# Subordination Method for the Estimation of Certain Subclass of Analytic Functions Defined by theq-Derivative Operator

Authors: Hormoz Rahmatan; Ali Shokri; Hijaz Ahmad; Thongchai Botmart
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we investigate an interesting class of analytic and biunivalent functions in the open unit diskΔ which is defined using the q-derivative operator. We apply the subordination method to the functional of coefficients problem. Furthermore, we obtain the bounds of the certain functional of coefficients for functions in the class HΣqλ,β, for real and complex parameters. Also, an estimation for the initial Taylor and Maclaurin coefficients of functions in HΣqλ,α was given.


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## Body

## 1. Introduction

The class of all analytic functions was denoted byA. Functions belonging to this class can be displayed in the form of the following power series:
(1)fz=z+∑n=1∞anzn.The class of univalent functions inA normalized with the conditions f0=f′0−1=0 was represented by S.Since each function which belongs to the classS has an inverse, we can easily calculate
(2)f−1w′=1f′z.Therefore, we conclude thatf−1 is analytic [1]. So, f−1 can also be displayed in the form of a power series as follows:
(3)f−1w=w−a2w2+2a22−a3w3−⋯,where
(4)f−1fz=zz∈Δ,ff−1w=ww∈Δ,w<r0f;r0f>14.A function is called biunivalent in open unit diskΔ, if f and f−1 are univalent in open unit disk Δ.Σ is considered a symbol of the class of biunivalent functions Δ. For more information on the class Σ, readers can refer to [2].In recent years, the subject of quantum calculus has been considered by many researchers. There are many applications of quantum calculus in various branches of mathematics and physics.q-derivatives and q-integrals by Jackson [3, 4] were first systematically way introduced and studied; also, he was the first to introduce applications of quantum calculus. For more information about quantum calculus, readers can refer to [5–8].The question that arises is whether it is possible to study concepts from a quantum calculus point of view in articles such as [9–14] that deal with equations numerically and topics related to numerical analysis.The theory post quantum calculus or (p,q)-calculus operators are used in various areas of science and also in the geometric function theory.Let0<q≤p≤1. The (p,q)-bracket np,q is defined by
(5)np,q=pn−qnp−q,p≠q,npn−1,p=q.Notice thatlimq⟶pnp,q=np,p.The (p,q)-derivative operator Dp,q of a function f∈A is given by
(6)Dp,qfz=1+∑n=2∞np,qanzn−1,p,q∈0,1.For a functionf∈A, it can be easily seen that
(7)Dp,qfz=fpz−fqzp−qz,z∈Δ,z≠0,p,q∈0,1,p≠q,Dp,qf0=1 and Dp,pfz=f′z.For definitions and properties of (p,q)-calculus, one may refer to [15].The (1,q)-derivative operator is known as the q-derivative operator and is denoted by Dq; for a function f∈A and z≠0 is defined by
(8)Dqfz=fz−fqzz1−q,z∈Δ,z≠0,q∈0,1.For more details, see [3, 4, 16]. Using equations (1) and (8), for n∈ℕ and z∈Δ, we obtained
(9)Dqfz=1+∑n=2∞nqanzn−1,q∈0,1,where
(10)nq=1−qn1−q=1+∑k=1n−1qk;0q=0.It is clear that1q=1.Also, thenth order q-derivative of f∈A is defined by
(11)Dqnfz=−1n1−q−nz−nq−nn−1/2×∑k=0∞nk−1kqkk−1/2fqn−kz.For this results, we refer to [17].One of the important subclass of analytic functions defined byq-derivative is Sq∗; the class Sq∗ of q-starlike functions consists of f∈A that satisfies
(12)f0=f′0−1=0,zDqfzfz−11−q≤11−q.Remark 1.
For functionsf∈A, the Alexander theorem [18] was used by Baricz and Swaminathan [19] for defining the class Cq of q-convex functions as follows:
(13)fz∈Cq⇐zDqfz∈Sq∗.Hence, the classCq of q-convex functions consisting of f∈A satisfies
(14)f0=f′0−1=0,zDq2fzDqfz−11−q≤11−q.Also, this study introduces a concept of the fractal derivative of a functiongt with respect to a fractal measure tα:
(15)∂gt∂tα=limt⟶t1gt−gt1tα−t1α,0<α;the fractal derivative (15) (see [20]) differs from the standard fractional derivative in that is does not involve the integral convolution and is local in nature.The elementary physical concepts such as velocity in a fractal spacetimexβ,tα can be defined by
(16)ν^=dx^dt^=dxβdtα,∀t^,x^∈Sα,β,0<α,0<β,where Sα,β represents time-space fabric having scaling indices α and β.Like the fractional derivative, the fractal derivative exists under a fractal metric spacetime [21].For instance,ν^=dt1/2/dt1/3t=0 exists while ν^=dt1/2/dtt=0 does not [20].For more information about fractal and fractional calculus, readers can refer to [22–26].We sayf is subordinate to g and is written as fz≺gz, if there exists a function wz belonging to class of Schwartz functions which are satisfying w0=0 and wz<1, such that
(17)fz=gwz,where f and g are analytic in Δ.Letg be univalent function, fz≺gz, if and only if f0=0 and fΔ⊂gΔ.One of the references that deals with differential subordination is the book of Miller and Mokanu; researchers can refer to [27].In the subordination method, we can use fixed point theory. In definition of subordination exists Schwarz functionwz, such that wz≺z. Researchers for more information about fixed point theory can refer to [28–30]. It seems quantum calculus has interesting applications in fixed point theory.The class of functions which are analytic inΔ and satisfying the following conditions Repz>0 and
(18)pz=1+p1z+p2z2+⋯,z∈Δ,denoted by P.In the sequel, for obtaining our main results, we used the following lemma [18].Lemma 2.
Ifp∈P, then pk≤2 for each k.Lemma 3.
Ifp∈P, then 2p2=p12+4−p12x for some x with x<1.Definition 4.
A functionfz of the form (1) is said to be in the class HΣqλ,β if
(19)f∈Σ,Re1−λfzz+λDqfz>β,(20)Re1−λgwz+λDqgw>β,where w=fz, g=f−1q∈0,1,0≤λ≤1,0≤β<1, and z,w∈Δ.Theorem 5.
If a functionfz of the form (1) is in the class HΣqλ,β, then
(21)a2≤min2δη+λ2q,2δη+λ3q,a3≤2δη+λ3q,where η=1−λ and δ=1−β.In this work, we investigate on the classesHΣqλ,β and HΣqλ,α of functions which are analytic and biunivalent in the open unit disk Δ. We obtain the bounds of the certain inequality for the class HΣqλ,β and provide estimates for the initial Taylor and Maclaurin coefficients of functions in HΣqλ,α.


## 2. Estimation of the Bounds of Certain Functional of Coefficients forf∈HΣqλ,β

In this section, we obtain the bounds of certain functional of coefficients forf∈HΣqλ,β.Theorem 6.
Iff∈HΣqλ,β and γ∈ℝ, then
(22)a3−γa22≤2δη+λ3q,hγ<1η+λ3q,δhγ,hγ>1η+λ3q,where
(23)hγ=1−γη+λ3q,η=1−λ,δ=1−β.Proof.
Our proof begins with the observation of the subordination method for conditions (19) and (20). It follows that
(24)ηfzz+λDqfz=β+δpz,z∈Δ,(25)ηgww+λDqgw=β+δqw,w∈Δ,g=f−1,respectively, where
(26)pz=1+p1z+p2z2+⋯,qw=1+q1w+q2w2+⋯,for z,w∈Δ.
Now, equating the coefficients in (24) and (25), we see that
(27)a2η+λ2q=δp1,(28)a3η+λ3q=δp2,(29)−a2η+λ2q=δq1,(30)2a22−a3η+λ3q=δq2.
From (27) and (29), we conclude that
(31)p1=−q1,(32)2a22η+λ2q2=δ2p12+q12.
Also, using (28) and (30), we obtain
(33)2a22η+λ3q=δp2+q2.
Subtracting (30) from (28) yields
(34)η+λ3q−2η+λ3qa22=δp2−q2.Thus, it follows from (33) and (34) that
(35)a3−γa22=a22+δp2−q22η+λ3q−γa22=δp2−q22η+λ3q+1−γa22=δ2p2−q2η+λ3q+p2+q2hγ,where hγ is given in (23). So, applying the triangle inequality and Lemma 2, we complete the proof by considering the following cases.Case 1.
(36)hγ<1η+λ3q.Case 2.
(37)hγ>1η+λ3q.Remark 7.
Note that lettingλ=1 in Definition 4, we obtain the class HΣqβ introduced by Bulut [31]. Also, we obtain the class
(38)limq⟶1−HΣqβ=HΣβintroduced by Sirivastava et al. [2].Corollary 8.
If a functionfz of the form (1) belongs to the class HΣq1,β and γ∈ℝ, then
(39)a3−γa22≤2δ3q,hγ<13q,δhγ,hγ>13q,where
(40)hγ=1−γ3q,δ=1−β.Theorem 9.
Iff∈HΣqλ,β and μ∈ℂ, then
(41)a3−μa22≤4δη+λ3q,1−μ∈0,ζ,4δ1−μη+λ2q2,1−μ∈ζ,∞,where
(42)ζ=η+λ2q2δη+λ3q,η=1−λ,δ=1−β.Proof.
From (31), (32), and (34), we deduce that
(43)a3−μa22=a22+δp2−q22η+λ3q−μa22=δp2−q22η+λ3q+1−μa22=δp2−q22η+λ3q+1−μδ2p12+q122η+λ2q2=1−μδ2p12η+λ2q2+δp2−q22η+λ3q.
Now, Lemma3 shows that
(44)p2−q2=124−p12x−y,x≤1,y≤1.
In (43), by (44), we conclude that
(45)a3−μa22=1−μδ2p12η+λ2q2+12δ4−p122η+λ3qx−y.
By choosingp1=p, we may assume without restriction that p∈0,2. Now, by using the triangle inequality and letting X=x≤1 and Y=y≤1, we obtain
(46)a3−μa22≤1−μδ2p2η+λ2q2+δ4−p22η+λ3qX+Y=FX,Y,for X,Y∈0,1:
(47)maxFX,Y=F1,1=δ2η+λ2q2×1−μ−η+λ2q2δη+λ3qp2+4δη+λ3q=Hp.
Forp∈0,2,
(48)H′p=2δη+λ2q21−μ−η+λ2q2δη+λ3qp,which proves the existence of a critical point at H′p=0, that is, at p=0.
Since(49)H′p<0;1−μ∈0,η+λ2q2δη+λ3q,
Hp is a strictly decreasing function of 1−μ. Now, (47) becomes
(50)maxHp:p∈0,2=H0=4δη+λ3q.
Also, since(51)H′p>0;1−μ∈η+λ2q2δη+λ3q,∞,
Hp is an increasing function of 1−μ. Thus, it follows from (47) that
(52)maxHp:p∈0,2=H2=4δ1−μη+λ2q2.
This completes the proof.Corollary 10.
If a functionfz of the form (1) belongs to the class HΣq1,β and μ∈ℂ, then
(53)a3−μa22≤4δ3q,1−μ∈0,ζ,4δ1−μ2q2,1−μ∈ζ,∞,where
(54)ζ=2q2δ3q,δ=1−β.


## 3. Coefficient Bounds for Functions inHΣqλ,α

Definition 11.
A functionfz of the form (1) is said to be in the class HΣqλ,α if
(55)f∈Σ,arg1−λfzz+λDqfz<απ2,(56)arg1−λgwz+λDqgw<απ2,where w=fz, g=f−1q∈0,1,0≤λ≤1,0<α≤1, and z,w∈Δ.Theorem 12.
If a functionfz of the form (1) is in the class HΣqλ,α, then
(57)a2≤2α2αη+λα3q+1−αη+λ2q2,(58)a3≤4α2η+λ2q2+2αη+λ3q,where η=1−λ.Proof.
It follows from (55) and (56) that
(59)ηfzz+λDqfz=pzα,(60)ηgwz+λDqgw=qwα,where
(61)pz=1+p1z+p2z2+⋯,qw=1+q1w+q2w2+⋯are in P.
Now, equating the coefficients in (59) and (60), we obtain
(62)a2η+λ2q=αp1,(63)a3η+λ3q=αp2+αα−12p12,(64)−a2η+λ2q=αq1,(65)2a22−a3η+λ3q=αq2+αα−12q12.
From (62) and (64), we deduce that
(66)p1=−q1,(67)2a22η+λ2q2=α2p12+q12.
Also, using (63), (65), and (67), we obtain
(68)2a22η+λ3q=αp2+q2+α−1αη+λ2q2a22.
Therefore,(69)a22=α2p2+q22αη+λα3q+1−αη+λ2q2.
Applying Lemma2 to the equality above, we obtain the desired estimate for a2, as asserted in (57).
Next, in order to find the bound fora3, we subtract (65) from (63). This gives
(70)2η+λ3qa3−2η+λ3qa22=αp2−q2+αα−12p12−q12.
It follows from (66), (67), and (70) that
(71)a3=α2p12+q122η+λ2q2+αp2−q22η+λ3q.
Applying Lemma2 to the equality above, we obtain the desired estimate for a3, as asserted in (58).Remark 13.
Note that lettingλ=1 in Definition 11, we obtain the class HΣqα introduced by Bulut [31]. Also, we obtain the class
(72)limq⟶1−HΣqα=HΣαintroduced by Srivastava et al. [2].In the next section, we introduce some methods for solving linear as well as nonlinear ordinary/partial differential equations. For example, the Exp-function method, homotopy perturbation method (HPM), and variational iteration method (VIM) are introduced.


## 4. Some Methods for Solving Linear and Nonlinear Differential Equations

### 4.1. Exp-Function Method

Suppose we are given a partial differential equation (PDE) for a functionux,t in the form
(73)χu,ut,ux,utt,utx,uxx,⋯=0,where χ is a polynomial in its arguments.The main idea of the Exp-function method for solving equation (73) proceeds in the following steps.Step 1.
Look for traveling wave solutions of equation (73) by taking the wave transformation:
(74)ux,t=Uρ,ρ=kx+wt,into account, where k and w are real nonzero constants.Step 2.
Substitution of equation (74) into (73) yields an ordinary differential equation (ODE) for Uρ. If possible, integrate the resulting ODE term by term one or more times. This introduces one or more integration constants.Step 3.
Introduce the form(75)Uρ=acexpcρ+⋯+a−dexp−dρbpexppρ+⋯+a−qexp−qρ,where c,d,p, and q are positive integers (to be determined). The coefficients ai and bj are arbitrary real constants (to be specified).Step 4.
Determine the highest order nonlinear term and the linear term of highest order in the ODE from Step2, and express them in terms of (75). Then, in the resulting terms, balance the highest order Exp-function to determine c and p and the lowest order Exp-function to determine d and q.Step 5.
Withc,d,p, and q as determined in Step 4, substitution of (75) into the ODE from Step 2 yields an algebraic equation involving the integer powers of expρ. Equating the coefficients of each power of expρ to zero results in a (highly nonlinear) system of algebraic equations for ai,bj,k, and w. If the original PDE contains some arbitrary parameters, these will, of course, also appear in the system.Step 6.
The original PDE has a solution in the form (75) if and only if there is a real nontrivial solution to the resulting system of algebraic equations, and using a computer can be very helpful.For more information about the Exp-function method, readers can refer to [32–38].


### 4.2. Homotopy Perturbation Method

The homotopy perturbation method is a semianalytical technique for solving linear and nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations [39–42].To illustrate the basic idea of the HPM, we consider the following differential equation:(76)Au−fr=0,r∈Ω,subject to the boundary condition
(77)Bu,∂u∂r,r∈Γ,where A represents a general differential operator, B is a boundary operator, Γ is the boundary of the domain Ω, and fr is a known analytic function.The operatorA can be decomposed into two parts viz linear L and nonlinear N. Therefore, equation (76) may be written in the following form:
(78)Lu+Nu−fr=0.Also, we have(79)Lu+pNu−fr=0,where p∈0,1 is the embedding parameter.Using homotopy technique, proposed by He [41], we construct a homotopy vr,p:Ω×0,1⟶ℝ to equation (79) which satisfies
(80)Hv,p=1−pLv−Lu0+pLv+nv−fr=0,(81)Hv,p=Lv−Lu0+pLu0+pNv−fr=0.Here,u0 is an initial approximation of equation (80) satisfying the given conditions.By substitutingp=0 and p=1 in equation (80), we may get following equations, respectively,
(82)Hv,0=Lv−Lu0,Hv,1=Av−fr=0.Asp changes from zero to unity, vr,p changes from u0r to ur. In topology, this is called deformation, and Lv−Lu0 and Av−fr are homotopic to each other. Due to the fact that p∈0,1 is a small parameter, we consider the solution of equation (80) as a power series in p as below:
(83)v=v0+v1p+v2p2+⋯.The approximate solution of equation (76) may then be obtained as
(84)u=limp⟶1v=v0+v1+v2+⋯.The convergence of the series solution (84) has been given in [41].For more information about the homotopy perturbation method, readers can refer to [43–45].


### 4.3. Variational Iteration Method

The variational iteration method is one of the well-known methods for solving linear and nonlinear ordinary as well as partial differential equations [46, 47]. He in [48] developed the VIM and successfully applied to ordinary and partial differential equations.To illustrate the basic idea of the VIM, we consider following the general nonlinear system as(85)Lu+Nu=gx,t,where L is a linear operator, N is a nonlinear operator, and gx,t is the given continuous function. The correction functional for the system (85)
(86)un+1x,t=unx,t+∫0tλLunx,s+Nu~x,s−gx,sds,n≥0,where λ is a Lagrange multiplier, which can be identified optimally via variational theory, un is the nth approximate solution, and u~n denotes a restricted variation, i.e., δu~n=0.The initial approximationu0 can be chosen freely if it satisfies the given conditions. The solution is approximated as
(87)ux,t=limn⟶∞unx,t.[49, 50].For more information about the variational iteration method, readers can refer to [51–53].


## 4.1. Exp-Function Method

Suppose we are given a partial differential equation (PDE) for a functionux,t in the form
(73)χu,ut,ux,utt,utx,uxx,⋯=0,where χ is a polynomial in its arguments.The main idea of the Exp-function method for solving equation (73) proceeds in the following steps.Step 1.
Look for traveling wave solutions of equation (73) by taking the wave transformation:
(74)ux,t=Uρ,ρ=kx+wt,into account, where k and w are real nonzero constants.Step 2.
Substitution of equation (74) into (73) yields an ordinary differential equation (ODE) for Uρ. If possible, integrate the resulting ODE term by term one or more times. This introduces one or more integration constants.Step 3.
Introduce the form(75)Uρ=acexpcρ+⋯+a−dexp−dρbpexppρ+⋯+a−qexp−qρ,where c,d,p, and q are positive integers (to be determined). The coefficients ai and bj are arbitrary real constants (to be specified).Step 4.
Determine the highest order nonlinear term and the linear term of highest order in the ODE from Step2, and express them in terms of (75). Then, in the resulting terms, balance the highest order Exp-function to determine c and p and the lowest order Exp-function to determine d and q.Step 5.
Withc,d,p, and q as determined in Step 4, substitution of (75) into the ODE from Step 2 yields an algebraic equation involving the integer powers of expρ. Equating the coefficients of each power of expρ to zero results in a (highly nonlinear) system of algebraic equations for ai,bj,k, and w. If the original PDE contains some arbitrary parameters, these will, of course, also appear in the system.Step 6.
The original PDE has a solution in the form (75) if and only if there is a real nontrivial solution to the resulting system of algebraic equations, and using a computer can be very helpful.For more information about the Exp-function method, readers can refer to [32–38].


## 4.2. Homotopy Perturbation Method

The homotopy perturbation method is a semianalytical technique for solving linear and nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations [39–42].To illustrate the basic idea of the HPM, we consider the following differential equation:(76)Au−fr=0,r∈Ω,subject to the boundary condition
(77)Bu,∂u∂r,r∈Γ,where A represents a general differential operator, B is a boundary operator, Γ is the boundary of the domain Ω, and fr is a known analytic function.The operatorA can be decomposed into two parts viz linear L and nonlinear N. Therefore, equation (76) may be written in the following form:
(78)Lu+Nu−fr=0.Also, we have(79)Lu+pNu−fr=0,where p∈0,1 is the embedding parameter.Using homotopy technique, proposed by He [41], we construct a homotopy vr,p:Ω×0,1⟶ℝ to equation (79) which satisfies
(80)Hv,p=1−pLv−Lu0+pLv+nv−fr=0,(81)Hv,p=Lv−Lu0+pLu0+pNv−fr=0.Here,u0 is an initial approximation of equation (80) satisfying the given conditions.By substitutingp=0 and p=1 in equation (80), we may get following equations, respectively,
(82)Hv,0=Lv−Lu0,Hv,1=Av−fr=0.Asp changes from zero to unity, vr,p changes from u0r to ur. In topology, this is called deformation, and Lv−Lu0 and Av−fr are homotopic to each other. Due to the fact that p∈0,1 is a small parameter, we consider the solution of equation (80) as a power series in p as below:
(83)v=v0+v1p+v2p2+⋯.The approximate solution of equation (76) may then be obtained as
(84)u=limp⟶1v=v0+v1+v2+⋯.The convergence of the series solution (84) has been given in [41].For more information about the homotopy perturbation method, readers can refer to [43–45].


## 4.3. Variational Iteration Method

The variational iteration method is one of the well-known methods for solving linear and nonlinear ordinary as well as partial differential equations [46, 47]. He in [48] developed the VIM and successfully applied to ordinary and partial differential equations.To illustrate the basic idea of the VIM, we consider following the general nonlinear system as(85)Lu+Nu=gx,t,where L is a linear operator, N is a nonlinear operator, and gx,t is the given continuous function. The correction functional for the system (85)
(86)un+1x,t=unx,t+∫0tλLunx,s+Nu~x,s−gx,sds,n≥0,where λ is a Lagrange multiplier, which can be identified optimally via variational theory, un is the nth approximate solution, and u~n denotes a restricted variation, i.e., δu~n=0.The initial approximationu0 can be chosen freely if it satisfies the given conditions. The solution is approximated as
(87)ux,t=limn⟶∞unx,t.[49, 50].For more information about the variational iteration method, readers can refer to [51–53].


## 5. Conclusion

The coefficient estimation problems have always been among the main interests of researchers in the study of univalent and biunivalent functions. Many studies related to this problem concern normalized analytic functions. In this paper, the bounds of the functional of coefficients and initial coefficient estimates were obtained for functions inHΣqλ,β, and estimates were found for the initial Taylor and Maclaurin coefficients of functions in HΣqλ,α. In a future research, we may obtain the bounds of the Teoplitz and Hankel determinants for classes of functions which are defined in terms of the (p,q)-derivative and q-derivative operators.



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