0 to any other point, lie entirely within D. Correspondingly, a domain E is convex if all possible lines which are obtained by connecting any two points in D lie thoroughly within D. More clearly, we can say that if the domain is starlike with respect to each of its points in D, then it is convex. If yE is starlike for y∈S with respect to the origin, then it is called a starlike function, whereas if yE is convex, then it is called a convex function. The class of all convex functions is represented by C, and the class of all starlike functions is represented by S∗. Analytically, these are defined as follows:(2)S∗:=y∈S:ℜzy′zyz>0,z∈E,C:=y∈S:ℜ1+zy″zy′z>0,z∈E.Forα∈0,1, suppose that S∗α and Cα are subclasses of S consisting of α-starlike functions and α-convex functions, respectively, defined analytically as follows:(3)S∗α:=y∈S:ℜzy′zyz>α,z∈E,Cα:=y∈S:ℜ1+zy″zy′z>α,z∈E.Forα=0, the class S∗α⇒S∗ and the class Cα⇒C. Moreover, the following two classes are closely related with their functions defined, respectively.(4)Sα∗:=y∈S:zy′zyz−1<1−α,z∈E,Cα:=y∈S:zy″zy′z<1−α,z∈E.Note thatSα∗⊆S∗α and Cα⊆Cα. The kth partial sum of the function y, denoted by yk, is the polynomial, defined by(5)ykz=z+∑n=2kbnzn.Generally, lower bounds on ratios such asℜyz/ykz or ℜykz/yz have been found to be sharp only when k=1, but Silverman determined sharpness ∀n∈ℕ; see [16, 17]. He investigated that lower bounds are strictly increasing functions of k. In the present article, by using Silverman’s technique [16], we will find the function’s ratio having Taylor series (1) to its sequence of partial sums ykz=z+∑n=2kbnzn when the coefficients of y are sufficiently small to fulfill the necessary and sufficient condition. In more details to clarify, we will find sharp lower bounds for yz/ykz, y′z/yk′z, ykz/yz, and yk′z/y′z. Indeed, we will use the familiar result, i.e., ℜwz−1/wz+1>0, z∈E, if and only if wz=∑n=1∞cnzn satisfies wz≤z. Unless otherwise stated, we will presume that y has form (1) and that its sequence of partial sums is represented by (5).Forα∈0,1, Ravichandran gave the sharp radius of starlike and convex functions of order α with form (1) whose Taylor series coefficients bn satisfy the conditions b2=2d,d∈0,1, and bn≤n; M or M/nM>0 for n≥3.Consider thaty1 and y2 are holomorphic functions in E with w0=0 and wz≤1, ∀z∈E, so that y1z=y2wz; y1 will be subordinated by y2 and denoted by y1≺y2. If y2 is holomorphic, then y1≺y2 iff y10=y20 and y1E⊆y2E.For two holomorphic functions(6)y1z=∑k=0∞akzkandy2z=∑k=0∞bkzkz∈Ε,the Hadamard product of y1z and y2z is defined as(7)y1z∗y2z=∑k=0∞akbkzk.We will define some notations and concepts of quantum calculus which are to be used in this article. All results can be found in [2, 3, 18]. For n∈ℕ, 0<q<1, we see the classical q-theory begins with the q-extension of the positive numbers. The expression(8)limq⟶11−qn1−q=n.proposes that we define the q-generalization of n, which is also called the q-bracket of n, given as(9)n,q=nq=1−qn1−q,and the q-generalization of the factorial which is called q-factorial given by(10)nq!=nqn−1q…1q,n=1,2,…,1,n=0.Theq-difference operator for y∈ℱ is defined as(11)∂qyz=yqz−yzzq−1,z∈Ε,and we can see that, for n∈ℕ and z∈Ε,(12)∂qzn=nqzn−1,∂q∑n−1∞bnzn=∑n−1∞nqbnzn−1.Foryz∈ℱ, the q-analogue of the Ruscheweyh differential operator is defined as(13)Rqλyz=φq,λ+1;z∗yz=z+∑n=2∞ψn−1bnzn,z∈Εandλ>−1,where(14)φq,λ+1;z=z+∑n=2∞ψn−1zn,and(15)ψn−1=Γqλ+nn−1q!Γqλ+1=λ+1,qn−1n−1q!,ψ0=1,where λ+1,qn−1 is a Pochhammer symbol, which is defined as follows:(16)n,qm=1,n=0,n,qn+1,qn+2,qn+3,q…m+n−1,q,n∈ℕ.From (13), it is clear that(17)Rq0yz=yzandRq1yz=z∂qyz,Rqmyz=z∂qmzm−1yzmq!,m∈ℕ,limq⟶1−φq,λ+1;z=z1−zλ+1,limq⟶1−Rqλyz=yz∗z1−zλ+1.It follows thatq⟶1−, and the Ruscheweyh q-differential operator converts into the Ruscheweyh differential operator Dδyz; for more details, see [19]. Using (13),(18)z∂Rqλyz=1+λqqλRqλ+1yz−λqqλRqλyz.Ifq⟶1−, then(19)zRλyz′=1+λRλ+1yz−λRλyz.Definition 1.
The functionpz will lie in the class β−PqA,B if and only if(20)pz≺A1+q+3−qp˜βz−A1+q−3−qB1+q+3−qp˜βz−B1+q−3−q,β≥0,where(21)p˜βz=1+z1−z,β=0,1+2π2log1+z1−z2,β=1,1+21−β2sinh22πarccosβarctanyz,0<β<1,1+1β2−1sinπ2Rn∫0uz/t11−x21−tx2dx+1β2−1,β>1.
For more details, see [20–24]. If p˜βz=1+δβz+⋯, then it is shown in [25] that, from (46), one can have(22)δβ=8arccosβ2π21−β2,0≤β<1,8π2,β=1,π24β2−1t1+tR2t,β>1.Definition 2.
A functionyz∈ℱ will lie in the class β−UCVqA,B, β≥0, −1≤B<A≤1, if and only if(23)ℜB1+q−3−qDqzDqyz/Dqyz−A1+q−3−qB1+q+3−qDqzDqyz/Dqyz−A1+q+3−q>βB1+q−3−qDqzDqyz/Dqyz−A1+q−3−qB1+q+3−qDqzDqyz/Dqyz−A1+q+3−q−1,or equivalently,(24)DqzDqyzDqyz∈β−PqA,B.For more details about the above classes and conic domain, we refer the readers to [20, 25–28]. Using the q-Ruscheweyh differential operator, we now define the following more general class β−UCVqλA,B of functions associated with the conic domain defined by Janowski functions.Definition 3.
A functionyz∈ℱ will lie in the class β−UCVqλA,B, β≥0,−1≤B<A≤1, if and only if(25)ℜB1+q−3−q∂qz∂qRqλyz/∂qRqλyz−A1+q−3−qB1+q+3−q∂qz∂qRqλyz/∂qRqλyz−A1+q+3−q>βB1+q−3−q∂qz∂qRqλyz/∂qRqλyz−A1+q−3−qB1+q+3−q∂qz∂qRqλyz/∂qRqλyz−A1+q+3−q−1,or equivalently,(26)∂qz∂qRqλyz∂qRqλyz∈β−PA,B.The above defined classβ−UCVqλA,B generalizes many known classes which can be obtained by setting suitable particular values to the parameters as follows.Special cases:(1)
β−UCV1−0A,B=β−UCVA,B, the well-known class of β-uniformly Janowski convex functions, introduced by Noor and Malik [27](2)
0−UCV1−0A,B=CA,B, the well-known class of Janowski convex functions, introduced by Janowski [20](3)
β−UCV1−01−2α,−1=KDβ,α, see [29](4)
0−UCV1−01−2α,−1=Cα, see [15]Lemma 1 (see [30]).
Letgz=1+∑n=1∞cnzn be subordinate to Gz=1+∑n=1∞Cnzn. If Gz is holomorphic in E and GE is convex, then(27)cn≤C1,n≥1.


## 2. Main Results

Theorem 1.
A functionyz∈ℱ with form (1) will lie in class β−UCVqλA,B, β≥0, −1≤B<A≤1, if it satisfies the condition(28)∑n=2∞Enεbn<1,where(29)En=nq23−qβ+1qn−1q+B1+q+3−qnq−A1+q+3−qψn−1,and(30)ε=1+qB−A.Proof.
Suppose that (28) holds; then, it is enough to show that(31)βB1+q−3−q∂qz∂qRqλyz/∂qRqλyz−A1+q−3−qB1+q+3−q∂qz∂qRqλyz/∂qRqλyz−A1+q+3−q−1−ℜB1+q−3−q∂qz∂qRqλyz/∂qRqλyz−A1+q−3−qB1+q+3−q∂qz∂qRqλyz/∂qRqλyz−A1+q+3−q−1<1.
We consider(32)βB1+q−3−q∂qz∂qRqλyz/∂qRqλyz−A1+q−3−qB1+q+3−q∂qz∂qRqλyz/∂qRqλyz−A1+q+3−q−1−ℜB1+q−3−q∂qz∂qRqλyz/∂qRqλyz−A1+q−3−qB1+q+3−q∂qz∂qRqλyz/∂qRqλyz−A1+q+3−q−1≤β+1B1+q−3−q∂qz∂qRqλyz−A1+q−3−q∂qRqλyzB1+q+3−q∂qz∂qRqλyz−A1+q+3−q∂qRqλyz−1=23−qβ+1∂qRqλyz−∂qz∂qRqλyzB1+q+3−q∂qz∂qRqλyz−A1+q+3−q∂qRqλyz=23−qβ+1∑n=2∞1−nqψn−1nqbnznzB−A1+q+∑n=2∞B1+q+3−qnq−A1+q+3−qψn−1nqbnzn≤23−qβ+1∑n=2∞qn−1qnqbn1+qB−A1/ψn−1−∑n=2∞B1+q+3−qnq−A1+q+3−qnqbn.
The last expression is bounded above by 1 if(33)23−qβ+1∑n=2∞qn−1qnqbn<1+qB−A1ψn−1−∑n=2∞B1+q+3−qnq−A1+q+3−qnqbn,which reduces to(34)∑n=2∞nq23−qβ+1qn−1q+B1+q+3−qnq−A1+q+3−qψn−1bn<1+qB−A.
This finalizes the proof. □Forq⟶1− and λ=0, we have the following known result, proved in [27].Corollary 1.
A functionyz∈ℱ with form (1) will lie in class β−UCVA,B, β≥0, −1≤B<A≤1, if it satisfies the condition(35)∑n=2∞n2β+1n−1+nB+1−A+1bn<B−A.Forq⟶1−, λ=0, and A=1−2αandB=−1, we have the following known result, proved in [29].Corollary 2.
A functionyz∈ℱ with form (1) will lie in class KDβ,α, β≥0 , 0≤α<1, if it satisfies the condition(36)∑n=2∞nnβ+1−β+αbn<1−α.Theorem 2.
Letyz∈β−UCVqλA,B, β≥0, −1≤B<A≤1, and be of form (1); then, for n≥2,(37)bn≤1nq∏j=0n−2A−Bq+1δβψj−4Bqjqψj4qj+1qψj+1,where ψ is defined by (15).Proof.
By the definition foryz∈β−UCVqλA,B, we have(38)∂qz∂qRqλyz∂qRqλyz=pz,where(39)pz≺A1+q+3−qp˜βz−A1+q−3−qB1+q+3−qp˜βz−B1+q−3−q.
Ifp˜βz=1+δβz+…, then(40)A1+q+3−qp˜βz−A1+q−3−qB1+q+3−qp˜βz−B1+q−3−q=1+14A−Bq+1δβ+14−14Aq−14A+14Bq+14BB+11+q+2−2qδβ2+⋯.
Now, ifpz=1+∑n=1∞cnzn, then by (27) and (39), we have(41)cn≤14A−Bq+1δβ,n≥1.
Now, from (38), we have(42)∂qz∂qRqλyz=pz∂qRqλyz.
Letpz=1+∑n=1∞cnzn, and using the Cauchy product formula, we obtain(43)1+∑n=2∞nqnqψn−1bnzn−1=1+∑n=1∞cnzn1+∑n=2∞nqψn−1bnzn−1∑n=2∞nqnqψn−1bnzn−1=∑n=2∞cn−1zn−1+∑n=2∞nqψn−1bnzn−1+∑n=2∞nqψn−1bnzn−1.∑n=2∞cn−1zn−1.
This implies that(44)∑n=2∞nq−1nqψn−1bnzn−1=∑n=2∞cn−1+∑n=2∞∑j=2njqψj−1bjcn−jzn−1.
Comparison of coefficients ofzn−1 gives us(45)nq−1nqψn−1bn=cn−1+∑j=2nj,qψj−1bjcn−jb1=1,or(46)bn=1nq−1nqψn−1cn−1+∑j=2njqψj−1bjcn−j.
Using (41), we have(47)bn≤A−Bq+1δβ4qn−1qnqψn−11+∑j=2n−1jqψj−1bj,or(48)bn≤A−Bq+1δβ4qn−1qnqψn−1∑j=1n−1jqψj−1bj.
Now, we prove that(49)A−Bq+1δβ4qn−1qnqψn−1∑j=1n−1jqψj−1bj≤1nq∏j=0n−2A−Bq+1δβψj−4Bqnqψj4qj+1qψj+1.
For this, we use the induction technique. Forn=2, we have from (46),(50)b2≤δβq+1A−B4q1q2qψ1∑j=12−1jqψj−1bj,or(51)b2≤δβA−Bq+14q1q2qψ1,ψ0=1.
Forn=3, we have from (46),(52)b3≤δβA−Bq+14q2q3qψ2∑j=12jqψj−1bj=δβA−Bq+14q2q3qψ21qψ0b1+2qψ1b2≤δβA−Bq+14q2q3qψ21+A−Bq+1δβ4q1q.
From (37), we have(53)b3≤13q∏j=01A−Bq+1δβψj−4Bqjqψj4qj+1qψj+1=13qA−Bq+1δβ4q1qψ1A−Bq+1δβψ1+4q1qψ14q2qψ2=δβA−Bq+14q2q3qψ21+A−Bq+1δβ4q1q.
Let the assumption be true forn=m+1. From (46), we have(54)bm≤δβA−Bq+14qm−1qmqψm−1∑j=1m−1jqψj−1bj.
From (37), we have(55)bm≤1mq∏j=0m−2A−Bq+1δβψj−4Bqjqψj4qj+1qψj+1.
By the induction hypothesis,(56)1mq∏j=0m−2A−Bq+1δβψj−4Bqjqψj4qj+1qψj+1≥δβA−Bq+14qm−1qmqψm−1∑j=1m−1jqψj−1bj.
Multiplying both sides by1/mqA−Bq+1δβψm−1+4qm−1qψm−1/4qmqψm, we have(57)1mq∏j=0m−2A−Bq+1δβψj−4Bqjqψj4qj+1qψj+1≥1mqA−Bq+1δβψm−1+4qm−1qψm−14qmqψm×.δβA−Bq+14qm−1qψm−1∑j=1m−1ψj−1bj=δβA−Bq+14qmqψm×.ψm−1δβA−Bq+14qm−1qmqψm−1∑j=1m−1ψj−1bj+1mq∑j=1m−1ψj−1bj≥δβA−Bq+14qmqψmψm−1bm+1mq∑j=1m−1ψj−1bj=δβA−Bq+14qmqmqψm∑j=1mψj−1bj.
That is,(58)δβA−Bq+14qm−1qmqψm−1∑j=1m−1ψj−1bj≤1mq∏j=0m−2A−Bq+1δβψj−4Bqjqψj4qj+1qψj+1.
Hence, the consequence is true forn=m+1. Therefore, using mathematical induction, we have proved that (37) is true ∀n, n≥2. □Forq⟶1− and λ=0, we have the following known result, proved in [27].Corollary 3.
Letyz∈β−UCVA,B, β≥0, −1≤B<A≤1, and be of form (1); then, for n≥2,(59)bn≤1n∏j=0n−2A−Bδβ−2Bj2j+1.Forq⟶1−, λ=0, and A=1−2αandB=−1, we have the following known result, proved in [29].Corollary 4.
Letyz∈KDβ,α, β≥0, 0≤α<1, and be of form (1); then, for n≥2,(60)bn≤1n∏j=0n−21−αδβ+jj+1.Using the already proven results of Silverman [16] and Silvia [17] on partial sums of holomorphic functions, we will find the fraction of (1) to its sequence of partial sums ykz=z+∑n=2kbnzn when the function yz has coefficients small enough to satisfy condition (28). We will investigate sharp lower bounds for ℜyz/ykz, ℜy′z/yk′z, ℜykz/yz, and ℜyk′z/y′z in the class β−UCVqλA,B.Theorem 3.
Ifyz∈β−UCVqλA,B, then(61)ℜyzykz≥1−εEk+1,where Ek+1 is defined by (29) and ε=1+qB−A. The extremal function(62)yz=z+εEk+1zk+1.gives the sharp result.Proof.
Define a functionwz:(63)wz=Ek+1ε.yzykz−1−εEk+1,and this will reduce to(64)=Ek+11+∑n=2∞bnzn−1ε1+∑n=2kbnzn−1−Ek+1ε+1wz=1+∑n=2kbnzn−1+Ek+1/ε∑n=k+1∞bnzn−11+∑n=2kbnzn−1.
We have(65)wz−1wz+1≤Ek+1/ε∑n=k+1∞bn2−2∑n=2kbn−Ek+1/ε∑n=k+1∞bn.
Now,(66)wz−1wz+1≤1if(67)∑n=2kbn+Ek+1ε∑n=k+1∞bn≤1.
It is sufficient to show that the left hand side of (28) is bounded above by ∑n=2∞En/εbn if(68)∑n=2kbn+Ek+1ε∑n=k+1∞bn≤∑n=2∞Enεbn.
This leads to the following expression:(69)∑n=2kEn−εεbn+En−Ek+1ε∑n=k+1∞bn≥0.
To ensure that the function defined by (62) gives the sharp outcome, we note that, for z=reiπ/n,(70)yzykz=1+εEk+1zn=1+εEk+1rneiπn=1+εrnEk+1cosπn+isinπn=1−εrnEk+1yzykz=Ek+1−εEk+1whenr⟶1.Theorem 4.
Ifyz∈β−UCVqλA,B, then(71)ℜykzyz≥Ek+1Ek+1+ε,where Ek+1 is defined by (29) and ε=1+qB−A. The result (71) is sharp with the function given by (62).Proof.
Define the functionwz:(72)wz=Ek+1+εε.ykzyz−Ek+1Ek+1+ε=1+∑n=2kbnzn−1−Ek+1/ε∑n=k+1∞bnzn−11+∑n=2∞bnzn−1.
This will become(73)wz−1wz+1=∑n=2kbnzn−1−∑n=2∞bnzn−1−Ek+1/ε∑n=k+1∞bnzn−12+∑n=2kbnzn−1+∑n=2∞bnzn−1−Ek+1/ε∑n=k+1∞bnzk−1=−1+Ek+1/ε∑n=k+1∞bnzn−12+2∑n=2kbnzn−1+1−Ek+1/ε∑n=k+1∞bnzk−1.
This implies that(74)wz−1wz+1≤1+Ek+1/ε∑n=k+1∞bn2−2∑n=2kbn−1−Ek+1/ε∑n=k+1∞bn.
Now,(75)wz−1wz+1≤1if(76)∑n=2kbn+∑n=k+1∞bn≤1.
It would be enough to show that the left side of (28) is bounded above by ∑n=2∞En/εbn if(77)∑n=2kbn+∑n=k+1∞bn≤∑n=2∞Enεbn,which leads to the following expression:(78)∑n=2kEnε−1bn+∑n=k+1∞Enε−1bn≥0or(79)∑n=2∞Enε−1bn≥0.
Consequently, the equality holds for the extreme functionyz given by (62). □We now turn to fractions related to the derivatives.Theorem 5.
Ifyz∈β−UCVqλA,B, then(80)ℜy′zyk′z≥Ek+1−εk+1Ek+1,where Ek+1 is defined by (29) and ε=1+qB−A. The result (80) is sharp with the function given by (62).Proof.
Define the functionwz:(81)wz=Ek+1εk+1.y′zyk′z−Ek+1−εk+1Ek+1=Ek+11+∑n=2∞nbnzn−1εk+11+∑n=2knbnzn−1−Ek+1−εk+1εk+1,and this will reduce to(82)wz=1+∑n=2knbnzn−1+Ek+1/εk+1∑n=k+1∞nbnzn−11+∑n=2knbnzn−1.
Now, we have(83)wz−1wz+1=Ek+1/εk+1∑n=k+1∞nbnzn−12+2∑n=2knbnzn−1+Ek+1/εk+1∑n=k+1∞nbnzn−1.
This implies that(84)wz−1wz+1≤Ek+1/εk+1∑n=k+1∞nbn2−2∑n=2knbn−Ek+1/εk+1∑n=k+1∞nbn.
Now,(85)wz−1wz+1≤1if(86)∑n=2knbn+Ek+1εk+1∑k=n+1∞nbn≤1.
It would be enough to show that the left side of (28) is bounded above by ∑n=2∞En/εbn if(87)∑n=2knbn+Ek+1εk+1∑n=k+1∞nbn≤∑n=2∞Ekεbn,which leads to the following expression:(88)∑n=2kEnε−nbn+∑n=k+1∞Enε−nEk+1εk+1bn≥0.
The result (80) is sharp with respect to the function given by (62). □Theorem 6.
Ifyz∈β−UCVqλA,B, then(89)ℜyk′zy′z≥Ek+1εk+1+Ek+1,where Ek+1 is defined by (29) and ε=1+qB−A. The result (89) is sharp with respect to the function given by (62).Proof.
Define the functionwz:(90)wz=εk+1+Ek+1εk+1.yk′zy′z−Ek+1εk+1+Ek+1=εk+1+Ek+11+∑n=2knbnzn−1εk+11+∑n=2∞nbnzn−1−Ek+1εk+1.
This will become(91)wz=1+∑n=2knbnzn−1−Ek+1/εk+1∑n=k+1∞nbnzn−11+∑n=2∞nbnzn−1.
This leads us to(92)wz−1wz+1=−∑n=k+1∞1+Ek+1/εk+1nbnzn−12+2∑n=2knbnzn−1+∑n=k+1∞1−Ek+1/εk+1nbnzn−1,which reduces to(93)wz−1wz+1≤1+Ek+1/εk+1∑n=k+1∞nbn2−2∑n=2knbn−1−Ek+1/εk+1∑n=k+1∞nbn.
Now,(94)wz−1wz+1≤1.if(95)∑n=2knbn+∑n=k+1∞nbn≤1.
It is sufficient to show that the left hand side of (28) is bounded above by ∑n=2∞En/εbn if(96)∑n=2knbn+∑n=k+1∞nbn≤∑n=2∞Enεbn,which leads to the following expression:(97)∑n=2∞Enε−nbn≥0.
The result (89) is sharp with respect to the function given by (62). □In the next theorem, we will find the radii of starlikeness for the classβ−UCVqλA,B.Theorem 7.
Letyz∈β−UCVqλA,B. Then, yz is a convex function of order α∈0,1 in z<r=r1α, where(98)r1α=En1−αεqn−1q+1−α1/n−1,n=2,3,….,where En is defined by (29) and ε=1+qB−A.Proof.
Letyz∈β−UCVqλA,B. Then, by the theorem,(99)∑n=2∞Enεbn<1,where En is defined by (29) and ε=1+qB−A. For α∈0,1, we need to show that(100)∂qz∂qRqλyz∂qRqλyz<1−α,that is,(101)∂qz∂qRqλyz−∂qRqλyz∂qRqλyz=−∑n=2∞qn−1qnqψn−1bnzn−11−∑n=2∞nqψn−1bnzn−1≤∑n=2∞qn−1qnqψn−1bnzn−11−∑n=2∞nqψn−1bnzn−1<1−α.
Thus,∂qz∂qRqλyz−∂qRqλyz/∂qRqλyz≤1−α if(102)qn−1q1−α+1nqψn−1bnzn−1≤1.
According to theorem (99), inequality (102) will be true if(103)qn−1q1−α+1zn−1≤Enε.
Solving (103) for z, we obtain(104)zn−1≤En1−αεqn−1q+1−α.
Settingz=rα in (104), we have(105)rα=En1−αεqn−1q+1−α1/n−1,which is the required result. □


## 3. Conclusion

In this article, we have applied theq-Ruscheweyh differential operator to define and study a new class β−UCVqλA,B of q-convex functions associated with the conic domain. This class generalizes the classes β−UCVA,B, CA,B, Kβ,α, Cα, and C which have been defined and studied earlier. This fact has been illustrated above with details and proper referencing. The results presented include sufficiency criteria related to Taylor series coefficients, the coefficient bounds, and the ratios of partial sums to their infinite sum for functions of the class β−UCVqλA,B.



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*Source: 2681789-2022-02-01.xml*