# Coefficient Estimates for Some Classes of Biunivalent Function Associated with Jacksonq-Difference Operator

Authors: Ekram E. Ali; Abdel Moneim Lashin; Abeer M. Albalahi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Using the Jacksonq-difference operator, we present two new subclasses of biunivalent functions. Furthermore, we estimate the initial Taylor-Maclaurin coefficients a2 and a3 of functions belonging to these new subclasses. Our results generalize some of the previously related works of several authors.


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## Body

## 1. Introduction

LetA denote the normalized analytical function family f of the formula
(1)fz=z+∑k=2∞akzk,in the open unit disc U=z∈ℂ:z<1. Further, by S we shall denote the class of all functions in A which are univalent in U. If f and g are analytic functions in U, we say that f is subordinate to g, written f≺g if there exists a Schwarz function w=∑k=1∞wkzk and wz<1 for all z∈U, such that fz=gwz,z∈U. Furthermore, if the function g is univalent in U, then we have the following equivalence (cf., e.g., [1, 2])
(2)f≺g⇔f0=g0,fU⊂gU.q-calculus plays an important role in the theory of hypergeometric series and quantum theory, number theory, statistical mechanics, etc. In the early 1900s, studies on q-difference equations were intensified by Jackson [3, 4], Carmichael [5], Mason [6], and Trjitzinsky [7]. It was Ismail et al. [8] who introduced geometric function theory and q-theory together for the first time. Following the same idea, the q-difference operator has been extensively investigated in the field of geometric function theory by many authors; for some recent works related to this operator on the classes of analytic functions, we refer to [9–11]. For any nonnegative integer k the q-number (or basic number) kqis defined by
(3)kq=1−qk1−q.0q=0.For nonnegative integerk, the q-factorial is defined by
(4)kq!=1q2q3q⋯.kq0q!=1.We note that whenq⟶1,k! reduces to classical definition of factorial. Throughout in this paper, we will assume q to be a fixed number between 0 and 1.Forfz∈A, the q-derivative operator or q-difference operator is defined as
(5)Dqfz=fqz−fzq−1zz≠0,f′zz=0,and Dq2fz=DqDqfz. From (5), we deduce that
(6)Dqfz=1+∑k=2∞kqakzk−1.Recently, Govindaraj and Sivasubramanian [12] defined Salagean q-derivative operator as follows:
(7)Sq0fz=fz,Sq1fz=zDqfz,Sqnfz=zDqSqn−1fz.A simple calculation implies(8)Sqnfz=f∗Gqnzn∈ℕ0=ℕ∪0=0,1,⋯,where
(9)Gqnz=z+∑k=2∞kqnzkn∈ℕ0,z∈U.Making use of (8) and (9), the power series of Sqnfz for f of the form (1) is given by
(10)Sqnfz=z+∑k=2∞kqnakzk,and limq⟶1−Sqnfz=Dnfz=z+∑k=2∞knakzk, which is the familiar Salagean derivative [13]. Shams et al. [14] introduced and investigated the class US∗k;β of parabolic starlike functions and the class UCk;β of parabolic convex functions of order β0≤β<1 as
(11)USk;β=f∈A:Rzf′zfz−β>kzf′zfz−1,k≥0,(12)UCk;β=f∈A:R1+zf′′zf′z−β>kzf′′zf′z,k≥0.SinceRew>αw−1+γ if and only if Rew1+αeiθ−αeiθ>γ(see [15]), then the conditions (11) and (12) can be written as
(13)USk;β=f∈A:R1+keiθzf′zfz−keiθ>β,k≥0,USk;β=f∈A:R1+keiθ1+zf′′zf′z−βeiθ>β,k≥0.Letφ be analytic function with positive real part and normalized by the conditions φ0=1,φ′0>0 and φ maps U onto a region starlike with respect to 1 and symmetric with respect to the real axis
(14)φz=1+B1z+B2z2+⋯,B1>0.Definition 1.
A functionfz∈A is said to be in the class Sγ,β,λ,ϕβ≥0,0≤λ≤1,−π≤γ<π if it satisfies
(15)1+βeiγ1−λzDqfzfz+λDqzDqfzDqfz−βeiγ≺φzz∈U.Remark 2.
Takingβ=0, φz=1+z/1−z, and q⟶1− in the class Sγ,β,λ,ϕ, we get the well-known class of λ-convex functions which was studied by [16].Definition 3.
A functionfz∈A is said to be in the class Bn,γ,β,λ,ϕβ≥0;n∈ℕ,0≤λ≤1,−π≤γ<π if it satisfies
(16)1+βeiγ1−λSqn+1fz+λSqn+2fz1−λSqnfz+λSqn+1fz−βeiγ≺φzz∈U.In Definition3, if we set Fz≔1−λSqnfz+λSqn+1fz, we obtain a new class USλ,γ,φ given below.Example 1.
A functionFz∈A is said to be in the class USλ,γ,φ0≤λ≤1,−π≤γ<π if it satisfies
(17)1+βeiγzDqFzFz−βeiγ≺φzz∈U.Remark 4.
Takingβ=0 and φz=1+1−2αz/1−z in the class Bλ,γ,φ, we get the class Sq∗α of q-starlike functions of order α0≤α<1 which was introduced by Seoudy and Aouf [17].The well-known Koebe one-quarter theorem [18] ensures the range of every function of the class S contains the disc w:w<1/4. Thus, every univalent function f∈S has an inverse f−1, which is defined by
(18)f−1fz=zz∈U,ff−1ω=ωω<14,where
(19)f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+⋯.If a functionf and its inverse f−1 are both univalent in U, then a member f of A is called biunivalent in U. We symbolize by ∑ the family of biunivalent functions in U given by (1). Lewin [19] examined the family ∑ and proved that a2<1.51 for elements of the family ∑. Later, Brannan et al. [20] claimed that a2<2 for f∈∑. Subsequently, Tan [21] obtained some initial coefficient estimates of functions belonging to the class ∑. Brannan and Taha in [22] proposed biconvex and bistarlike functions, which are similar to well-known subfamilies of S. The research trend in the last decade was the study of subfamilies of ∑. Generally, interest was shown to obtain the initial coefficient bounds for certain subfamilies of ∑. In 2010, Srivastava et al. [23] introduced two interesting subfamilies of the function family and found bounds for a2 and a3 of functions belonging to these subfamilies. Subsequently, other writers explored related problems in this direction (see [9, 10, 24–30]).Definition 5.
A functionf∈∑ given by (1) is said to be in the class S∑γ,β,λ,ϕ if both f and its inverse map g=f−1 are in Sγ,β,λ,ϕ.Remark 6.
Note the following:(1)
Ifβ=0 and ϕ=1+τ2z2/1−τz−τ2z2, then the class S∑γ,0,λ,1+τ2z2/1−τz−τ2z2 is equivalent to the class SLM∑q,α,0 introduced by [10](2)
Ifq⟶1−, then the class S∑γ,β,λ,ϕ is equivalent to the class the class obtained by Attiya et al. [32](3)
limq⟶1−S∑γ,1,0,ϕ≡Rσϕ (see Darwish et al. [33])(4)
limq⟶1−S∑γ,0,0,1+Az/1+Bz≡SA,B (see Hamidi and Jahangiri [34])(5)
limq⟶1−S∑γ,0,λ,ϕ≡M∑qλ,ϕφz=1 (see Goyal and Kumar [35] and also Zireh et al. [36])Definition 7.
A functionf∈∑ given by (1) is said to be in the class B∑n,γ,β,λ,ϕ if both f and its inverse map g=f−1 are in Bn,γ,β,λ,ϕ.In Definition7, if we set Fz≔1−λSqnfz+λSqn+1fz and Gz≔1−λSqngz+λSqn+1gz, we obtain a new class US∑λ,γ,φ given below.Example 2.
A functionF∈∑ given by (1) is said to be in the class US∑λ,γ,φ if
(20)1+βeiγzDqFzFz−βeiγ≺φzz∈U,1+βeiγwG′wGw−βeiγ=ϕww∈U.Remark 8.
Note the following:(1)
Ifβ=0, then the class B∑n,γ,0,λ,ϕ is equivalent to the class P∑qkλ,ϕ introduced by Murugusundaramoorthy et al. [31](2)
Ifq⟶1−, then the class B∑0,γ,β,λ,ϕ is equivalent to the class introduced by Attiya et al. [32](3)
limq⟶1−B∑0,γ,1,0,ϕ≡Rσϕ (see Darwish et al. [33])(4)
limq⟶1−B∑0,γ,0,0,1+Az/1+Bz=SA,B (see Hamidi and Jahangiri [34])(5)
limq⟶1−B∑0,γ,0,λ,ϕ≡M∑qλ,ϕφz=1 (see Goyal and Kumar [35] and also Zireh et al. [36])In order to prove our main results, we will need the following result.Lemma 9 (see [37]).
Ifh∈P, then ck≤2 for each k, where P is the family of all functions h analytic in U for which Rehz>0,hz=1+c1z+c2z2+c3z3+⋯ for z∈U.The aim of this paper is to find bounds on first two coefficients in the Taylor-Maclaurin expansion functional problem for functions belonging to the classesS∑γ,β,λ,ϕ and B∑n,γ,β,λ,ϕ. We also indicate interesting cases of the main results.


## 2. Coefficient Estimates

Unless otherwise mentioned, we shall assume throughout the remainder of this paper thatβ≥0,0≤λ≤1,−π≤γ<π,n∈ℕ0, and z∈U.Theorem 10.
Let the functionf given by (1) be in the class S∑γ,β,λ,ϕ. Then
(21)a2≤B1B1q1+βeiγ1−λ+λ2qB12+1+βeiγ1+λq2B1−B2,(22)a3≤B1q2q1+βeiγ1+λq2q+B122q21+βeiγ21+λq2.Proof.
Letf∈S∑γ,β,λ,,ϕ; then f and its inverse map g=f−1 are in the class Sγ,β,λ,ϕ; there exist two analytic functions u;v:U⟶U with u0=v0=0,uz<1 and vz<1, such that
(23)1+βeiγ1−λzDqfzfz+λDqzDqfzDqfz−βeiγ=φuz,(24)1+βeiγ1−λwDqgwgw+λDqwDqgwDqgw−βeiγ=φvw.
Define the functionsp and q by
(25)pz=1+uz1−uz=1+p1z+p2z2+⋯,qw=1+vw1−vw=1+q1w+q2w2+⋯,or equivalently,
(26)uz=pz−1pz+1=12p1z+p2−p122z2+⋯,(27)vw=qw−1qw+1=12q1w+q2−q122w2+⋯.
We observe thatp,q∈P, and in view of Lemma 9, we have that pn≤2 and qn≤2, for n≥2. Further, using (26) and (27) together with (14), it is evident that
(28)φuz=1+12B1p1z+12B1p2−p122+14B2p12z2+⋯,(29)φvw=1+12B1q1w+12B1q2−q122+14B2q12w2+⋯.
Therefore, in view of (23), (24), (28), and (29), we have
(30)1+βeiγ1−λzDqfzfz+λDqzDqfzDqfz−βeiγ=1+12B1p1z+12B1p2−p122+14B2p12z2+⋯,(31)1+βeiγ1−λwDqgwgw+λDqwDqgwDqgw−βeiγ=1+12B1q1w+12B1q2−q122+14B2q12w2+⋯.
Sincef∈∑ has the Taylor series expansion (1) and g=f−1 the series (19), we have
(32)1+βeiγ1−λzDqfzfz+λDqzDqfzDqfz−βeiγ=1+q1+βeiγ1+λqa2z+1+λq2q2qa3−1+λq2q+1a22z2+⋯,(33)1+βeiγ1−λwDqgwgw+λDqwDqgwDqgw−βeiγ=1+q1+βeiγ−1+λqa2z+−1+λq2q2qa3+22q1+λq2q−1+λq2q+1a22z2+⋯.
Comparing the corresponding coefficients of (30) and (32) yields
(34)q1+βeiγ1+λqa2=12B1p,(35)q1+βeiγ1+λq2q2qa3−1+λq2q+1a22=12B1p2−p122+14B2p12.
Similarly, from (31) and (33), we have
(36)−q1+βeiγ1+λqa2=12B1q1,(37)q1+βeiγ−1+λq2q2qa3+22q1+λq2q−1+λq2q+1a22=12B1q2−q122+14B2q12.
From (34) and (36), it follows that
(38)p1=−q1,(39)2q21+βeiγ21+λq2a22=14B12p12+q12.
Adding (35) and (37) yields
(40)2q21+βeiγ1−λ+λ2qa22=12B1p2+q2+14B2−B1p12+q12.
From (39) and (40), we get
(41)a22=B13p2+q24q21+βeiγ1−λ+λ2qB12+1+βeiγ1+λq2B1−B2.
Applying Lemma9 for the coefficients p2 and q2, we immediately have
(42)a2≤B1B1q1+βeiγ1−λ+λ2qB12+1+βeiγ1+λq2B1−B2.
This gives the bound ona2 as asserted in (21). Next, in order to find the bound on a3, by subtracting (37) from (35), we get
(43)a3=a22+B1p2−q24q2q1+βeiγ1+λq2q.Upon substituting the value of a22 from (39), we obtain
(44)a3=B1p2−q24q2q1+βeiγ1+λq2q+B12p12+q128q21+βeiγ21+λq2.
Applying Lemma9 for the coefficients p1,p2,q1, and q2, we get
(45)a3≤B1q2q1+βeiγ1+λq2q+B122q21+βeiγ21+λq2,which yield the estimate given by (22), and so the proof of Theorem 10 is completed.If we set(46)φz=1+z1−zη=1+2ζz+2ζ2z2+⋯0<ζ≤1,z∈U,in Definition 5, of the biunivalent function class S∑γ,β,λ,ϕ, we obtain a new class S∑ηγ,β,λ given by Definition 11.Definition 11.
For0<ζ≤1, a function fz∈A is said to be in the class S∑ηγ,β,λ if it satisfies the following conditions:
(47)1+βeiγ1−λzDqfzfz+λDqzDqfzDqfz−βeiγ≺1+z1−zηz∈U,1+βeiγ1−λwDqgwgw+λDqwDqgwDqgw−βeiγ≺1+w1−wηw∈U,where g=f−1.Using the parameter setting of Definition11, in Theorem 10, we get the following corollary.Corollary 12.
For0<ζ≤1, let the function fz∈A be in the class S∑ηγ,β,λ. Then
(48)a2≤2ζq1+βeiγ1−λ+λ2q2ζ−1−ζ1+βeiγ1+λq2,a3≤2ζq2q1+βeiγ1+λq2q+2ζ4q21+βeiγ21+λq2.
Let(49)φz=1+1−2ϑz1−z=1+21−ϑz+21−ϑz2+⋯0<ϑ≤1,z∈U,in Definition 5, of the biunivalent function class S∑γ,β,λ,ϕ, we obtain a new class Sϑ∑γ,β,λ given by Definition 13.Definition 13.
For0<ϑ≤1, a function fz∈A is said to be in the class Sϑ∑γ,β,λ if it satisfies the following conditions:
(50)1+βeiγ1−λzDqfzfz+λDqzDqfzDqfz−βeiγ≺1+1−2ϑz1−zz∈U,1+βeiγ1−λwDqgwgw+λDqwDqgwDqgw−βeiγ≺1+1−2ϑw1−ww∈U,where g=f−1.Using the parameter setting of Definition13 in Theorem 10, we get the following corollary.Corollary 14.
For0<ϑ≤1, let the function fz∈A be in the class Sϑ∑γ,β,λ. Then
(51)a2≤1q21−ϑ1+βeiγ1−λ+λ2q,a3≤21−ϑq2q1+βeiγ1+qλ2q+21−ϑ2q21+λq21+βeiγ2.Theorem 15.
Let the functionf given by (1) be in the class B∑n,γ,β,λ,ϕ. Then
(52)a2≤B1B1q1+βeiγ2q3qn1+λq2q−2q2n1+λq2B12+B1−B21+βeiγ2q22q2n1+λq2,(53)a3≤B1q2q3qn1+βeiγ1+λq2q+B12q22q2n1+βeiγ21+λq2.Proof.
Letf∈B∑n,γ,β,λ,ϕ; there are two Schwarz functions u and vdefined by (26) and (27), respectively, such that
(54)1+βeiγ1−λSqn+1fz+λSqn+2fz1−λSqnfz+λSqn+1fz−βeiγ=φuz,1+βeiγ1−λSqn+1gw+λSqn+2gw1−λSqngw+λSqn+1gw−βeiγ=φvw.
Since(55)1+βeiγ1−λSqn+1fz+λSqn+2fz1−λSqnfz+λSqn+1fz−βeiγ=1+q1+βeiγ2qn1+λqa2z+2q3qn1+λq2qa3−2q2n1+λq2a22z2+⋯,(56)1+βeiγ1−λwDqgwgw+λDqwDqgwDqgw−βeiγ=1+q1+βeiγ−2qn1+λqa2z+22q3qn1+λq2q−2q2n1+λq2a22−2q3qn1+λq2qa3z2+⋯.
Now, upon equating the coefficients in (30) and (55) and in (31) and (56), we get
(57)q2qn1+βeiγ1+λqa2=12B1p1,(58)q1+βeiγ2q3qn1+λq2qa3−2q2n1+λq2a22=12B1p2−p122+14B2p12,(59)−1+βeiγq2qn1+λqa2=12B1q1,(60)q1+βeiγ22q3qn1+λq2q−2q2n1+λq2a22−2q3qn1+λq2qa3=12B1q2−q122+14B2q12.
From (57) and (59), it follows that
(61)p1=−q1,and after a few additional measurements using (58)–(60), we find
(62)a22=B13p2+q24q1+βeiγ2q3qn1+λq2q−2q2n1+λq2B12+B1−B21+βeiγ2q22q2n1+λq2,a3=B1p2−q24q2q3qn1+βeiγ1+λq2q+B12p12+q128q22q2n1+βeiγ21+λq2.
Applying Lemma9 is followed by the estimates in (52) and (53).If we setφz=1+z/1−zη in Definition 7, of the biunivalent function class B∑n,γ,β,λ,ϕ, we obtain a new class B∑ηn,γ,β,λ given by Definition 16.Definition 16.
For0<ζ≤1, a function fz∈A is said to be in the class B∑ηn,γ,β,λ if it satisfies the following conditions:
(63)1+βeiγ1−λSqn+1fz+λSqn+2fz1−λSqnfz+λSqn+1fz−βeiγ≺1+z1−zηz∈U,1+βeiγ1−λSqn+1gw+λSqn+2gw1−λSqngw+λSqn+1gw−βeiγ≺1+w1−wηw∈U,where g=f−1.Using the parameter setting of Definition16, in Theorem 15, we get the following corollary.Corollary 17.
For0<η≤1, let the function fz∈A be in the class B∑ηn,γ,β,λ. Then
(64)a2≤2ζq1+βeiγ2q3qn1+λq2q−2q2n1+λq22ζ+q22q2n1−ζ1+βeiγ21+λq2,a3≤2ζq2q3qn1+βeiγ1+λq2q+4ζ2q22q2n1+βeiγ21+λq2.If we setφz=1+1−2ϑz/1−z in Definition 7, of the biunivalent function class B∑n,γ,β,λ,ϕ, we obtain a new class B∑ϑn,γ,β,λ given by Definition 18.Definition 18.
For0<ϑ≤1, a function fz∈A is said to be in the class B∑ϑn,γ,β,λ if it satisfies the following conditions:
(65)1+βeiγ1−λSqn+1fz+λSqn+2fz1−λSqnfz+λSqn+1fz−βeiγ≺1+1−2ϑz1−zz∈U,1+βeiγ1−λSqn+1gw+λSqn+2gw1−λSqngw+λSqn+1gw−βeiγ≺1+1−2ϑw1−ww∈U,where g=f−1.Using the parameter setting of Definition18 in Theorem 15, we get the following corollary.Corollary 19.
For0<ϑ≤1, let the function fz∈A be in the class B∑ϑn,γ,β,λ. Then
(66)a2≤21−ϑq1+βeiγ2q3qn1+λq2q−2q2n1+λq2,a3≤21−ϑq2q3qn1+βeiγ1+λq2q+41−ϑ2q22q2n1+βeiγ21+λq2.


## 3. Conclusions

This study introduces two new subclasses of biunivalent functions associated with the Jacksonq-difference operator in the open unit disc. We have determined upper bounds for the Taylor-Maclaurin coefficients a2 and a3 of functions belonging to these new subclasses. Our results generalize some of the earlier work of several authors.



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