# Properties of Meromorphic Spiral-Like Functions Associated with Symmetric Functions

Authors: Daniel Breaz; Luminița-Ioana Cotîrlă; Elangho Umadevi; Kadhavoor R. Karthikeyan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

To consolidate or adapt to many studies on meromorphic functions, we define a new subclass of meromorphic functions of complex order involving a differential operator. The defined function class combines the concept of spiral-like functions with other studies pertaining to subclasses of multivalent meromorphic functions. Inclusion relations, integral representation, geometrical interpretation, coefficient estimates and solution to the Fekete-Szegö problem of the defined classes are the highlights of this present study. Further to keep up with the present direction of research, we extend the study using quantum calculus. Applications of our main results are given as corollaries.


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## Body

## 1. Introduction

LetA be the class of function of the form
(1)χξ=ξ+∑n=2∞anξn,which are analytic in the unit disc E=ξ:ξ<1. Also let S denote the class of functions χ∈A which are univalent in E. The subclasses of S consisting of functions which map unit disc onto a star-like and convex domain will be symbolized by S∗ and C, respectively. Also let P denote the class of functions h analytic in the unit disc, given by
(2)hξ=1+∑n=1∞Rnξn,ξ∈E,R1>0,and satisfies Rehξ>0, ξ∈E. For p∈ℕ=1,2,⋯, we let Lp to denote the class of functions χ of the form
(3)χξ=ξ−p+∑n=1∞dn−pξn−p,which are analytic in E∗=ξ:ξ∈ℂand0<ξ<1. Shi et al. [1] defined the class χξ∈MSpσ,τ if and only if
(4)−eiσξχ′ξχξ≺peiσ−2τ−pe−iσξ1−ξχ∈Lp,where σ<λ/2 and τ>pcosσ. Here, ≺ denotes the usual subordination of analytic function. The class MSpσ,τ is the meromorphic analogue of the class of p-valent spiral-like functions defined by Uyanik et al. in [2]. Similarly, we let MCpσ,τ to denote the class of function in Lp satisfying the condition
(5)−eiσ1+ξχ′′ξχ′ξ≺peiσ−2τ−pe−iσξ1−ξ.Extending the class of Janowski function ([3]), Aouf [4] (Equation (4)) (also see [5]) defined the class hξ∈PX,Y,p,τ if and only if
(6)hξ=p+pY+X−Yp−τwξ1+Ywξ,−1≤Y<X≤1,0≤τ<1,where wξ is the Schwartz function. Motivated by the recent study of Breaz et al. [5] and in view generalizing the superordinate function in (4), Cotîrla˘ and Karthikeyan in [6] defined and studied the following relation
(7)Δστξ=1+Xe−2iσpeiσ+τY−Xhξ+1−Xe−2iσpeiσ−τY−XY+1hξ+1−Y,where −1≤Y<X≤1,−π/2<σ<π/2,τ>pcosσ and hξ∈P.It is well-known that the functionhξ=1+ξ/1−ξ maps the unit disc onto the right half plane. For an admissible choice of the parameter X=0.5,Y=−0.5, p=1,σ=π/3, and τ=0.6, Δστξ maps unit disc onto a domain which is convex with respect to point 0.5 if hξ=1+ξ/1−ξ (see Figure 1). Similarly, the function hξ=ξ+1+ξ33 which is related to the class of functions associated with leaf-like domain (see [7–9]) gets rotated and translated on the impact of Δστξ (see Figure 2) for a choice of the parameter X=0.5,Y=−0.5, p=1,σ=π/3, and τ=0.6.Remark 1.
The purpose to studyΔστξ was mainly motivated by the study of Karthikeyan et al. [10] and Noor and Malik [11]. Here, we will list some recent studies.
(1)
If we letσ=0 in (7), then, Δστξ reduces to(8)ℵξ=1+Xp+τY−Xhξ+1−Xp−τY−XY+1hξ+1−Y.
The functionℵξ was defined and studied by Breaz et al. in [5].
(2)
If we letX=1,Y=−1 and hξ=1+ξ/1−ξ in (7), then, Δστξ reduces to 2τ−pe−iσ+2pcosσ−τ/1−ξ (see the superordinate function in (4)).Figure 1
The image of the unit disc under the mapping ofΔστξ, if hξ=1+ξ/1−ξ.Figure 2
The image of the unit disc under the mapping ofΔστξ, if hξ=ξ+1+ξ33.It is well-known that ifχξ given by (1) is in S, then, the ℓ-symmetrical function χξℓ1/ℓ, (ℓ is a positive integer) is also in S. Let ℓ be a positive integer and ε=exp2πi/ℓ. For χ∈A, let
(9)χℓξ=1ℓ∑ν=0ℓ−1χενξεν.The functionχ is said to be star-like with respect to ℓ-symmetric points if it satisfies the condition
(10)Reξχ′ξχℓξ>0.Here, we will letSℓs to denote the class of star-like functions with respect to ℓ-symmetric points. The class Sℓs was introduced by Sakaguchi [12] in which he showed that all functions in Sℓs are univalent. Note that S1s=S∗.A functionχ∈Lp is said to be ℓ-symmetrical if for each ξ∈E(11)χεξ=ε−pχξ,Forχ∈Lp, Equation (9) can be defined by the following equality
(12)χℓξ=1ℓ∑ν=0ℓ−1χενξε−νp,ℓ=1,2,3,⋯.Now, we extend the operator defined by Selvaraj and Karthikeyan in [13]. Using Hadamard product (or convolution), we define a operator for functions χ∈Lp as follows:
(13)Iμma1,a2,⋯,ar,c1,c2,⋯,csχ=1ξp+∑n=1∞μn+μma1na2n⋯arnc1nc2n⋯csndn−pξn−pn!,where xn is the Pochhammer symbol defined by
(14)xn=1ifn=0xx+1x+2⋯x+n−1ifn∈N0=1,2,,⋯.For convenience, we shall henceforth denote(15)Iμma1,a2,⋯,ar,c1,c2,⋯,csχ=Iμma1,c1χ.Note that in [13], Iμma1,c1χ was defined for χ∈L1. Here, we skip the discussion on the necessity of using differential or integral operator, refer to [13–17] and reference provided therein for detailed properties of Iμma1,c1χ.Throughout this paper, we assume that−1≤Y<X≤1,−π/2<σ<π/2,τ>pcosσ,λ≥1, ℓ∈ℕ, ε=exp2πi/ℓ and
(16)χℓm,μ,a1,c1;ξ=1ℓ∑ν=0ℓ−1ενpIμma1,c1χενξ=ξ−p+⋯,(17)χ∈Lp;ℓ=2,3,⋯.


### 1.1. Short Introduction to Quantum Calculus

For0<q<1, the Jacksons q-derivative operator is defined by (see [18, 19])
(18)Dqχξ≔χ′0,ifξ=0,χξ−χqξ1−qξ,ifξ≠0.From (18), if χ has the power series expansion (3), we can easily see that Dqχξ=−pqξ−p−1+∑n=1∞n−pqdn−pξn−p−1, for ξ≠0, where the q-integer numbernq is defined by
(19)nq≔1−qn1−q,and note that limq⟶1−Dqχξ=χ′ξ. Throughout this paper, we let denote
(20)nqk≔nqn+1qn+2q⋯n+k−1q.Theq-Jackson integral is defined by (see [20])
(21)Iqχξ≔∫0ξχtdqt=ξ1−q∑n=0∞qnχξqn,provided the q-series converges. Further observe that
(22)DqIqχξ=χξandIqDqχξ=χξ−χ0,where the second equality holds if χ is continuous at ξ=0. For details pertaining to the significance of univalent function theory in dual with quantum calculus, refer to [21, 22] (also see [23–26]).Meromorphic multivalent functions have been extensively studied by various authors, but motivation and references of this study are [1, 13, 27–36].Definition 2.
For−π/2<σ<π/2, λ≥1, τ≥pcosσ, b∈ℂ\0 and Iμma1,c1χ defined as in (13), a function χ belongs to the class MSℓm,λa1,c1;b;h;X,Y if it satisfies
(23)eiσp−1bξp+1λ−pIμma1,c1χ′ξλχℓm,μ,a1,c1;ξ−−pλ≺Δστξ,where ≺ denotes subordination and hζ is defined as in (2).Now, we will define a class replacing ordinary derivative with a quantum derivative inMSℓm,λa1,c1;b;h;X,Y.Definition 3.
For−π/3<σ<π/2, 0≤η≤1, τ≥pcosσ, b∈ℂ\0 and Iμma1,c1χ defined as in (13), a function χ belongs to the class QMSℓm,λa1,c1;b;h;X,Y if
(24)eiσpq−1bξp+1λ−pDqIμma1,c1χξλχℓm,μ,a1,c1;ξ−−pqλ≺Yqσ,τ;ξ,where Yqσ,τ;ξ is the q−analogue of Δστξ, which is defined by
(25)Yqσ,τ;ξ=1+Xe−2iσpqeiσ+τY−Xhξ+1−Xe−2iσpqeiσ−τY−XY+1hξ+1−Y.Remark 4.
We note that in the definition ofQMSℓm,λa1,c1;b;h;X,Y, the operator Iμma1,c1χ and χℓm,μ,a1,c1;ξ are the same as used in MSℓm,λa1,c1;b;h;X,Y. We have not used the q-analogue operator as it would require the reader to contend with additional set of parameters.


## 1.1. Short Introduction to Quantum Calculus

For0<q<1, the Jacksons q-derivative operator is defined by (see [18, 19])
(18)Dqχξ≔χ′0,ifξ=0,χξ−χqξ1−qξ,ifξ≠0.From (18), if χ has the power series expansion (3), we can easily see that Dqχξ=−pqξ−p−1+∑n=1∞n−pqdn−pξn−p−1, for ξ≠0, where the q-integer numbernq is defined by
(19)nq≔1−qn1−q,and note that limq⟶1−Dqχξ=χ′ξ. Throughout this paper, we let denote
(20)nqk≔nqn+1qn+2q⋯n+k−1q.Theq-Jackson integral is defined by (see [20])
(21)Iqχξ≔∫0ξχtdqt=ξ1−q∑n=0∞qnχξqn,provided the q-series converges. Further observe that
(22)DqIqχξ=χξandIqDqχξ=χξ−χ0,where the second equality holds if χ is continuous at ξ=0. For details pertaining to the significance of univalent function theory in dual with quantum calculus, refer to [21, 22] (also see [23–26]).Meromorphic multivalent functions have been extensively studied by various authors, but motivation and references of this study are [1, 13, 27–36].Definition 2.
For−π/2<σ<π/2, λ≥1, τ≥pcosσ, b∈ℂ\0 and Iμma1,c1χ defined as in (13), a function χ belongs to the class MSℓm,λa1,c1;b;h;X,Y if it satisfies
(23)eiσp−1bξp+1λ−pIμma1,c1χ′ξλχℓm,μ,a1,c1;ξ−−pλ≺Δστξ,where ≺ denotes subordination and hζ is defined as in (2).Now, we will define a class replacing ordinary derivative with a quantum derivative inMSℓm,λa1,c1;b;h;X,Y.Definition 3.
For−π/3<σ<π/2, 0≤η≤1, τ≥pcosσ, b∈ℂ\0 and Iμma1,c1χ defined as in (13), a function χ belongs to the class QMSℓm,λa1,c1;b;h;X,Y if
(24)eiσpq−1bξp+1λ−pDqIμma1,c1χξλχℓm,μ,a1,c1;ξ−−pqλ≺Yqσ,τ;ξ,where Yqσ,τ;ξ is the q−analogue of Δστξ, which is defined by
(25)Yqσ,τ;ξ=1+Xe−2iσpqeiσ+τY−Xhξ+1−Xe−2iσpqeiσ−τY−XY+1hξ+1−Y.Remark 4.
We note that in the definition ofQMSℓm,λa1,c1;b;h;X,Y, the operator Iμma1,c1χ and χℓm,μ,a1,c1;ξ are the same as used in MSℓm,λa1,c1;b;h;X,Y. We have not used the q-analogue operator as it would require the reader to contend with additional set of parameters.


## 2. Preliminaries and some Supplementary Results

Here, we will discuss the results which would help us to obtain our main results.We note that everything in classical calculus cannot be generalized to quantum calculus, notably the chain rule needs adaptation. Hence, logarithmic differentiation needs some application of analysis. In [37], Agrawal and Sahoo obtained the following result on logarithmic differentiation. For χ∈A and 0<q<1, we have
(26)IqDqχξχξ=q−1lnqlogχξ,where Iqχ is the Jackson q-integral, defined as in (21). Similarly, we can see that
(27)Dqχξλ−1/λ=λ−1λDqχξχξ−1/λ.Ifν is an integer, then the following identities follow directly from (16):
(28)χℓm,μ,a1,c1;ενξ=ε−νpχℓm,μ,a1,c1;ξ.(29)χℓ′m,μ,a1,c1;ενξ=ε−νp−νχℓ′m,μ,a1,c1;ξ=1ℓ∑ν=0ℓ−1εν+νpIμma1,c1χ′ενξ.Sinceq-derivative satisfies the linearity condition, (29) holds if the classical derivative is replaced with quantum derivative. That is,
(30)Dqχℓm,μ,a1,c1;ενξ=ε−νp−νDqχℓm,μ,a1,c1;ξ.We now state the following result which will be used to establish the coefficient inequalities.Lemma 5 (see [38]).
Letϑξ=1+∑n=1∞ϑnξn∈P and also let v be a complex number, then
(31)ϑ2−vϑ12≤2max1,2v−1,the result is sharp for functions given by
(32)ϑξ=1+ξ21−ξ2,ϑξ=1+ξ1−ξ.The Maclaurin series for the functionΔστξ (see [6]) for the function is given by
(33)Δστξ=peiσ+Xpe−iσ−τ−Ypeiσ−τR12ξ+⋯.If we define the functionϑξ by
(34)ϑξ=1+ϑ1ξ+ϑ2ξ2+⋯=1+wξ1−wξ≺1+ξ1−ξ,ξ∈E.We note thatϑ0=1 and ϑ∈P. Using (34), we have
(35)wξ=ϑξ−1ϑξ+1=12ϑ1ξ+ϑ2−ϑ122ξ2+ϑ3−ϑ1ϑ2+ϑ134ξ3+⋯.For somehξ=1+R1ξ+R2ξ2+⋯, we have
(36)p+be−iσΔστwξ−p=p+be−iσR1ϑ1Xpe−iσ−τ−Ypeiσ−τ4ξ+be−iσXpe−iσ−τ−Ypeiσ−τR14ϑ2−ϑ12Y+1R1+21−R2/R14ξ2+⋯.


## 3. Integral Representations and Closure Properties

We begin with the following.Theorem 6.
Letχ∈MSℓm,λa1,c1;b;h;X,Y, then for λ=1, we get for ξ∈E∗(37)χℓm,μ,a1,c1;ξ=ξpexp1ℓ∑ν=0ℓ−1∫0ενξ1tbp−e−iσΔστwt−2pdt.
And forλ>1, we have for ξ∈E∗(38)χℓm,μ,a1,c1;ξ=λ−1λ1ℓ∑ν=0ℓ−1∫0ξbp−e−iσΔστwενt+−pλtp+1λ−p1/λdtλ−1/λ,where χℓm,μ,a1,c1;ξ is defined by equality (16) and wξ is analytic in E with w0=0 and wξ<1.Proof.
Letχ∈MSℓm,λa1,c1;b;h;X,Y. In view of (23), we have
(39)ξp+1λ−pIμma1,c1χ′ξλχℓm,μ,a1,c1;ξ=bp−e−iσΔστwξ+−pλ,where wξ is analytic in E and w0=0, wξ<1. Substituting ξ by ενξ in the equality (39), respectively, (ν=0,1,2,⋯,ℓ−1,εℓ=1), we have
(40)ενξp+1λ−pIμma1,c1χ′ενξλχℓm,μ,a1,c1;ενξ=bp−e−iσΔστwενξ+−pλ.
Using (28) in (40), we get
(41)ξp+1λ−pενp+1λIμma1,c1χ′ενξλχℓm,μ,a1,c1;ξ=bp−e−iσΔστwενξ+−pλ.
Using the equality (29) in (42), we can get
(42)εν+νpIμma1,c1χ′ενξχℓm,μ,a1,c1;ξ1/λ=bp−e−iσΔστwενξ+−pλξp+1λ−p1/λ.
Letν=0,1,2,⋯,ℓ−1 in (42), respectively, and summing them we get
(43)χℓ′m,μ,a1,c1;ξχℓm,μ,a1,c1;ξ1/λ=1ℓ∑ν=0ℓ−1bp−e−iσΔστwενξ+−pλξp+1λ−p1/λ.Case 1.
Letλ=1 in (43). We need to integrate from 0 to ξ to find χℓm,μ,a1,c1;ξ. But from (43), we notice the presence of the first-order pole at the origin, the difficulty to integrate the above equality is avoided by integrating from ξ0 to ξ with ξ0≠0, and then, let ξ0⟶0. Therefore, on applying integration, we get
(44)logχℓm,μ,a1,c1;ξξp=1ℓ∑ν=0ℓ−1∫0ενξ1tbp−e−iσΔστwt−2pdt.
Hence, the proof of (37).Case 2.
Ifλ>1, (43) can be rewritten as
(45)χℓm,μ,a1,c1;ξ1−1/λ′=1−1λ1ℓ∑ν=0ℓ−1bp−e−iσΔστwενξ+−pλξp+1λ−p1/λ.
On integrating the above expression we obtain (38). Hence, the proof of Theorem 6.Theorem 7.
Letχ∈QMSℓm,λa1,c1;b;h;X,Y, then for λ=1, we get
(46)χℓm,μ,a1,c1;ξ=ξpexplnqq−1ℓ∑ν=0ℓ−1∫0ενξ1tbpq−e−iσYqσ,τ;wt−2pqdt.
And forλ>1, we have
(47)χℓm,μ,a1,c1;ξ=λ−1λ1ℓ∑ν=0ℓ−1∫0ξbpq−e−iσΔστwενt+pqλtp+1λ−p1/λdtλ−1/λ,where χℓm,μ,a1,c1;ξ is defined by equality (16) and wξ is analytic in E with w0=0 and wξ<1.Proof.
In view of (24), (30), and (43), we have
(48)Dqχℓm,μ,a1,c1;ξχℓm,μ,a1,c1;ξ1/λ=1ℓ∑ν=0ℓ−1bpq−e−iσYqσ,τ;wενξ+−pqλξp+1λ−p1/λ.Case 1.
Letλ=1 in (48). Using the definition of logarithmic differentiation for q-derivative operator (see (26)) in (48), we get 0<q<1(49)logχℓm,μ,a1,c1;ξξp=lnqq−1ℓ∑ν=0ℓ−1∫0ενξ1tbpq−e−iσYqσ,τ;wt−2pqdqt,where the integral is q-Jackson integral. Hence, the proof of (37).Case 2.
Ifλ>1, using chain rule (see (27)) for the q-difference operator defined in the previous section, (43) can be rewritten as
(50)Dqχℓm,μ,a1,c1;ξ1−1/λ=1−1λ1ℓ∑ν=0ℓ−1bpq−e−iσYqσ,τ;wενξ+−pqλξp+1λ−p1/λ.
On applyingq-Jackson integral in the above expression, we obtain (47).Corollary 8 (see [1, Theorem 1]).
Letχξ∈MSpσ,τ, then
(51)χξ=ξ−pexp2τ−pcosσe−iσ∫0ξwtt1−wtdt,ξ∈E∗,where wξ is analytic in E with w0=0 and wξ<1.Proof.
Lettingm=2, s=1, a1=c1, a2=1, X=1,Y=−1, ℓ=λ=b=1, and hξ=1+ξ/1−ξ in Theorem 6, then (43) reduces to the form
(52)−eiσξχ′ξχξ=peiσ−2τ−pcosσwξ1−wξ.
Retracing the steps as in Theorem6, we can establish the assertion of the corollary.Settingm=0, r=2, s=1, a1=c1, and a2=1 in Theorem 6, we get the followingCorollary 9.
Letχ∈MSℓ0,λ2,1;b;h;X,Y, then, for λ=1, we get for ξ∈E∗(53)χℓξ=−p∫0ξu−p−1exp1ℓ∑ν=0ℓ−1∫0ενu1tbp−e−iσΔστwtdtdu.
And forλ>1, we have for ξ∈E∗(54)χℓξ=−pλ−1λ∫0ξu1ℓ∑ν=0ℓ−1∫0ubp−e−iσΔστwενt+−pλtp+1λ−p1/λdtλ−1/λdu,where χℓξ is defined by equality (12) and wξ is analytic in E with w0=0 and wξ<1.Lettingλ=1X=1,Y=−1, b=1, and hξ=1+ξ/1−ξ in Corollary 9, we get the following result.Corollary 10. (see [1]).
Letχξ∈MCpσ,τ, then for ξ∈E∗(55)χξ=−p∫0ξu−p−1exp2τ−pcosσe−iσ∫0uwtt1−wtdtdu,where wξ is analytic in E with w0=0 and wξ<1.


## 4. Fekete-Szegö Inequality ofMSℓm,λa1,c1;b;h;X,Y and QMSℓm,λa1,c1;b;h;X,Y

Very few researchers have attempted at finding solution to the Fekete-Szegö problem for class of functions with respect toℓ-symmetric points, as it is computational tedious. Notable among those works on coefficient inequalities of classes of functions with respect to ℓ-symmetric points were done by Aouf et al. [39].Throughout this section, we let(56)Ψn=1ℓ∑ν=0ℓ−1ενn,ℓ∈ℕ;n≥1;εℓ=1,Θn=a1na2n⋯arnn!c1nc2n⋯csnandΩnm=μn+μm,n∈ℕ.Theorem 11.
Ifχξ∈MSℓm,λa1,c1;b;h;X,Y, then, we have for all μ∈ℂ(57)d2−p−μd1−p2≤bXpe−iσ−τ−Ypeiσ−τR12−pλ−1pΨ2+2−pλΘ2Ω2mmax1,2Q1−1,where Q1 is given by
(58)Q1=14Y+1R1+21−R2R1−e−iσbXpe−iσ−τ−Ypeiσ−τ2p2Ψ12+2p1−pΨ1+λλ−11−p2R12−pλpΨ1+1−pλ2−μe−iσbR1Xpe−iσ−τ−Ypeiσ−τpΨ2+2−pλΘ2Ω2m−pλ−1pΨ1+1−pλ2Θ12Ω12m.The inequality is sharp for eachμ∈ℂ.Proof.
Asχ∈MSℓm,λa1,c1;b;h;X,Y, by (23), we have
(59)ξp+1λ−pIμma1,c1χ′ξλχℓm,μ,a1,c1;ξ−−pλ=−be−iσΔστwξ−p.
Thus, letϑ∈P be of the form ϑξ=1+∑ℓ=1∞ϑnξn and defined by
(60)ϑξ=1+wξ1−wξ,ξ∈Ω.
On computation, we have(61)wξ=12ϑ1ξ+12ϑ2−12ϑ12ξ2+12ϑ3−ϑ1ϑ2+14ϑ13ξ3+⋯,ξ∈Ω.
The right hand side of (58)
(62)−be−iσΔστwξ−p=−be−iσR1ϑ1Xpe−iσ−τ−Ypeiσ−τ4ξ−be−iσXpe−iσ−τ−Ypeiσ−τR14ϑ2−ϑ12Y+1R1+21−R2/R14ξ2+⋯.
From the left hand side of (58) is given by
(63)ξp+1λ−pIμma1,c1χ′ξλχℓm,μ,a1,c1;ξ−−pλ=−pλ−1pΨ1+1−pλΘ1Ω1md1−pξ+−pλ−12p2ppΨ2+2−pλΘ2Ω2md2−p−2p2Ψ12+2p1−pΨ1+λλ−11−p2d1−p2Θ12Ω12mξ2+⋯.
From (61) and (62), we obtain
(64)d1−p=−e−iσbR1ϑ1Xpe−iσ−τ−Ypeiσ−τ4−pλ−1pΨ1+1−pλΘ1Ω1m,d2−p=−e−iσbXpe−iσ−τ−Ypeiσ−τR14−pλ−1pΨ2+2−pλΘ2Ω2mϑ2−14Y+1R1+21−R2R1−e−iσbXpe−iσ−τ−Ypeiσ−τ2p2Ψ12+2p1−pΨ1+λλ−11−p2R12−pλpΨ1+1−pλ2ϑ12.
Now we consider(65)d2−p−μd1−p2=−e−iσbXpe−iσ−τ−Ypeiσ−τR14−pλ−1pΨ2+2−pλΘ2Ω2mϑ2−14Y+1R1+21−R2R1−e−iσbXpe−iσ−τ−Ypeiσ−τ2p2Ψ12+2p1−pΨ1+λλ−11−p2R12−pλpΨ1+1−pλ2ϑ12−μe−2iσb2R12ϑ12Xpe−iσ−τ−Ypeiσ−τ216−p2λ−2pΨ1+1−pλ2Θ12Ω12m=−e−iσbXpe−iσ−τ−Ypeiσ−τR14−pλ−1pΨ2+2−pλΘ2Ω2mϑ2−ϑ124Y+1R1+21−R2R1−e−iσbXpe−iσ−τ−Ypeiσ−τ2p2Ψ12+2p1−pΨ1+λλ−11−p2R12−pλpΨ1+1−pλ2−μe−iσbR1Xpe−iσ−τ−Ypeiσ−τpΨ2+2−pλΘ2Ω2m−pλ−1pΨ1+1−pλ2Θ12Ω12m.
On applying Lemma5, we get the assertion.To demonstrate the applications of our results, here, we provide the most simple special case of our result. Note that the following result was obtained [[40], Theorem 6] for functions in χ∈A.Corollary 12.
Ifχξ∈L1 satisfies
(66)−ξχ′ξχξ≺hξ,and hξ=1+R1ξ+R2ξ2+⋯, with R1,R2∈ℝ, R1>0, then for all μ∈ℂ we have
(67)d1−μd02≤R12max1;R2R1−R1+2μR1.
The inequality is sharp for the functionχ∗ given by
(68)χ∗ξ=ξexp∫0ξ−ht+1tdt,ifR2R1−R1+2μR1≥1,ξexp∫0ξ−ht2+1tdt,ifR2R1−R1+2μR1≤1..Proof.
In Theorem11, taking r=2,s=1, a1=b1, a2=1, X=1, Y=−1, m=σ=τ=0, and ℓ=λ=p=1, we get the inequality
(69)d1−μd02≤R12,ifR2R1−R1+2μR1≤1,R12R2R1−R1+2μR1,ifR2R1−R1+2μR1≥1.Analogous to Theorem11, we can prove the following.Theorem 13.
Ifχξ∈QMSℓm,λa1,c1;b;h;X,Y, then, we have for all μ∈ℂ(70)d2−p−μd1−p2≤bXpqe−iσ−τ−Ypqeiσ−τR12−pqλ−12−pqλ−−pqΨ2Θ2Ω2mmax1,2Q2−1,where Q2 is given by
(71)Q2=14Y+1R1+21−R2R1−e−iσbXpqe−iσ−τ−Ypqeiσ−τ2−pq2Ψ12−2−pq1−pqΨ1+λλ−11−pq2R12−pλ1−pqλ−−pqΨ12−μe−iσbR1Xpqe−iσ−τ−Ypqeiσ−τ2−pqλ−−pqΨ2Θ2Ω2m−pλ−11−pqλ−−pqΨ12Θ12Ω12m.
The inequality is sharp.


## 5. Conclusions

The defined function classMSℓm,λa1,c1;b;h;X,Y though familiar with so called pseudo-star-like functions required lots of adaptation since it involves functions with a removable singularity of order p at the origin. Integral representation and Fekete-Szegö inequalities have been established. Further, we extend the class MSℓm,λa1,c1;b;h;X,Y by replacing the classical derivative with q-derivative. Since all the results involving classical derivative does not get translated to the results involving q-derivative, we used some modified conditions to obtain our main results. We note that these adaptation are essential for future research.



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*Source: 3444854-2022-08-28.xml*