# Certain Classes of Analytic Functions Bound with Kober Operators inq-Calculus

Authors: S. D. Purohit; M. M. Gour; S. Joshi; D. L. Suthar
Journal: Journal of Mathematics
 (2021)

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## Abstract

Through applying the Kober fractionalq-calculus apprehension, we preliminary implant and introduce new types of univalent analytical functions with a q-differintegral operator in the open disk U=ξ∈ℂ:∣ξ|<1. The coefficient inequality and distortion theorems are among the results examined with these forms of functions. Specific cases are responded and addressed immediately. The findings include an expansion of the numerous established results in the q-theory of analytical functions.


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## Body

## 1. Introduction and Preliminary

Theq-analysis theory has been applied in recent times in several fields of science and engineering. The fractional q-calculus is indeed an analog of the conventional fractional calculus in q-theory. Very recently, Wang et al. [1] and Yan et al. [2] investigated the properties of subclasses of multivalent analytic or meromorphic functions expressed with q-difference operators. Furthermore, Srivastava [3] investigated the excellent work with q-calculus and fractional q-calculus operators, which is quite valuable for academics working on these issues. The applications of fractional q-calculus operators have been investigated by Purohit and Raina [4] to describe several new classes of analytic functions in open disk U=ξ∈ℂ:ξ<1. Moreover, Murugusundaramoorthy et al. [5], Purohit [6], and Purohit and Raina [4, 7] gave related work and added various classes of univalent and multivalently analytic functions in open unit disk U. Several others have also released new classes of analytical functions with the resources of q-calculus operators. For any more inquiries on the analytic functions classes, we refer to [1, 2, 8–13] for functions described by applying q-calculus operators and subject related to this work.In the current inquiry, we are planning to develop few additional families of analytic functions applying the Kober differential and integral operators inq-calculus. The results obtained must also provide the coefficient inequalities and distortion theorems for the subclasses established here below. First, we use the main notations and definitions in the q-calculus which are relevant to grasp the object of the study.For each complex numberP, the q-shifted factorials are delimited by(1)P;qm=∏j=0m−11−qjP;m∈ℕ,P;q0=1,and with regard to the basic analog of the gamma function,(2)qP;qm=ΓqP+m1−qmΓqP;m>0,in which the q-gamma function is set by (see [14])(3)ΓqP=q;q∞1−q1−PqP;q∞;0<q<1.The recurrence relationship specified by Gaspar and Rahman [15] for the q-gamma function is(4)Γq1+P=1−qPΓqP1−q.Ifq<1, then equation (1) shall continue to play a role m=∞ as an infinite product of convergence:(5)P;q∞=∏j=0∞1−Pqj,and we have(6)P;qm=P;q∞Pqm;q∞;m∈ℕ∪∞.Theq-binomial expansion is now as follows:(7)a−bτ=aτ−ba;qτ=aτϕ10q−τ;−;q,bqτa.[15] accounts for Jackson’s q-integral and q-derivative of a function f, which are described on a subset of ℂ, as(8)Dqfξ=fξ−fqξξ1−q;ξ≠0,q≠0,with(9)Dξμ=1−qμξμ−11−q,∫0ξfydqy=ξ1−q∑k=0∞qkfξqk.


## 2. The Fractionalq-Calculus Operators

Purohit and Raina [4] described the fractional q-integral operator of function fξ given by(10)Iq,ξPfξ=1ΓqP∫0ξξ−yqP−1fydqy,where P>0 is the order of integral and fξ is an analytic function in U, and (7) the ξ−yqP−1 be expressed as(11)ξ−yqP−1=ξP−1ϕ10q1−P;−;q,yqPξ,where ϕ10P;−;q,ξ is special case of basic hypergeometric series ϕ21P;J;γ;q,ξ for γ=J is single valued for argξ<π and ξ<1 (see [15]).Purohit and Raina [4] defined the Dq,ξPfξ fractional q-derivative operator of a function fξ by(12)Dq,ξPfξ=1Γq1−PDq,ξ∫0ξξ−yq−Pfydqy,where 0≤P<1 and fξ is suitably constrained with Dq,ξ−Pfξ=Iq,ξPfξ.The Kober fractionalq-integral operator for a real valued function fx is determined by Garg and Chanchalani [16] as(13)Iqγ,Pfx=x−γ−PΓqP∫0xx−yqP−1yγfydqy,where γ being real or complex and P is an absolute order of integration with ℜP>0. For q⟶1, operator (13) is reduced to Kober operator Iγ,Pfx as defined in [17]. For γ=0, this operator is converted to Riemann–Liouville fractional q-integral operator with a power weight function as Iq0,Pfx=x−PIqPfx.The Kober fractionalq-derivative operator for a real valued function fx is detailed by Garg and Chanchalani [16] as(14)Dqγ,Pfx=∏j=1mγ+jq+xqγ+jDqIqγ+P,m−Pfx,where P is order of derivative with ℜP>0 and m=ℜP+1,m∈ℕ. For q⟶1, operator (14) is reduced to Kober operator Dγ,Pfx as defined in [17].We are now definingq-calculus operators with a view to applying these operators to the geometric function theory of complex analysis.Definition 1.
Kober fractionalq-integral operator:
For the functionfξ, the Kober fractional q-integral operator is demarcated by(15)Iqγ,Pfξ=x−γ−PΓqP∫0ξξ−yqP−1yγfydqy,where γ is the real or complex, P is an absolute order of integration with ℜP>0, and the q-binomial ξ−yqP−1is expressed as in (11).
Forq⟶1, operator (15) is reduced to Kober integral operator Iγ,Pfξ as defined in [17].Definition 2.
Kober fractionalq-derivative operator:
The Kober fractionalq-derivative operator for the function fξ is demarcated by(16)Dqγ,Pfξ=∏j=1mγ+jq+ξqγ+jDqIqγ+P,m−Pfξ,where P is the order of derivative with ℜP>0 and m=ℜP+1,m∈ℕ. For q⟶1, operator (16) is reduced to Kober derivative operator Dγ,Pfξ as defined in [17].
Under Koberq-integral and q-derivative operators fixed by (15) and (16), we offer the following image formulae for function ξμ.Remark 1.
IfP,γ,μ∈ℂ, ℜγ+P+μ+1>0, and ℜγ+μ+1>0, then(17)Dqγ,Pξμ=Γqγ+P+μ+1Γqγ+μ+1ξμ.Remark 2.
IfP,γ,μ∈ℂ, ℜγ+μ+1>0, and ℜγ+P+μ+1>0, then(18)Iqγ,Pξμ=Γqγ+μ+1Γqγ+P+μ+1ξμ.


## 3. New Classes of Functions

LetAm represent the function class of the form(19)fξ=ξ+∑k=m+1∞akξk;m∈ℕ,which are analytic and univalent in open unit disk U. Above, let Am− highlight the subclass of Am imposing of analytical and univalent functions articulated in the form(20)fξ=ξ−∑k=m+1∞akξk;ak≥0,m∈ℕ.For the dedication of this work, we describe a fractionalq-differintegral operator Ωqγ,P for a function fξ of the form (20) by(21)Ωqγ,Pfξ=Γqγ+2Γqγ+P+2ξ−1Dqγ,Pfξ=1−∑k=m+1∞Γqγ+2Γqγ+P+k+1Γqγ+P+2Γqγ+k+1,Ωqγ,Pfξ=1−∑k=m+1∞AP,γ,k,qakξk−1,where(22)AP,γ,k,q=Γqγ+2Γqγ+P+k+1Γqγ+P+2Γqγ+k+1,and ℜγ+2>0, 0<q<1,  ξ∈U,  m∈ℕ,  ℜP>0, and Dqγ,Pfξ represent a fractional q-derivative of fξ of order P. We announce here the alike classes of functions connecting operator (21):(23)Sqγ,PT=f∈Am−,Ωqγ,Pfξ−1Ωqγ,Pfξ−2T+1<J,where ℜP>0,ℜγ+2>0,0≤T<1,0≤J<1,0<q<1,ξ∈U.And(24)Tqγ,Pτ=f∈Am−,ℜ1−τΩqγ,Pfξ+τ1−q1−P1−qΩqγ,P+1fξ>J.The subsequent coefficient bounds for functions of the form (20) that belong to the classes Sqγ,PT and Tqγ,Pτ are now obtained (interpreted above).Theorem 1.
A functionf defined by (20) is connected to the class Sqγ,PT if and only if(25)∑k=m+1∞AP,γ,k,qak1+J≤2J1−T,where(26)AP,γ,k,q=Γqγ+2Γqγ+P+k+1Γqγ+P+2Γqγ+k+1.
The result is sharp.Proof.
Let us consider that inequality (25) holds, and for ξ=1, we have(27)Ωqγ,Pfξ−1−JΩqγ,Pfξ−2T+1=−∑k=m+1∞AP,γ,k,qakξk−1−J21−T−∑k=m+1∞AP,γ,k,qakξk−1≤∑k=m+1∞AP,γ,k,qak1+J−2J1−T≤0,and by our assumption, this indicates that fξ∈Sqγ,PT.
For the proof of converse part, suppose thatfξ∈Sqγ,PT, and then it follows that(28)Ωqγ,Pfξ−1Ωqγ,Pfξ−2T+1<J,which implies that(29)Ωqγ,Pfξ−1Ωqγ,Pfξ−2T+1=−∑k=m+1∞AP,γ,k,qakξk−121−T−∑k=m+1∞AP,γ,k,qakξk−1−1<J.
Sinceℜξ≤ξ for any ξ, therefore on choosing values of ξ on the real axis so that Ωqγ,Pfξ is real and allowing ξ⟶1 all through real values, we obtain from above inequality(30)∑k=m+1∞AP,γ,k,qak≤2J1−T−J∑k=m+1∞AP,γ,k,qak,which implies that(31)∑k=m+1∞AP,γ,k,qak1+J≤2J1−T,which is desired result. Here, we notice that assumption (25) of Theorem 1 is sharp and the external function is assumed by(32)fξ=ξ−2J1−T1+JAP,γ,m+1,qξm+1;m∈ℕ,where AP,γ,k,q is defined in (26).Theorem 2.
A functionf defined by (20) is connected to the class Tqγ,Pτ if and only if(33)∑k=m+1∞AP,γ,k,qAk,qP,γ,τak≤1−J−τ+τ1−q1−P,where(34)Ak,qP,γ,τ=1−τ1−q+τ1−q1−P1−qγ+P+k+11−qγ+P+2.
The accomplishment is sharp.Proof.
To prove above theorem, we address the elementary assertion that(35)ℜgξ≥J⇔1−J+gξ≥1+J−gξ.
Now,(36)gξ=1−τΩqγ,Pfξ+τ1−q1−P1−qΩqγ,P+1fξ=1−τ1−∑k=m+1∞AP,γ,k,qakξk−1+τ1−q1−P1−q1−∑k=m+1∞AP+1,γ,k,qakξk−1=1−τ+τ1−q1−P1−q−11−q∑k=m+1∞Ak,qP,γ,τakξk−1,where(37)Ak,qP,γ,τ=1−τ1−q+τ1−q1−P1−qγ+P+k+11−qγ+P+2.
In (35), it then suffices to show that(38)2−J−τ+τ1−q1−P1−q−11−q∑k=m+1∞AP,γ,k,qAk,qP,γ,τakξk−1−J+τ−τ1−q1−P1−q+11−q∑k=m+1∞AP,γ,k,qAk,qP,γ,τakξk−1≥0.
Now,(39)2−J−τ+τ1−q1−P1−q−11−q∑k=m+1∞AP,γ,k,qAk,qP,γ,τakξk−1−J+τ−τ1−q1−P1−q+11−q∑k=m+1∞AP,γ,k,qAk,qP,γ,τakξk−1≥21−q1−J−τ1−q+τ1−q1−P−∑k=m+1∞AP,γ,k,qAk,qP,γ,τakξk−1≥0.
This accomplishes the proof of theorem.
We accommodate that answer (33) is sharp. The external function is assumed by(40)fξ=ξ−1−J−τ+τ1−q1−PAP,γ,m+1,qAm+1,qP,γ,τξm+1;m∈ℕ,where Ak,qP,γ,τ is given by (34).


## 4. Distortion Theorems

Theorem 3.
Suppose that the functionf is defined by (20) in the class Sqγ,PT, then(41)ξ−ξm+12J1−T1+JBP,γ,m,q≤fξ≤ξ+ξm+12J1−T1+JBP,γ,m,q,where(42)BP,γ,m,q=1AP,γ,m+1,q=Γqγ+P+2Γqγ+m+2Γqγ+2Γqγ+P+m+2.
Furthermore,(43)ξ−ξm+12J1−T1+J≤ξΩqγ,Pfξ≤ξ+ξm+12J1−T1+J,where γ>−2,ℜP>0,0<q<1.Proof.
Sincefξ∈Sqγ,PT, then in interpretation of Theorem 1, we first show that the function(44)AP,γ,k,q=Γqγ+2Γqγ+P+k+1Γqγ+P+2Γqγ+k+1=ϕk,letis an increasing function of k for γ>−2 and ℜP>0.
It follows that(45)ϕk+1ϕk=Γqγ+P+k+2Γqγ+k+1Γqγ+P+k+1Γqγ+k+2=1−qγ+P+k+11−qγ+k+1,0<q<1.
Takingk=m+1, then(46)ϕm+2ϕm+1=1−qγ+P+m+21−qγ+m+2,0<q<1.
The functionϕk is an increasing function of k if ϕm+2/ϕm+1≥1, and this gives(47)1−qγ+P+m+21−qγ+m+2≥1;0<q<1,which implies(48)qP≤1,0<q<1.
This inequality abides forℜP>0.
Thus,ϕk,k≥m+1,m∈ℕ is an increasing function of k for ℜγ+2>0,ℜP>0 and 0<q<1.
Now, (25) gives the alike inequality:(49)AP,γ,m+1,q∑k=m+1∞ak1+J≤∑k=m+1∞AP,γ,k,qak1+J≤2J1−T,which implies that(50)∑k=m+1∞ak≤2J1−T1+JBP,γ,m,q,where BP,γ,m,q is defined in (42), and this last inequality is in the conjunction with the alike inequality (easily obtained from (20)):(51)ξ−ξm+1∑k=m+1∞ak≤fξ≤ξ+ξm+1∑k=m+1∞ak,and using (50), we have(52)ξ−ξm+12J1−T1+JBP,γ,m,q≤fξ≤ξ+ξm+12J1−T1+JBP,γ,m,q,which is result (41) of Theorem 3.
Now, on using (21), we observe that for functions of form (20),(53)ξΩqγ,Pfξ≥ξ−∑k=m+1∞AP,γ,k,qakξk≥ξ−ξm+1∑k=m+1∞AP,γ,k,qak,which on using Theorem 1 gives(54)ξΩqγ,Pfξ≥ξ−2J1−T1+Jξm+1,and similarly,(55)ξΩqγ,Pfξ≤ξ+2J1−T1+Jξm+1,which implies that(56)ξ−2J1−T1+Jξm+1≤ξΩqγ,Pfξ≤ξ+2P1−T1+Jξm+1.Corollary 1.
Let the function detailed in (20) be in the class Sqγ,PT, then(57)Γqγ+P+2Γqγ+2|ξ|−2P1−T1+Jξm+1≤Dqγ,Pfξ≤Γqγ+P+2Γqγ+2ξ+2J1−T1+Jξm+1,where ℜγ+2>0,ℜP>0, and ξ∈U.Corollary 2.
Let the function detailed in (20) be in the class Sqγ,PT, then(58)Γqγ+2Γqγ+P+2ξ−2J1−T1+JΓqγ+P+2Γqγ+P+m+22ξm+1≤Iqγ,Pfξ≤Γqγ+2Γqγ+P+2ξ+2J1−T1+JΓqγ+P+2Γqγ+P+m+22ξm+1,where ℜγ+2>0,ℜP>0, and ξ∈U.Theorem 4.
Suppose that the functionfξ detailed in (20) be in the class Tqγ,Pτ, then for ℜP>0,ℜγ+2>0,ξ∈U,0<q<1,(59)ξ−Bγ,P,m,qCξm+1≤fξ≤ξ+Bγ,P,m,qCξm+1,also(60)ξ−Cξm+1≤ξΩqγ,Pfξ≤ξ+Cξm+1,where(61)C=1−J−τ1−q+τ1−q1−PAm+1,qP,γ,τ,and Ak,qP,γ,τ and Bγ,P,m,q are detailed in (34) and (42), respectively.Proof.
SinceTqγ,Pτ, then under the hypothesis of Theorem 2, we have(62)∑k=m+1∞AP,γ,k,qak≤1−J−τ1−q+τ1−q1−PAm+1,qP,γ,τ,which implies that(63)∑k=m+1∞AP,γ,k,qak≤C,where Ak,qP,γ,τ and C are given by (34) and (61), respectively, and this last inequality, when combined with the following inequality (which is conveniently obtained from (20)),(64)ξ−ξm+1∑k=m+1∞ak≤fξ≤ξ+ξm+1∑k=m+1∞ak,and using (63), we have(65)ξ−Bγ,P,m,qCξm+1≤fξ≤ξ+Bγ,P,m,qCξm+1,which is result (59) of Theorem 4.
Now, from (21), we obtain(66)ξ−ξm+1∑k=m+1∞Aλ,γ,k,qak≤ξΩqγ,λfξ,on using (63), this implies that(67)ξ−Cξm+1≤ξΩqγ,λfξ,similarly, we have(68)ξΩqγ,λfξ≤ξ+Cξm+1,and on combining above two results, we have(69)ξ−Cξm+1≤ξΩqγ,λfξ≤ξ+Cξm+1.Corollary 3.
Let the function detailed in (20) be in the class Tqγ,Pτ, then for all ξ∈U,ℜγ+λ+2>0, ℜγ+2>0,(70)Γqγ+λ+2Γqγ+2ξ−Cξm+1≤Dqγ,λfξ≤Γqγ+λ+2Γqγ+2ξ+Cξm+1.The fractionalq-calculus operators presented in Section 2 may be used to explore numerous different multivalent (or meromorphic) analytic function subclass and geometric characteristics which includes coefficient estimates, distortion bounds, radii of starlikeness, convexity, and so forth. The concept of fractional q-calculus can also be used to again with considerations.



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