# Certain Class of Analytic Functions with respect to Symmetric Points Defined byQ-Calculus

Authors: K. R. Karthikeyan; G. Murugusundaramoorthy; S. D. Purohit; D. L. Suthar
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard toj,k-symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to j,k-symmetric points, we successfully obtained the coefficient bounds for functions in the newly specified class. We also quantified few applications as special cases which are new (or known).


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## Body

## 1. Definitions and Preliminaries

The set of all analytic functions constructed on the unit discU=z∈ℂ:z<1 is symbolised by HU. Also, A indicates the subclass of HU that has a Taylor series representation:(1)fξ=ξ+∑n=2∞anξn,ξ∈U=ξ:ξ<1.The family of functionsf∈A that are univalent in U is represented by S. This is well established that if fξ, assume by (1), is in S, then fξk1/k (k is a positive integer) is consequently in S.Definition 1.
(see [1], Definition 3). Assume k is a positive integer. A domain D is known to be k-fold symmetric if a rotation of D about the origin through an angle 2π/k carries D onto itself. For U, a function f is said to be k-fold symmetric if and only if for each ξ in U(2)fe2πi/kξ=e2πi/kfξ.
ℱk represents the family including all k-fold symmetric functions.The concept ofk-symmetrical function was protracted to so-called j,k-symmetrical function by Liczberski and Połubiński in [2]. To be specific, a function fξ is reported for being j,k-symmetrical if(3)fεξ=εjfξ,ξ∈U,where k≥2 is a fixed integer, j=0,1,2,…,k−1 and ε=exp2πi/k. The family of j,k-symmetrical functions will indeed be indicated by ℱkj. We believe that ℱ21, ℱ20, and ℱk1 are quite well groups of odd functions, even functions, and k-symmetrical functions. Consider the subsequent equivalence demarcate fj,kξ as well(4)fj,kξ=1k∑ν=0k−1fενξενj,f∈A;k=1,2,…;j=0,1,2,…,k−1.It is evident thatfj,kξ is a linear operator from U into U. If ν is an integer, then the subsequent assumptions result directly from (4):(5)fj,kενξ=ενjfj,kξ,fj,k′ενξ=ενj−νfj,k′ξ=1k∑ν=0k−1f′ενξενj−ν,fj,k″ενξ=ενj−2νfj,k″ξ=1k∑ν=0k−1f″ενξενj−2ν.Let the functionf∈A provided by (1) and g∈A of the form gξ=ξ+∑n=2∞Υnξn, the Hadamard product (or convolution) of these two functions is indicated by(6)Hξ≔f∗gξ≔ξ+∑n=2∞anΥnξn,ξ∈U.Using Hadamard product, various authors studied the univalent function theory in dual with the theory of special functions, see [3–5] and references provided therein. Throughout this whole article, we will assume that k∈ℕ, ε=exp2πi/k, and(7)Hj,kξ=1k∑ν=0k−1ε−νjf∗gενξ=ξ+⋯,where(8)f,g∈A;k=1,2,…;j=0,1,2,…,k−1.From (7), we, thus, have(9)Hj,kξ=∑n=1∞anΥnΔn,jξn,a1=Υ1=1,Λn,j=1k∑ν=0k−1εn−jν.The investigation ofq-calculus (q stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [6, 7] was among the key contributors of all the scientists who introduced and developed the q-calculus theory. Just like q-calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. Though it is the first article in which a link was established between certain geometric nature of the analytic function and the q-derivative operator and the usage of q-calculus in function theory, a solid and comprehensive foundation is given in [8] by Srivastava. After this development, many researchers introduced and studied some useful operators in q-analog with the applications of convolution concepts. For example, Kanas and Raducanu [9] established the q-differential operator and then examined the behavior of this operator in function theory. For more applications of this operator, see [10, 11].Forf∈A assumed by (1) and 0<q<1, the Jackson’s q-derivative operator or q-difference operator for f∈A is specified under (see [12–14])(10)Dqfξ≔f′0,ifξ=0,fξ−fqξ1−qξ,ifξ≠0.From (10), if f is assumed as in (1), we can effortlessly see that(11)Dqfξ=1+∑n=2∞nqanξn−1,for ξ≠0, provided the q-integer number nq is represented by(12)nq≔1−qn1−q,and take into consideration limq⟶1−Dqfξ=f′ξ. During our study, we let signify(13)nqκ≔nqn+1qn+2q…n+κ−1q.Theq-Jackson integral is defined by (see [6])(14)Iqfξ≔∫0ξftdqt=ξ1−q∑k=0∞qkfξqk.If theq-series converges, further witness that(15)DqIqfξ=fξ,IqDqfξ=fξ−f0,where the second equality grasps if f is continuous at ξ=0.Let the classes of star-like functions of orderη0≤η<1 and convex functions of order η0≤η<1 are symbolised by S∗η and Cη, respectively. In A, we categorize the collection P of functions pξ∈A with p0=1 and ℜpξ>0. The functions in the P class are not univalent.WithU, let f,g be analytic. The function f is said to be subordinate to g in U if the Schwarz functionωξ exists in U such that ωξ<ξ and fξ=gωξ, as shown through f≺g. Whenever g is univalent in U, consequently the subordination is identical to f0=g0 and fU⊂gU.Using the concept of subordination for holomorphic functions, Ma and Minda [15] proposed the classes:(16)S∗ψ=f∈A:ξf′ξfξ≺ψ,Cψ=f∈A:1+ξf″ξf′ξ≺ψ,where ψ∈P with ψ′0>0 maps U onto a region star-like with respect to 1 and symmetric with respect to the real axis. By making a choice ψ to map unit disc on to some specific regions such as cardioid, parabolas, lemniscate of Bernoulli, and booth lemniscate in the right-half of the complex plane, various interesting subclasses of star-like and convex functions could be gained well.Lots of fascinating subclasses of star-like and convex functions may be constructed by usingψ to map unit disc on to particular areas such as cardioid, parabolas, lemniscate of Bernoulli, and booth lemniscate on the right-half of the complex plane.For arbitrary fixed numbersC,D, −1<C≤1,and−1≤D<C, we express through PC,D the family of functions pξ=1+p1ξ+p2ξ2+⋯ analytic in the unit disc and pξ∈PC,D if and only if(17)pξ=1+Cwξ1+Dwξ,where wξ is the Schwarz function. Geometrically, pξ∈PC,D if and only if p0=1 and pU lies inside an open disc centred with center 1−CD/1−D2 on the real axis having radius C−D/1−D2 with diameter end points p1−1=1−C/1−Dandp11=1+C/1+D. On observing that wξ=pξ−1/pξ+1 for pξ∈P, we have Sξ∈PC,D if and only if for some pξ∈P(18)Sξ=1+Cpξ+1−C1+Dpξ+1−D.For detailed study on the class of Janowski functions, we refer [16]. The class of Janowski star-like functions and Janowski convex functions is defined as follows:(19)S∗C,D≔f∈A:ξf′ξfξ≺1+Cξ1+Dξ,−1≤D<C≤1CC,D≔f∈A:1+ξf′′ξf′ξ≺1+Cξ1+Dξ,−1≤D<C≤1.Inspired by the theory familiarized by Sakaguchi [17], and the study on analytic functions with respect to j,k-symmetrical points by various authors (see [18–22]), under this article, we formulate new subclasses listed in Definition 2.Definition 2.
For−π/2<θ<π/2, b∈ℂ∖0, and Hj,kξ/ξ≠0 be defined as in (7). We say that f∈Ksbϑ;θ;ψ;g;C,D if Hξ=f∗gξ satisfies the subordination condition:(20)1+1+itanθbϑξ2H″ξ+ξH′ξ1−ϑHj,kξ+ϑξHj,k′ξ−1≺C+1ψξ−C−1D+1ψξ−D−1,where ψ∈P and is given by(21)ψξ=1+L1ξ+L2ξ2+L3ξ3+⋯,ξ∈U,L1≠0.Remark 1.
Here, we list few exceptional cases of the defined classKsbϑ;θ;ψ;g;C,D.(1)
If we letC=1,D=−1, θ=0, b=1 and gξ=ξ+∑n=2∞ξn, then Ks10;0;ψ;g;1,−1≡Ssj,kψ[19] and Ks11;0;ψ;g;1,−1≡Csj,kψ [19](2)
Fixingθ=0, b=1, gξ=ξ+∑n=2∞ξn and ψξ=1+ξ/1−ξ, then Ksbϑ;0;ψ;g;C,D reduces to the class Sj,kC,D ([18], Definition 5)For completeness, we will now defineq-analogue of the as follows.Definition 3.
For−π/2<θ<π/2, b∈ℂ∖0 and Hj,kξ/ξ≠0 be defined as in (7). We say that f∈QKsbϑ;θ;ψ;g;C,D if Hξ=f∗gξ holds the subordination condition:(22)1+1+itanθbϑqξ2Dq2Hξ+ξDqHξ1−ϑHj,kξ+ϑξDqHj,kξ−1≺C+1ψξ−C−1D+1ψξ−D−1,where ψ∈P and ψ is defined as in (21).By lettingψξ=1+ξ/1−qξ,q∈0,1 in QKsbϑ;θ;ψ;g;C,D, we have(23)1+1+itanθbϑqξ2Dq2Hξ+ξDqHξ1−ϑHj,kξ+ϑξDqHj,kξ−1=C+1wξ+2+C−1qwξD+1wξ+2+D−1qwξ,where q∈0,1, wξ is analytic in U, and w0=0, wξ<1.Remark 2.
The impact of Janowski functions on a particular conic region was initiated by Noor and Malik [23] and was subsequently studied by various authors (see [11, 24, 25] and references provided therein).


## 2. Inclusion Relationships and Integral Representations of the ClassesKsbϑ;θ;ψ;g;C,D and QKsbϑ;θ;ψ;g;C,D

Let us begin with the following.Theorem 1.
LetFj,kϑ;ξ=1−ϑHj,kξ+ϑξHj,k′ξ. If f∈Ksbϑ;θ;ψ;g;C,D, then(24)ℜ1+k1+itanθbξFj,k′ϑ;ξFj,kϑ;ξ−1k>0.Proof. From the definition ofKsbϑ;θ;ψ;h;C,D and (18), we have(25)ℜ1+1+itanθbϑξ2H″ξ+ξH′ξ1−ϑHj,kξ+ϑξHj,k′ξ−1>0.Replacingξ by ενξ in (25), then for all ν=0,1,2,…,k−1, we have(26)ℜ1+1+itanθbε2νϑξ2H″ενξ+ενξH′ενξ1−ϑHj,kενξ+ϑενξHj,k′ενξ−1>0,ξ∈U.Using (5) in (26), we get(27)ℜ1+1+itanθbε2νϑξ2H″ενξ+ενξH′ενξ1−ϑενjHj,kξ+ϑενjξHj,k′ξ−1>0,ξ∈U.Supposeν=0,1,2,…,k−1 in (27), respectively, and summing them, we arrive at(28)ℜ1+1+itanθbϑξ2∑ν=0k−1ε2ν−νjH″ενξ+ξ∑ν=0k−1εν−νjH′ενξ1−ϑHj,kξ+ϑξHj,k′ξ−1>0.or equivalently,(29)ℜ1+k1+itanθbξFj,k′ϑ;ξFj,kϑ;ξ−1k>0,ξ∈U.Hence the proof.Now, by using the following two equivalent forms (see ([14], page 3)) of product rule of the q-difference operator,(30)Dqfξgξ=gξDqfξ+fqξDqgξ=gqξDqfξ+fξDqgξ,we can establish the following result by retracing the steps as in Theorem 1.Theorem 2.
LetFj,kϑ;ξ=1−ϑHj,kξ+ϑξDqHj,kξ, where Hj,kξ/ξ≠0. If f∈QKsbϑ;θ;ψ;g;C,D, then(31)ℜ1+k1+itanθbξDqFj,kϑ;ξFj,kϑ;ξ−1k>0.Theorem 3.
Letf∈Ksbϑ;θ;ψ;g;C,D, then(32)Hj,kξ=ξexp1k∑ν=0k−1∫0ενξbC−Dψwt−11+itanθtD+1ψwt−D−1dt,ifϑ=0,Hj,kξ=∫0ξexp1k∑ν=0k−1∫0ενζbC−Dψwt−11+itanθtD+1ψwt−D−1dtdζ,ifϑ=1,where Hj,kξ is given by (7), wξ is analytic in U, and w0=0, wξ<1.Proof.
Letf∈Ksbϑ;θ;ψ;g;C,D. In view of (20), we have(33)ϑξ2H″ξ+ξH′ξ1−ϑHj,kξ+ϑξHj,k′ξ−1=bC−Dψwξ−11+itanθD+1ψwξ−D−1,where wξ is analytic in U and w0=0, wξ<1. Substituting ξ by ενξ in equality (33) and ensuing the steps as in Theorem 1, we get(34)ξFj,k′ϑ;ξFj,kϑ;ξ−1=1k∑ν=0k−1bC−Dψwενξ−11+itanθD+1ψwενξ−D−1.
From this equality, we get(35)ξFj,k′ϑ;ξFj,kϑ;ξ−1ξ=1k∑ν=0k−1bC−Dψwενξ−11+itanθξD+1ψwενξ−D−1.
Upon integration, we get(36)logFj,kϑ;ξξ=1k∑ν=0k−1∫0ξbC−Dψwενζ−11+itanθζD+1ψwενζ−D−1dζ.or equivalently,(37)1−ϑHj,kξ+ϑξHj,k′ξ=ξexp1k∑ν=0k−1∫0ενξbC−Dψwt−11+itanθtD+1ψwt−D−1dt.
This concludes the proof of Theorem3.Theorem 4.
Letf∈QKsbϑ;θ;ψ;g;C,D, then we have(38)Hj,kξ=ξexplnqq−1k∑ν=0k−1∫0ενξbC−Dψwt−11+itanθtD+1ψwt−D−1dqt,ifϑ=0,Hj,kξ=∫0ξexplnqq−1k×∑ν=0k−1∫0ενζbC−Dψwt−11+itanθtD+1ψwt−D−1dqtdqζ,ifϑ=1.where Hj,kξ defined by equality (7), wξ is analytic in U, and w0=0, wξ<1.Proof. Letf∈QKsbϑ;θ;ψ;g;C,D. In sight of Theorems 2 and 3, we have(39)ξFj,k′ϑ;ξFj,kϑ;ξ−1ξ=1k∑ν=0k−1bC−Dψwενξ−11+itanθξD+1ψwενξ−D−1,where wξ is analytic in U and w0=0, wξ<1. For f∈HU and 0<q<1, we obtain (see [10])(40)IqDqfξfξ=q−1lnqlogfξ,where Iqf is the Jackson q-integral, defined as in (14). Integrating the above equality, we get(41)q−1lnqlogFj,kϑ;ξξ=1k∑ν=0k−1∫0ξbC−Dψwενζ−11+itanθζD+1ψwενζ−D−1dqζ,or equivalently,(42)1−ϑHj,kξ+ϑξHj,k′ξ=ξexplnqq−1k∑ν=0k−1∫0ενξbC−Dψwt−11+itanθtD+1ψwt−D−1dqt.This concludes the proof of Theorem3.By fixingC=1,D=−1, θ=0, b=1, and gξ=ξ+∑n=2∞ξn in Theorem 3, we state the subsequent result.Corollary 1 (see ([19], Theorems 3 and 4)).
Letfj,kξ≠0 be assumed as in (4).(i)
Iff∈Ssj,kϕ, then(43)fj,kξ=ξexp1k∑ν=0k−1∫0ενξϕwt−1tdt.(ii)
Iff∈Csj,kϕ, then(44)fj,kξ=∫0ξexp1k∑ν=0k−1∫0ενξϕwt−1tdtdxieta,where wξ is analytic in U and w0=0, wξ<1.Corollary 2.
Letfj,kξ≠0 be assumed as in (4). If f∈Sj,kC,D, then(45)fj,kξ=ξexp1k∑ν=0k−1∫0ενξ1t1+Cwt1+Dwtdt,where wξ is analytic in U and w0=0, wξ<1.


## 3. Coefficient Inequalities forKsbϑ;θ;ψ;g;C,D and QKsbϑ;θ;ψ;g;C,D

The coefficient estimatean of the defined function classes is determined in this section.Theorem 5.
Letψξ<D−1/D+1,∀ξ∈U and Υn’s be real. If f∈Ksbϑ;θ;ψ;g;C,D, then for n≥2,(46)an≤1Υn∏m=1n−11−ϑ+ϑmΛm,jC−DL1b−21+itanθϑmm−1−Λm,j+m−1−ϑΛm,jD2secθϑm+1m−Λm+1,j+m+1−1−ϑΛm+1,j.Proof.
By the definition ofKsbϑ;θ;ψ;g;C,D, we have(47)1+1+itanθbϑξ2H″ξ+ξH′ξ1−ϑHj,kξ+ϑξHj,k′ξ−1=pξ,where pξ∈P and satisfies the condition pξ≺C+1ψξ−C−1/D+1ψξ−D−1.
Equivalently, (47) can be rewritten as(48)1+itanθ∑n=2∞ϑnn−1−Λn,j+n−1−ϑΛn,jΥnanξn=b∑n=1∞1−ϑ+ϑnΛn,janΥnξn∑n=1∞pnξn,a1=Υ1=Λ1,j=1.
On equating the coefficient ofξn, we get(49)1+itanθϑnn−1−Λn,j+n−1−ϑΛn,jϒnan=bpn−1Λ1,j+pn−21+ϑΛ2,jΥ2a2+⋯+p11+n−2ϑϒn−1an−1.
From Lemma 6 of [26], we have pn≤L1C−D/2, n≥1. On computation, we have(50)an≤bC−DL12secθϑnn−1−Λn,j+n−1−ϑΛn,jϒn∑m=1n−11−ϑ+ϑmΛm,jϒmam.
Takingn=2, in (50), we get(51)a2≤bC−DL12secθ2ϑ1−Λ2,j+2−1−ϑΛ2,jΥ2.
On substitutingn=2 in (46), we can see the hypothesis is true for n=2. Now, taking n=3 in (50), we get(52)a3≤bC−DL12secθ3ϑ2−Λ3,j+3−1−ϑΛ3,jΥ31+1+ϑΛ2,jΥ2a2≤bC−DL12secθ3ϑ2−Λ3,j+3−1−ϑΛ3,jΥ31+1+ϑΛ2,jbC−DL12secθ2ϑ1−Λ2,j+2−1−ϑΛ2,j.
If we letn=3, in (46), we have(53)a3≤1Υ3bC−DL1Λ1,j2secθ2ϑ1−Λ2,j+2−1−ϑΛ2,j×C−DL1b1+ϑΛ2,j−21+itanθ2ϑ1−Λ2,j+2−1−ϑΛ2,jD2secθ3ϑ2−Λ3,j+3−1−ϑΛ3,j≤1Υ3bC−DL1Λ1,j2secθ3ϑ2−Λ3,j+3−1−ϑΛ3,j×bC−DL11+ϑΛ2,j+2secθ2ϑ1−Λ2,j+2−1−ϑΛ2,jD2secθ2ϑ1−Λ2,j+2−1−ϑΛ2,j≤bC−DL12secθ3ϑ2−Λ3,j+3−1−ϑΛ3,jΥ31+1+ϑΛ2,jbC−DL12secθ2ϑ1−Λ2,j+2−1−ϑΛ2,j.
Hence, the hypothesis of the theorem is true forn=3. Now, let us suppose (46) is valid for n=2,3,…r. On using triangle inequality in (46), we get(54)ar≤1Υr∏m=1r−11−ϑ+ϑmΛm,jbC−DL1+2secθϑmm−1−Λm,j+m−1−ϑΛm,j2secθϑm+1m−Λm+1,j+m+1−1−ϑΛm+1,j.
By induction hypothesis, we have(55)bC−DL12secθϑrr−1−Λr,j+r−1−ϑΛr,jΥr∑m=1r−11−ϑ+ϑmΛm,jΥmam≤1Υr∏m=1r−11−ϑ+ϑmΛm,jbC−DL1+2secθϑmm−1−Λm,j+m−1−ϑΛm,j2secθϑm+1m−Λm+1,j+m+1−1−ϑΛm+1,j.
From the above inequality, we have(56)∏m=1r1−ϑ+ϑmΛm,jbC−DL1+2secθϑmm−1−Λm,j+m−1−ϑΛm,j2ϒr+1secθϑm+1m−Λm+1,j+m+1−1−ϑΛm+1,j≥bC−DL12secθϑr+1r−Λr+1,j+r+1−1−ϑΛr+1,jϒr+1∑m=1r1−ϑ+ϑmΛm,jϒmam,which implies that inequality (46) is true for n=r+1. Hence the proof of the theorem.Theorem 6.
Letψξ<D−1/D+1 for all ξ∈U and Υn be real. If f∈QKsbϑ;θ;ψ;g;C,D, then for n≥2,(57)an≤1Υn∏m=1n−11−ϑ+ϑmqΛm,jC−DL1b−21+itanθqϑmqm−1q−Λm,j+mq−1−ϑΛm,jD2secθϑqm+1qmq−Λm+1,j+m+1q−1−ϑΛm+1,j.Proof.
From the definition ofQKsbϑ;θ;ψ;g;C,D, we have(58)1+itanθ∑n=2∞ϑnqqn−1q−Λn,j+nq−1−ϑΛn,jΥnanξn=b∑n=1∞1−ϑ+ϑnqΛn,janΥnξn∑n=1∞pnξn.
Equating the coefficient ofξn and retracing the steps as in Theorem 5, we get(59)an≤bC−DL12secθϑnqqn−1q−Λn,j+nq−1−ϑΛn,jΥn×∑m=1n−11−ϑ+ϑmqΛm,jΥmam.
Now, by repeating the processes in Theorem5, we acquire the required outcome. □If we letψξ=1+ξ/1−qξ in Theorem 6, we have the following.Corollary 3.
Letf∈QKsbϑ;θ;ψ;g;C,D and Υn be real, then for n≥2,(60)an≤1Υn∏m=1n−11−ϑ+ϑmqΛm,jC−D1+qb−21+itanθqϑmqm−1q−Λm,j+mq−1−ϑΛm,jD1+q+1−q2secθϑqm+1qmq−Λm+1,j+m+1q−1−ϑΛm+1,j.If we letθ=0, b=1, gξ=ξ+∑n=2∞ξn, and ψξ=1+ξ/1−ξ in Theorem 5, we have the following.Corollary 4.
(see [18], Theorem 2) Let f∈Sj,kC,D and Υn be real, then for n≥2,(61)an≤∏m=1n−1Λm,jC−D−1+mm+1−Λm+1,j.


## 4. Conclusion

Very few studies have been showed on analytic functions with regard toj,k-symmetric points. Since we have articulated the problem differently so as to deviate from the similar studies, only few special cases could be discussed. Furthermore, by swapping the ordinary differentiation with quantum differentiation, we have tried at the discretization of some of the familiar findings.



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