# New Subclass of Analytic Function Related with Generalized Conic Domain Associated withq− Differential Operator

Authors: Shahid Khan; Saqib Hussain; Ilyas Khan; Amnah s. Al-johani; Mulugeta Andualem
Journal: Journal of Mathematics
 (2022)

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## Abstract

The quantum (orq-) calculus is widely applied in various operators which include the q-difference (q-derivative) operator, and this operator plays an important role in geometric function theory (GFT) as well as in the theory of hypergeometric series. In our present investigation, we introduce and study q-differential operator associated with q-Mittag–Leffler function which is an extension of the Salagean q-differential operator. By using this newly defined operator, we define a new subclass of analytic function and studied certain subclass of analytic function in generalized conic domain Ωk,q,γ. For this class, we investigate structural formula, coefficient estimates, sufficient condition, Fekete–Szegö problem, and also some subordination results.


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## Body

## 1. Introduction and Definition

Let▽ denote the set of all analytic function h^z in the open unit disk O¨=z∈ℂ:z<1 with normalized condition h^0=0 and h^′0=1, and series expansion of h^z∈▽ is given as(1)h^z=z+∑n=2∞anzn.The star-like and convex functionsh^∈▽ of order α can be defined as(2)S∗α=h^∈▽:Rezh^′zh^z>α,0≤α<1,z∈O¨,(3)Cα=h^∈▽:Rezh^′z′h^′z>α,0≤α<1,z∈O¨.Note that, if we takeα=0, then the above defined classes of star-like and convex functions of order α convert into the well-known classes of star-like S∗ and convex C functions.In 1991, the uniformly star-like functionsUS are introduced by Goodman [1] and defined as(4)US=h^∈▽: Rezh^′zh^z>zh^′zh^z−1,z∈O¨.After that, uniformly star-like functionsUS are studied by Ronning in [2] and Ma and Minda in [3].Furthermore, in [4], Kanas and Wisniowska introduced the class k-uniformly star-like k−UST and k-uniformly convex functions k−UCV, k≥0, as(5)k−UST=h^∈▽: Rezh^′zh^z>kzh^′zh^z−1,z∈O¨,(6)k−UCV=h^∈▽: Rezh^′z′h^′z>kzh^′z′h^′z−1,z∈O¨.Also, they defined these classes subject to the conic domainΩk (k≥0) (see [5]) as follows:(7)Ωk=u+iv:u2>k2u−12+v2,=w:Rew>kw−1.This domain represents the right-half plane fork=0, hyperbola for 0<k<1, parabola for k=1, and ellipse k>1. Furthermore, in [6], Shams et al. gave the generalization of k-uniformly star-like and k-uniformly convex functions and introduced a new subclasses DDk,γ and SDk,γ of analytic functions subject to the conic domain Ωk,γk≥0, 0≤γ<1, as follows:(8)Ωk,γ=u+iv:u−γ2>k2u−12+v2.The functionpk,γz is extremal function for domain Ωk,γ.(9)pk,γz=Ψ1zfork=0Ψ2zfork=1,Ψ3zfor0<k<1,Ψ4zfork>1,,where(10)Ψ1z=1+z1−z,Ψ2z=1+2γπ2log1+z1−z2,Ψ3z=1+2γ1−k2sinh22πarccoskarctanhz,Ψ4z=1+γk2−1sinπ2Ki∫0uz/t11−x21−ix2dx+γ1−k2,and i∈0,1; k=coshπK′i/4Ki, Ki is the first kind of Legendre’s complete elliptic integral. For details, see [5]. Indeed, from (9), we have(11)pk,γz=1+Q1z+Q2z2+…,where(12)Q1=2γ2/πarccosk21−k2for 0≤k<1,8γπ2for k=1,π2γ41+ttK2tk2−1for k>1,,(13)Q2=2/πarccosk2+23Q1for 0≤k<1,23Q1for k=1,4K2tt2+6t+1−π224K2t1+ttQ1for k>1..The convolution of two functionsh^ and g can be defined as(14)h^∗gz=z+∑n=2∞anbnzn,where h^z is given by (1) and gz=z+∑n=2∞bnzn,z∈O¨.Ifh^ and g∈▽, we call that h^ is subordinate to g, written as h^z≺gz, such that h^z=guz, where uz∈▽ is the Schwarz function in O¨ with conditions u0=0 and uz<1 (see [7]).LetP represent Carathéodory class of p∈▽, in the open unit disk O¨, and given by(15)pz=1+∑n=1∞cnzn,such that(16)Repz>0.In geometric function theory (GFT), the quantum (q-) calculus used as an important tools to study different families of analytic function and due to its application in mathematics and some related areas it has inspired the researcher very much. The quantum (or q-) calculus is widely applied in various operators which include the q-difference (q-derivative) operator, and this operator plays an important role in GFT, quantum theory, number theory and statistical mechanics, etc. Jackson [8, 9] was among the few researchers who defined the q-derivative and q-integral operator as well as provided some of their applications. Also, Ismail et al. [10] introduced research work in connection with function theory and q-theory. Later on, by using q-beta function, Gupta [11] introduced q-Baskakov–Durrmeyer operator, while q-Picard and q-Gauss–Weierstrass singular integral operators were introduced and studied by Aral, in [12]. Recently, Srivastava is one of the few researchers who studied univalent function theory by using a q-calculus (see, for details, [13, 14]). Kanas and Raducanu [15] introduced Ruscheweyh q-differential operator, and Arif et al. [16] discussed some of its applications for multivalent functions. For more studies on q-analogous of operator, we refer [15, 17–19].(17)tq=1+q+q2+…+qn−1t=n∈ℕ,(18)nq!=∏k=1nkq,n∈ℕ,where 0q!=1 and tq=1−qt/1−q, t∈ℂ.(19)tn,q=∏k=tn−1kq,n∈ℕ,(20)tq=Γqt+1ΓqtandΓq1=1.Definition 1 (see [6]).
Theq-number tq and q-factorial c for q∈0,1 is defined asDefinition 2.
Theq-generalized Pochhammer symbol and q-Gamma function be defined asDefinition 3 (see [9]).
Forh^∈▽, the q-derivative or (q -difference) operator be defined as(21)∂qh^z=−h^z+h^qzq−1z,z∈O¨.
Combining (1) and (21), we have(22)∂qh^z=1+∑n=2∞nqanzn−1.
Note that(23)∂qzn=nqzn−1,∂q∑n=1∞anzn=∑n=1∞nqanzn−1.
We can observe that(24)limq⟶1−∂qh^z=h^′z.Definition 4 (see [10]).
Let an analytic functionh^∈Sq∗ if(25)z∂qh^zh^z−11−q≤11−q,(26)h^0=h^′0=1.
Equivalently, we can rewrite (26) and (25) as follows (see [20]):(27)z∂qh^zh^z≺1+z1−qz.In [21], Zhang et al. defined the class k−Pq,γ and generalized conic domain Ωk,q,γ.Definition 5 (see [21]).
Letk∈0,∞, q∈0,1, and γ∈ℂ/0. A function pz∈k−Pq,γ if and only if(28)pz≺pk,γ,qz=2pk,γz1+q+1−qpk,γz,where pk,γz is given by (9).Geometrically, the functionpz∈k−Pq,γ takes all values from the domain Ωk,q,γ which is defined as follows:(29)Ωk,q,γ=γΩk,q+1−γ,where(30)Ωk,q=w:Re1+qwq−1w+2>k1+qwq−1w+2−1.Remark 1.
Whenq⟶1−, then Ωk,q,γ=Ωk,γ, where Ωk,γ is considered by Shams et al. [6].Remark 2.
Whenγ=1 and q⟶1−, then Ωk,q,γ=Ωk, where Ωk is the conic domain considered by Kanas and Wisniowska [4].Remark 3.
Forγ=1 and q⟶1−, then k−Pq,γ=Ppk, where Ppk is the well-known class introduced by Kanas and Wisniowska [4].Remark 4.
Forγ=1, k=0, and q⟶1−, then k−Pq,γ=P, where P is the well-known Carathéodory class of analytic functions p.Mittag–Leffler introduced Mittag–Leffler functionℋαz in [22, 23] as(31)ℋαz=∑n=0∞1Γαn+1zn,α∈ℂ,Reα>0,and its generalization ℋα,βz is introduced by Wiman [24] as(32)ℋα,βz=∑n=0∞1Γαn+βzn,α,β∈ℂ,Reα,Reβ>0.For more study about Mittag–Leffler functions (see [25–30]).Theq-Mittag–Leffler function defined by (see [31])(33)ℋα,βz,q=∑n=0∞1Γqαn+βzn,α,β∈ℂ,Reα,Reβ>0.Note thatq-Mittag–Leffler function is the specialized case of the q-Fox–Wright function rΦsz,q (see, for details, [32, 33]). Since the q-Mittag–Leffler function ℋα,βz,q defined by (33) does not belong to the normalized analytic function class ▽.Now, we define the normalization of thisq-Mittag–Leffler function ℛα,βz as(34)ℛα,βz,q=zΓqβℋα,βz,ℛα,βz,q=z+∑n=2∞φnq,βzn,(35)φnq,β=ΓqβΓqαn−1+β,where z∈O¨, Reα>0, and β∈ℂ/0,−1,−2,…. Corresponding to ℛα,βz,q and for h^∈▽, we define the following q-differential operator Dqmα,β:▽⟶▽ by(36)Dq0α,βh^z=h^z∗ℛα,βz,q,Dq1α,βh^z=zDqh^z∗ℛα,βz,q,Dq2α,βh^z=DqDq1α,βh^z,Dqmα,βh^z=DqDqm−1α,βh^z.We note that(37)Dqmα,βh^z=z+∑n=2∞nqmφnq,βanzn.Note that,(i)
Forα=0 and β=1, we have Salagean q-differential operator [34](ii)
Forq⟶1−, α=0, and β=1, we have Salagean differential operator [35](iii)
Form=0, we have Eα,βz,q (see [31])(iv)
Form=0, we have Eα,βz (see [28])We introduce a new subclass of analytic functions associated with newly definedq-differential operator (37).Definition 6.
Leth^z∈▽; then, h^z is in the class USα,βq,mγ,k, γ∈ℂ/0, if it satisfies the condition(38)Re1+1γz∂qDqmα,βh^zDqmα,βh^z−1>k1γz∂qDqmα,βh^zDqmα,βh^z−1,z∈O¨,or equivalently(39)z∂qDqmα,βh^zDqmα,βh^z∈k−Pq,γ.Remark 5.
Forα=0 and β=1, the class USα,βq,mγ,k=k−USq,γ,m studied by Hussain et al. [17].Remark 6.
Form=0, α=0, β=1, q⟶1−, and γ=1/1−η, η∈ℂ/1, the class USα,βq,mγ,k=SDk,η is studied by Shams et al. [6].Remark 7.
Form=0, α=0, β=1,q⟶1−, and γ=2/1−η, η∈ℂ/1, the class USα,βq,mγ,k=DDk,η is studied by Owa et al. [36].Remark 8.
Fork=0, m=0, α=0, β=1, q⟶1−, and γ=1/1−η, ℂ\1, the class USα,βq,mγ,k=S∗η is the well-known subclass of star-like function.Geometrically, an analytic functionh^∈USα,βq,mγ,k if and only if(40)z∂qDqmα,βh^zDqmα,βh^z,take all values in the generalized conic domain Ωk,q,γ given by (29). Taking geometrical interpretation into consideration, one can rephrase the above definition as follows.Definition 7.
Let an analytic functionh^∈USα,βq,mγ,k if and only if(41)z∂qDqmα,βh^zDqmα,βh^z≺pk,γ,qz,where pk,γ,qz is given by (28).


## 2. Set of Lemmas

Lemma 1 (see [37]).
IfFz is convex univalent in O¨ and let pz=∑n=1∞pnzn≺Fz=∑n=1∞dnzn in O¨, then(42)pn≤d1,n≥1.Lemma 2 (see [21]).
Letk∈0,∞ be fixed and(43)pk,γ,qz=2pk,γz1+q+1−qpk,γz.Then,(44)pk,γ,qz=1+21+qQ1z+21+qQ2−21−q1+qQ12z2+⋯,where Q1 and Q2 are given by (12) and (13).Lemma 3 (see [21]).
Letpz=1+∑n=1∞pnzn∈k−Pq,γ; then,(45)pn≤21+qQ1,n≥1.Lemma 4 (see [38]).
Let an analytic functionpz=1+∑n=1∞cnzn∈P and Repz>0 for all z in O¨; then,(46)c2−μc12≤2max1,2μ−1,∀μ∈ℂ.


## 3. Main Results

Theorem 1.
Let an analytic functionh^z∈USα,βq,mγ,k; then,(47)Dqmα,βh^z≺zexp∫0zpk,γ,quξ−1ζdξ,where uz is analytic in O¨ having condition u0=0 and uz<1. Furthermore, for z=ρ, we have(48)exp∫01pk,γ,q−ρ−1ρdρ≤Dqmα,βh^zz≤exp∫01pk,γ,qρ−1ρdρ,where pk,γ,qz is defined by (28).Proof.
Ifh^z∈USα,βq,mγ,k, then, from identity (41), we have(49)z∂qDqmα,βh^zDqmα,βh^z=pk,γ,quz,(50)∂qDqmα,βh^zDqmα,βh^z−1z=pk,γ,quz−1z.
Integrating (50) and after some simplification, we have(51)Dqmα,βh^z≺zexp∫0zpk,γ,quξ−1ζdξ.
Hence, (47) is proved. Noting that the univalent function pk,γz maps the disk z<ρ0<ρ≤1 onto a region which is convex and symmetric with respect to the real axis, we see(52)pk,γ,q−ρz≤Repk,γ,quρz≤pk,γ,qρz0<ρ≤1,z∈O¨.
Using (51) and (52) gives(53)∫01pk,γ,q−ρz−1ρdρ≤Re∫01pk,γ,quρz−1ρdρ≤∫01pk,γ,qρz−1ρdρ,for z∈O¨. Thus, inequality (51) leads us to(54)∫01pk,γ,q−ρz−1ρdρ≤logDqmα,βh^zz≤∫01pk,γ,qρz−1ρdρ,(55)pk,γ,q−ρ≤pk,γ−ρz,pk,γ,qρz≤pk,γ,qρ,which implies that(56)exp∫01pk,γ,q−ρ−1ρdρ≤Dqmα,βh^zz≤exp∫01pk,γ,qρ−1ρdρ.
Hence, the proof is complete.Forα=0 and β=1, then we have the following known corollary proved by Hussain et al. in [17].Corollary 1.
Let an analytic functionh^z∈k−USq,γ,m; then,(57)Dqmh^z≺zexp∫0zpk,γuξ−1ζdξ,where uz is analytic in O¨ having condition u0=0 and uz<1. Furthermore, for z=ρ, we have(58)exp∫01pk,γ−ρ−1ρdρ≤Dqmh^zz≤exp∫01pk,γρ−1ρdρ.where pk,γz is defined by (9).Theorem 2.
Ifh^z∈USα,βq,mγ,k, then(59)a2≤δ2qm2q−1φ2q,β,(60)an≤δnqmnq−1φnq,β∏j=1n−21+δj+1q−1,forn≥3,where δ=2Q1/1+q with Q1 and φnq,β is given by (12) and (35).Proof.
Let(61)z∂qDqmα,βh^zDqmα,βh^z=pz,where pz=1+∑n=1∞cnzn and Dqmα,βh^z is given by (37); then, (61) becomes(62)z+∑n=2∞nqm+1φnq,βanzn=∑n=0∞cnznz+∑n=2∞nqmφnq,βanzn.
Now, comparing the coefficients ofzn, we obtain(63)nqm+1φnq,βan=nqmφnq,βan+∑j=1n−1Jqmφjq,βajcn−j,which implies(64)an=1nqmnq−1φnq,β∑j=1n−1jqmφjq,βajcn−j.
Using Lemma4, we have(65)an≤2Q11+qnqmnq−1φnq,β∑j=1n−1jqmφjq,βaj.
Let us takeδ=2Q1/1+q. Then, we have(66)an≤δnqmnq−1φnq,β∑j=1n−1jqmφjq,βaj.
Forn=2, in (66), we have(67)a2≤δ2qm2q−1φ2q,β,which shows that (60) holds for n=2. To prove (60), we use the principle of mathematical induction; for this, consider the case n=3:(68)a3≤δ3qm3q−1φ3q,β1+2qmφ2q,βa2.
Using (67), we have(69)a3≤δ3qm3q−1φ3q,β1+δ2q−1,which shows that (60) holds for n=3. Let us assume that (60) is true for n≤t, that is,(70)at≤δtqmtq−1φtq,β∏j=1t−21+δj+1q−1,forn=3,4,….
Consider(71)at+1≤δt+1qmt+1q−1φt+1q,β≤δt+1qmt+1q−1φt+1q,β,×1+2qmφ2q,βa2+3qmφ3q,βa3+4qmφ4q,βa4+⋯+tqmφtq,βat≤δt+1qmt+1q−1φt+1q,β×1+δ2q−1+δ3q−11+δ2q−1+⋯+δtq−1∏j=1t−21+δj+1q−1=δt+1qmt+1q−1φt+1q,β∏j=1t−11+δj+1q−1,which shows that (60) holds for n=t+1. Hence, theorem is completed.Forα=0 and β=1, then we have the following known corollary proved by Hussain et al., in [17].Corollary 2.
Ifh^z∈USα,βq,mγ,k, then(72)a2≤δ2qm2q−1φ2,(73)an≤δnqmnq−1∏j=1n−21+δj+1q−1,forn≥3,where δ=2Q1/1+q and Q1 is given by (12).Theorem 3.
Leth^z∈USα,βq,mγ,kof form (1) and0≤k<∞be fixed andμ∈ℂ; then,(74)a3−μa22≤Q13qmφ3q,β1+q3q−1max1,2v−1,where(75)v=121−Q2Q1+1−qQ1−2Q11+q2q−1+μQ13qm3q−1φ2q,β1+q2qm2q−12,where Q1 and Q2 are given by (12) and (13).Proof.
Leth^z∈USα,βq,mγ,k; then, from (12), we have(76)z∂qDqmα,βh^zDqmα,βh^z=pk,γuzz∈O¨.
Let(77)pz=11−uz1+uz−1=1+c1z+c2z2+…,and this gives(78)uz=c12z+12c2−c122z2+…,(79)pk,γ,quz=1+Q1c11+qz+11+qQ2c122+c2−c122Q1−1−qQ12c122z2+⋯,(80)z∂qDqmα,βh^zDqmα,βh^z=1+2qmφ2q,β2q−1a2z+3qmφ3q,β3q−1a3−2qmφ2q,β22q−1a22z2.
Using (79) in (76) and comparing with (80), we obtain(81)a2=Q1c12qmφ2q,β1+q2q−1,a3=13qmφ3q,β3q−1×Q1c2/1+q+11+qQ22−Q12−1−qQ122+Q121+q2q−1c12,a3−μa22=Q13qmφ3q,β3q−11+q×c2−121−Q2Q1+1−qQ1−2Q11+q2q−1+μQ13qm3q−1φ3q,β1+q2qm2q−12c12.
For any complex numberμ and after some calculation, we have(82)a3−μa22=Q13qmφ3q,β1+q3q−1c2−vc12,where v is given by (75). Using Lemma 4 on (82), we have the required results.Forα=0 and β=1, then we have the following known corollary proved by Hussain et al. in [17].Corollary 3.
Leth^z∈USα,βq,mγ,k of form (1) and 0≤k<∞ be fixed and μ∈ℂ; then,(83)a3−μa22≤Q123qm3q−1max1,2v−1,where(84)v=121−Q2Q1+1−qQ1−Q12q−1+μQ13qm3q−122qm2q−12,where Q1 and Q2 are given by (12) and (13), where pk,γz is defined by (9)).Theorem 4.
If an analytic functionh^z of form (1) satisfies the condition,(85)∑n=2∞nq−1k+1+γφnq,βnqman≤γ,then h^z∈USα,βq,mγ,k.Proof.
Let we note that(86)z∂qDqmα,βh^zDqmα,βh^z−1=z∂qDqmα,βh^z−Dqmα,βh^zDqmα,βh^z=∑n=2∞nqmφnq,βnq−1anznz+∑n=2∞nqmφnq,βanzn≤∑n=2∞nqmφnq,βnq−1an1−∑n=2∞nqmφnq,βan.
From (85), it follows that(87)1−∑n=2∞nqmφnq,βan>0.
To show thath^z∈USα,βq,mγ,k, it is suffices that(88)kγz∂qDqmα,βh^zDqmα,βh^z−1−Re1γz∂qDqmα,βh^zDqmα,βh^z−1≤1.
From (86), we have(89)≤kγz∂qDqmα,βh^zDqmα,βh^z−1−Re1γz∂qDqmα,βh^zDqmα,βh^z−1≤kγz∂qDqmα,βh^zDqmα,βh^z−1+1γz∂qDqmα,βh^zDqmα,βh^z−1≤k+1γz∂qDqmα,βh^zDqmα,βh^z−1=k+1γz∂qDqmα,βh^z−Dqmα,βh^zDqmα,βh^z≤k+1γ∑n=2∞nqmφnq,βnq−1an1−∑n=2∞nqmφnq,βan1,from (85).Whenq⟶1−, m=0, α=0, β=1, and γ=1−η, with 0≤η<1, then we have the following.Corollary 4 (see [6]).
An analytic functionh^ of form (1) belongs to the class k−US1−2η if it holds the following condition:(90)∑n=2∞nk+1−k+ηan≤1−η,where 0≤η<1 and k≥0.Whenq⟶1−, m=0, α=0, β=1, and γ=1−η, with 0≤η<1 and k=0, then we have the following.Corollary 5 (see [39]).
An analytic functionh^∈US1−η of form (1) it holds(91)∑n=2∞n−ηan≤1−δ,0≤η<1.Theorem 5.
Leth^z∈USα,βq,mγ,k; then, h^O¨ contains an open disk of radius:(92)2mφ2q,β2q−122qm2q−1φ2q,β+δ,where δ=2Q1/1+q with Q1 is given by (12).Proof.
Letu0≠0 be a complex number such that h^z≠u0 for z∈O¨. Then,(93)h^1z=u0h^zu0−h^z=z+a2+1u0z2+….
Sinceh^1z is univalent, so(94)a2+1u0≤2.
Now, using (59), we have(95)1u0≤22qm2q−1φ2q,β+δ2qm2q−1φ2q,β.
Hence, we have(96)u0≥2qm2q−1φ2q,β22qm2q−1φ2q,β+δ.Forα=0 and β=1, then we have the following known corollary proved by Hussain et al. in [17].Corollary 6.
Leth^z∈USα,βq,mγ,k; then, h^O¨ contains an open disk of radius:(97)2m2q−122qm2q−1+δ,where pk,γz is defined by (9) and δ=Q1 with Q1 is given by (12).


## 4. Conclusion

Here, we have introduced and defined new subclass of analytic functions in generalized conic domain by using newly definedq-differential operator. This newly defined operator for analytic functions is the extension of q-Salagean operator. We investigated structural formula, coefficient estimates, and Fekete–Szegö problem, and also, we have successfully derived subordination result.



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