# New Subclass of Analytic Function Involvingq-Mittag-Leffler Function in Conic Domains

Authors: Saima Noor; Asima Razzaque
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we formulate theq-analogus of differential operator associated with q-Mittag-Leffler function. By using this newly defined operator, we define a new subclass k−USq,γmα,β, of analytic functions in conic domains. We investigate the number of useful properties such that structural formula, coefficient estimates, Fekete–Szego problem and subordination result. We also highlighted some known corollaries of our main results.


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## Body

## 1. Introduction Definition

LetA denote the class of functions lz which are analytic in the open unit disk E=z∈ℂ:z<1, satisfying the condition l0=0 and l′0=1, and for every l∈A has the series expansion of the form
(1)lz=z+∑n=2∞anzn.LetS⊂A be the class of all functions which are univalent in E (see [1]). Also, P denotes the well-known Carathéodory class of functions p which are analytic in open unit disk E and has the series expansion of the form
(2)pz=1+∑n=1∞cnzn,and satisfying the condition
(3)p0=1andRepz>0.For the functionl given by (1) and the function g defined by gz=z+∑n=2∞bnzn, the Hadamard product (convolution) l∗g of the functions l and g stated by
(4)l∗gz=z+∑n=2∞anbnzn.For the analytic functionsl,g, l is said to be subordinate to g (indicated as l≺g), if there exists a Schwarz function
(5)wz=∑n=1∞cnzn,with
(6)w0=0andwz<1,such that
(7)lz=gwz.Furthermore, ifg is univalent in E, (see [2]); then, we have
(8)lz≺gzifandonlyifl0=g0andlE⊂gE,z∈E.The class of starlike functions of orderαS∗α in E and the class of convex functions of order αKα, 0≤α<1, were defined as follows:
(9)S∗α=l:l∈SandRezl′zlz>α,0≤α<1,z∈E,Kα=l:l∈SandRezzl′z′l′z>α,0≤α<1,z∈E.It should be noted that(10)S∗0=S∗andK0=K,where S∗ and K are the well-known function classes of starlike and convex functions, respectively.In the year of 1991, Goodman [3] introduced the class UCV of uniformly convex functions which was extensively studied by Ronning [4], and its characterization was given by Ma and Minda [5]. After that, Kanas and Wisniowska [6] defined the class k-uniformly convex functions (k-UCV) and a related class k−ST was defined by
(11)l∈k−UCV⟺zl′∈k−ST⟺l∈AandRezl′z′l′z>zl′′zl′z,k≥0.From different viewpoints, the various subclasses of the normalized analytic function of classA have been studied in the field of Geometric Function Theory. To investigate various subclasses of A, many authors have been used the q-calculus as well as the fractional q-calculus. In 1910, Jackson [7] was among the one of few researchers who studied q-calculus operator theory on q-definite integrals and also Trjitzinsky in [8] studied about analytic theory of linear q-difference equations. Curmicheal [9] studied general theory of linear q-difference equations and the first use of q-calculus operator theory in Geometric Function Theory in a book chapter by Srivastava (see, for details, [10]). Recently, Hussain et al. discussed the some applications of q-calculus operator theory in [11], while in [12, 13], Ibrahim et al. used the notion of quantum calculus and the Hadamard product to improve an extended Sàlàgean q-differential operator and defined some new subclasses of analytic functions in open unit disk E. Govindaraj and Sivasubramanian [14] as well as Ibrahim et al. [15, 16] employed the quantum calculus and the Hadamard product to defined some new subclasses of analytic functions involving the Sàlàgean q-differential operator and the generalized symmetric Sàlàgean q-differential operator, respectively. Furthermore, Srivastava et al. [17] defined q-Noor integral operator by using q-calculus operator theory and investigated some subclasses of biunivalent functions in open unit disk.Here, we give some basic definitions and details of theq-calculus and suppose that 0<q<1.For any nonnegative integern, the q-integer number nq is defined by
(12)nq=1−qn1−q,0q=0,and for any nonnegative integer n, the q-number shift factorial is defined by
(13)nq!=1q2q3q⋯nq,0q!=1.We note that whenq⟶1−, then n!=n.Theq-difference operator was introduced by Jackson (see in [7]). For l∈A, the q-derivative operator or q-difference operator is defined as
(14)∂qlz=lqz−lzzq−1,z∈E,z≠0,q≠1.It is readily deduced from (1) and (14) that
(15)∂qzn=nqzn−1,∂qlz=1+∑n=2∞nqanzn−1.We can observe that(16)limq⟶1−∂qlz=l′z.The familiar Mittag-Leffler functionHαz introduced by Mittag-Leffler [18] and its generalization Hα,βz introduced by Wiman [19] which are defined by
(17)Hαz=∑n=0∞1Γαn+1zn,α∈ℂ,Reα>0,Hα,βz=∑n=0∞1Γαn+βzn,α,β∈ℂ,Reα,Reβ>0.Recently, Attiya [20] investigated some applications of Mittag-Leffler functions and generalized k-Mittag-Leffler studied by Rehman et al. in [21]. Moreover, Srivastava et al. [22, 23] introduced the generalization of Mittag-Leffler functions.Theq-Mittag-Leffler function was defined by (see [24]):
(18)Hα,βz,q=∑n=0∞1Γqαn+βzn,α,β∈ℂ,Reα,Reβ>0.Theq-Mittag-Leffler function has also been investigated in [25, 26]. Since the q-Mittag-Leffler function Hα,βz,q defined by (18) does not belong to the normalized analytic function class A. Hence, we define the normalization of q-Mittag-Leffler function as
(19)Mα,βz,q=zΓqβHα,βz=z+∑n=2∞ΓqβΓqαn−1+βzn,where z∈E,Reα>0, β∈ℂ\0,−1,−2,−3⋯. Corresponding to Mα,βz,q and for l∈A, we define the following q-analogous of differential operator Dqmα,β:A⟶A by
(20)Dq0α,βlz=lz∗Mα,βz,q,Dq1α,βlz=z∂qlz∗Mα,βz,q,Dq2α,βlz=DD1α,βlz,Dqmα,βlz=DDm−1α,βlz.We note that(21)Dqmα,βlz=z+∑n=2∞nqmTnα,qanzn,where
(22)Tnα,q=ΓqβΓqαn−1+β.Note that(i)
Forα=0 and β=1, we get Salagean q-differential operator [14](ii)
Forq⟶1−,α=0, and β=1, we get Salagean differential operator [27](iii)
Form=0, we get Eα,βz,q (see [24])(iv)
Form=0, we get Eα,βz (see [22])Definition 1.
Letlz∈A, then lz is in the class k−USq,γmα,β,γ∈ℂ\0, if it satisfies the condition
(23)Re1+1γz∂qDqmα,βlzDqmα,βlz−1>k1γz∂qDqmα,βlzDqmα,βlz−1,z∈E.Remark 2.
(i)
Forα=0 and β=1, the class k−USq,γmα,β=k−USq,γ,m studied in [11](ii)
Form=0,α=0,β=1,q⟶1−, and γ=1/1−η,η∈ℂ\1, the class k−USq,γmα,β=SDk,η studied in [28](iii)
Form=0,α=0,β=1,q⟶1−, and γ=2/1−η,η∈ℂ\1, the class k−USq,γmα,β=KDk,η, studied in [29](iv)
Fork=1,m=0,α=0,β=1,q⟶1−, and γ=1−η,η∈ℂ\1, the class k−USq,γmα,β=Spη studied in [30](v)
Fork=1,m=0,α=0,β=1,q⟶1−, and γ=2/1−η,η∈ℂ\1,k−USq,γmα,β=Kpη, studied in [30]


## 2. Geometric Interpretation

A functionlz∈A, belongs to k−USq,γmα,β if and only if z∂qDqmα,βlz/Dqmα,βlz takes all the values in the conic domain Ωk,γ=pk,γE, such that
(24)Ωk,γ=γΩk+1−γ,0≤γ<1,k≥0,where
(25)Ωk=u+iv:u2>k2u−12+v2.The domainΩk,γ is not always well defined because in general 1,0∉Ωk,γ (for example, in particular 1,0∉Ω2,0.5). We see that in [31], the conic domain Ωk0,b concides with Ωk,b only when b is chosen according to
(i)
Fork=0, we take b=0(ii)
Fork∈0,1/2, we take b∈1/2k2−1,1(iii)
Fork∈1/21/2,1, we take b∈−∞,1(iv)
Fork∈1,∞, we take b∈−∞,1/2k2−1This means that forΩk,γ to contain the point 1,0,γ must be chosen according as follows:
(26)γ∈0,1if0≤k≤1,0,1−k2−1kifk≥0.Sincepk,γz is convex univalent, the above definition can be written as
(27)z∂qDqmα,βlzDqmα,βlz≺pk,γz,where
(28)pk,γz=1+z1−z,fork=0,U1γ,k,fork=1,U2γ,k,for0<k<1,U3γ,k,fork>1,(29)U1γ,k=1+2γπ2log1+z1−z2′,(30)U2γ,k=1+2γ1−k2sinh22πarccoskarctanhz,(31)U3γ,k=1+γk2−1sinπ2Rt∫0uzt11−x21−tx2dx+γ1−k2.For more detail (see [32, 33]).


## 3. Set of Lemmas

Lemma 3. (see [34]).
Letpz=∑n=1∞pnzn≺Fz=∑n=1∞dnzn in E. If Fz is convex univalent in E, then
(32)pn≤d1,n≥1.Lemma 4. (see [35]).
Letk∈0,∞ be fixed and let pk,γz of the form (28). If
(33)pk,γz=1+Q1z+Q2z2+⋯,where
(34)Q1=2γA21−k2,0≤k<1,8γπ2,k=1,π2γ41+ttK2tk2−1,k>1,(35)Q2=A2+23Q1,0≤k<1,23Q1,k=1,4K2tt2+6t+1−π224K2t1+ttQ1,k>1.Lemma 5. (see [36]).
Letpz=1+∑n=1∞cnzn∈P and let pz be analytic in E and satisfy Repz>0 for z in E, then
(36)c2−μc12≤2max1,2μ−1,∀μ∈ℂ.


## 4. Main Results

Theorem 6.
Letlz∈k−USq,γmα,β. Then,
(37)Dqmα,βlz≺zexp∫0zpk,γwξ−1ζdξ,where wz is a Schwarz function given in (5). Moreover, for z=ρ, we have
(38)exp∫01pk,γ−ρ−1ρdρ≤Dqmα,βlzz≤exp∫01pk,γρ−1ρdρ,where pk,γz is given by (28).Proof.
Iflz∈k−USq,γmα,β, then by using (27), we obtain
(39)∂qDqmα,βlzDqmα,βlz−1z=pk,γwz−1z.
Integrating (39) and after some simplification, we have
(40)Dqmα,βlz≺zexp∫0zpk,γwξ−1ζdξ.
This proves (37). We know that
(41)pk,γ−ρz≤Repk,γwρz≤pk,γρz0<ρ≤1,z∈E.
Using (40) and (41), we have
(42)∫01pk,γ−ρz−1ρdρ≤Re∫01pk,γwρz−1ρdρ≤∫01pk,γρz−1ρdρ,for z∈E. From (40), we have
(43)Dqmα,βlzz≺exp∫0zpk,γwξ−1ζdξ,∫01pk,γ−ρz−1ρdρ≤logDqmα,βlzz≤∫01pk,γρz−1ρdρ,exp∫01pk,γ−ρz−1ρdρ≤explogDqmα,βlzz≤exp∫01pk,γρz−1ρdρ,
which implies that(44)exp∫01pk,γ−ρ−1ρdρ≤Dqmα,βlzz≤exp∫01pk,γρ−1ρdρ.Corollary 7. (see [11]).
Letlz∈k−USq,γm0,1. Then,
(45)Dqmlz≺zexp∫0zpk,γwξ−1ζdξ,where wz is a Schwarz function given in (5). Moreover, for z=ρ, we have
(46)exp∫01pk,γ−ρ−1ρdρ≤Dqmlzz≤exp∫01pk,γρ−1ρdρ.Theorem 8.
Iflz∈k−USq,γmα,β. Then,
(47)a2≤δ2qm2q−1T2α,q,(48)an≤δnqmnq−1Tnα,q∏j=1n−21+δj+1q−1,forn≥3,where δ=Q1 with Q1 and Tnα,q are given by (34) and (22).Proof.
Let(49)z∂qDqmα,βlzDqmα,βlz=pz,where pz is the analytic in E and p0=1. Let pz=1+∑n=1∞cnzn and Dqmα,βlz is given by (21). Then, (49) implies that
(50)z+∑n=2∞nqm+1Tnα,qanzn=∑n=0∞cnznz+∑n=2∞nqmTnα,qanzn,=∑n=0∞cnzn+1+∑n=0∞cnzn∑n=2∞nqmTnα,qanzn.
Now comparing the coefficients ofzn, we obtain
(51)nqm+1Tnα,qan=nqmTnα,qan+∑j=1n−1jqmΓβΓqαj−1+βajcn−j,nqm+1Tnα,qan−nqmTnα,qan=∑j=1n−1jqmΓβΓqαj−1+βajcn−j,nqmnq−1Tnα,qan=∑j=1n−1jqmΓβΓqαj−1+βajcn−j,which implies
(52)an=1nqmnq−1Tnα,q∑j=1n−1jqmΓβΓqαj−1+βajcn−j.
Using the results thatcn≤Q1 given in ([33]), we have
(53)an≤Q1nqmnq−1Tnα,q∑j=1n−1jqmΓβΓqαj−1+βaj.
Let us takeδ=Q1. Then, we have
(54)an≤δnqmnq−1Tnα,q∑j=1n−1jqmΓβΓqαj−1+βaj.
Forn=2 in (54), we have
(55)a2≤δ2qm2q−1T2α,q∑j=11jqmΓβΓqαj−1+βaj=δ2qm2q−1T2α,q,
Hence, forn=2 the inequality (48) holds. To prove (48), we use mathematical induction, for n=3(56)a3≤δ3qm3q−1T3α,q1+2qmT2α,qa2.
By using (55), we have
(57)a3≤δ3qm3q−1T3α,q1+2qmT2α,qδ2qm2q−1T2α,q.
Therefore,(58)a3≤δ3qm3q−1T3α,q1+δ2q−1.
Hence, (48) holds for n=3. Now, we suppose that (48) is true for n=t+1, that is
(59)at≤δtqmtq−1Ttα,q∏j=1t−21+δj+1q−1,n≥3.
Consider(60)at+1≤δt+1qmt+1q−1Tt+1α,q×1+2qmT2α,qa2+3qmT3α,qa3+4qmT4α,qa4+⋯+tqmTtα,qat≤δt+1qmt+1q−1Tt+1α,q1+δ2q−1+δ3q−11+δ2q−1+⋯+δtq−1∏j=1t−21+δj+1q−1=δt+1qmt+1q−1Tt+1α,q∏j=1t−11+δj+1q−1.
Hence, (48) holds for n=t+1. Hence, proof is complete.Corollary 9. (see [11]).
flz∈k−USq,γm0,1. Then,
(61)a2≤δ2qm2q−1an≤δnqmnq−1∏j=1n−21+δj+1q−1,forn≥3.Theorem 10.
Let0≤k<∞ be fixed and let lz∈k−USq,γmα,β with the form (1), then, for μ∈ℂ(62)a3−μa22≤Q123qmT3α,q3q−1max1,2v−1,
whereQ1 and Q2 are given by (34) and (35).Proof.
Letlz∈k−USq,γmα,β, then there exists a Schwarz function wz given by (5), such that
(63)z∂qDqmα,βlzDqmα,βlz≺pk,γz,z∈E(64)z∂qDqmα,βlzDqmα,βlz=pk,γwz,z∈E.
Letpz∈P be defined as
(65)pz=1+c1z+c2z2+⋯.
This gives(66)wz=c12z+12c2−c122z2+⋯(67)pk,γwz=1+Q1c12z+Q2c124+12c2−c122Q1z2+⋯.(68)z∂qDqmα,βlzDqmα,βlz=1+2qmT2α,q2q−1a2z+3qmT3α,q3q−1a3−2qmT2α,q22q−1a22z2.
Using (67) in (64) and comparing with (68), we obtain
(69)a2=Q1c122qmT2α,q2q−1,a3=13qmT3α,q3q−1Q1c22+c124Q2−Q1+Q122q−1.a3−μa22=13qmT3α,q3q−1Q1c22+c124Q2−Q1+Q122q−1−μQ1c122qmT2α,q2q−12.
For any complex numberμ and after some calculation we have
(70)a3−μa22=Q123qmT3α,q3q−1c2−vc12,where
(71)v=121−Q2Q1−Q112q−1−μ3qm3q−12T2α,q2qm2q−12.
Using a lemma (36) on (70), we have the required result.Corollary 11. (see [11]).
Let0≤k<∞ be fixed and let lz∈k−USq,γm0,1 with the form (1.1), then, for μ∈ℂ(72)a3−μa22≤Q123qm3q−1max1,2v−1,where
(73)v=121−Q2Q1−Q112q−1−μ3qm3q−122qm2q−12.Theorem 12.
Letlz∈A of the form (1) and satisfy the condition
(74)∑n=2∞nq−1k+1+γTnα,qnqman≤γ,
then,lz∈k−USq,γmα,β.Proof.
Let we note that(75)z∂qDqmα,βlzDqmα,βlz−1=z∂qDqmα,βlz−Dqmα,βlzDqmα,βlz=∑n=2∞nqmTnα,qnq−1anznz+∑n=2∞nqmTnα,qanzn≤∑n=2∞nqmTnα,qnq−1an1−∑n=2∞nqmTnα,qan.
From (74), we get
(76)1−∑n=2∞nqmTnα,qan>0.
To show thatlz∈k−USq,γmα,β, it suffices that
(77)kγz∂qDqmα,βlzDqmα,βlz−1−Re1γz∂qDqmα,βlzDqmα,βlz−1≤1.
From (Proof), we have
(78)kγz∂qDqmα,βlzDqmα,βlz−1−Re1γz∂qDqmα,βlzDqmα,βlz−1≤kγz∂qDqmα,βlzDqmα,βlz−1+1γz∂qDqmα,βlzDqmα,βlz−1≤k+1γz∂qDqmα,βlzDqmα,βlz−1=k+1γz∂qDqmα,βlz−Dqmα,βlzDqmα,βlz≤k+1γ∑n=2∞nqmTnα,qnq−1an1−∑n=2∞nqmTnα,qan≤1.
Because of (74).Corollary 13. (see [11]).
If a functionlz∈A of the form (1) and satisfy the condition
(79)∑n=2∞nqmnq−1k+1+γan≤γ,then, lz∈k−USq,γm0,1.Corollary 14. (see [28]).
A functionl∈A of the form (1) belongs to k−US1−2η, if
(80)∑n=2∞nk+1−k+ηan≤1−η,where 0≤η<1 and k≥0. Then, lz∈k−USq⟶1−,1−η00,1.
Whenq⟶1−, then, m=0,α=0,β=1,γ=1−η, with 0≤η<1 and k=0.Corollary 15. (see [37]).
A functionl∈A of the form (1) is in the class 0−US1−η, if(81)∑n=2∞n−ηan≤1−δ,0≤η<1.Theorem 16.
Letlz∈k−USq,γmα,β. Then, lE includes an open disk of radius
(82)2mT2α,q2q−122qm2q−1T2α,q+δ,where Q1 is given by (34).Proof.
Let a nonzero complex numberw0, such that lz≠w0 for z∈E. Then,
(83)l1z=w0lzw0−lz=z+a2+1w0z2+⋯.
Sincel1z is univalent, therefore
(84)a2+1w0≤2.
Now using (47), we have
(85)1w0≤2+δ2qm2q−1T2α,q=22qm2q−1T2α,q+δ2qm2q−1T2α,q.
Hence we have(86)w0≥2qm2q−1T2α,q22qm2q−1T2α,q+δ.
Whenα=0 and β=1, then we have known result [11].Corollary 17.
Letlz∈k−USq,γm0,1. Then, lE includes an open disk of radius
(87)2m2q−122qm2q−1+δ.


## 5. Conclusion

In this paper, we formulate theq-analogous of differential operator associated with q-Mittag-Leffler function. By applying newly defined operator, we defined and investigated a new subclass k−USq,γmα,β, of analytic functions in conic domains. We investigated the number of useful properties such that structural formula, coefficient estimates, Fekete–Szego problem, and subordination results. We also highlighted some known consequences of our main result. For future work, one can employ the q-analogous of differential operator (21) in different classes of analytic functions such as the meromorphic and multivalent functions (see [38–42].



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