# Coefficients Bounds for Certain Subclass of Biunivalent Functions Associated with Ruscheweyhq-Differential Operator

Authors: Saqib Hussain; Shahid Khan; Muhammad Asad Zaighum; Maslina Darus; Zahid Shareef
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

We introduce in our present investigation a new subclass of analytic and biunivalent functions associated with Ruscheweyhq-differential operator in open unit disk E. We use the Faber polynomial expansions to find nth coefficients bounds of class of bisubordinate functions and also find initial coefficient estimates.


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## Body

## 1. Introduction

LetA denote the class of all function f(z) which is analytic in the open unit disk E={z:z<1} and has the Taylor series expansion of the following form:(1)fz=z+∑n=2∞anzn.By S we mean the class of all functions in A which are univalent in E. The Koebe one-quarter theorem [1] states that the image of E under every function f from S contains a disk of radius 1/4. It is well known that every univalent function f∈A has an inverse f-1 which is defined as (2)f-1fz=z,z∈E,ff-1w=w,w<r0f,r0f≥14,where(3)gw=f-1w=w-a2w2+2a22-a3w3-5a23-5a2a3+a4w4+⋯.A function f∈A is said to be biunivalent in E if both f and f-1 are univalent in E.If the functionsf and g are analytic in E, then f is said to be subordinate to g, written as (4)fz≺gz,z∈E if there exists a Schwarz function w(z), analytic in E, with w(0)=0 and w(z)<1, such that (5)fz=gwz,z∈E.Let Σ denote the class of analytic and biunivalent functions in E given by the Taylor-Maclaurin series expansion (1). Here we give few examples of functions in the class Σ such that(6)u1z=z1-z,u2z=-log⁡1-z,u3z=12log⁡1+z1-z,z∈E. However, the famous Koebe function k(z)=z/1-z2 is not in Σ; for more details we refer to [2]. For f∈Σ, the class of biunivalent analytic functions was first introduced and studied by Lewin where it was proved to show that a2<1.51. Brannan and Clunie [3] proved that a2≤2. Netanyahu [4] showed that maxa2=4/3. Brannan and Taha [5] introduced certain subclass of the biunivalent functions class Σ. For a brief history and interesting examples of biunivalent functions we refer to [3, 5–11].The Faber polynomials introduced by Faber [12]. Gong [13] and Schiffer [14] demonstrated the significance of the Faber polynomials in mathematical sciences, especially in geometric function theory. In the literature, there are only a few works determining the general coefficient bounds |an| for the analytic biunivalent functions given by (1) using Faber polynomial expansions. A very little is known about the bounds of Maclaurin’s series coefficient an for n≥3 by using a Faber polynomials. For more study about Faber polynomials we refer to [15–21].Using the technique of convolution, Ruscheweyh [22] defined the operator Rλ on the class of analytic functions A as (7)Rλfz=fz∗z1-zλ+1,z∈E,λ∈R,λ>-1.For λ=m∈N0=NU{0}, we obtain (8)Rmfz=zzm-1fzmm!.The expression Rmf(z) is called an mth-order Ruscheweyh derivative of f(z) and the symbol ∗ stands for Hadamard product (or convolution).Fort∈R and q>0, q≠1, the number [t] is defined in [23] as (9)t=1-qt1-q,0=0. For any nonnegative integer n the q-number shift factorial is defined as (10)n!=123⋯n,0!=1. We have limq→1[n]=n. Throughout in this paper we will assume q to be fixed number between 0 and 1.Theq-derivative operator or q-difference operator for f∈A is defined as (11)∂qfz=fqz-fzzq-1,z∈E. It can easily be seen that for n∈N={1,2,3,…} and z∈E.(12)∂qzn=nzn-1,∂q∑n=1∞anzn=∑n=1∞nanzn-1 The q-generalized Pochhammer symbol for t∈R and n∈N is defined as (13)tn=tt+1t+2⋯t+n-1, and, for t>0, let q-gamma function be defined as (14)Γqt+1=tΓqt,Γq1=1. For f∈A Ruscheweyh q-differential operator was defined by Aldweby and Darus [24] (see also [23]), as (15)Rqλfz=fz∗Fq,λ+1z,z∈E,λ>-1.Fq,λ+1z=z+∑n=2∞Γqn+λn-1!Γq1+λzn=z+∑n=2∞λ+1n-1n-1!zn=z+∑n=2∞Ψn-1zn,(16)Ψn-1=Γqn+λn-1!Γq1+λ=λ+1n-1n-1!,(17)z∂qRqλfz=1+λqλRqλ+1fz-λqλRqλfz.If q→1, equality (17) implies (18)zRλfz′=1+λRλ+1fz-λRλfz. which is the well known recurrent formula for Ruscheweyh differential operator.in the present paper we introduce new subclass of the function classΣ, involving Ruscheweyh q-differential operator Rqλf(z). By using Faber polynomial coefficient techniques we determine estimates for the general coefficient bounds an for n≥3 and also estimates on the coefficients a2 and a3 for functions in the new subclass of function class Σ. Several related classes are also considered, and connections to earlier known results are also defined.Definition 1.
A functionf∈Σ, 0≤γ, -1<λ, 1≤β, and 0≤α<1; we introduce a new class of biunivalent functions NΣq(α,β,γ,λ) as f∈NΣq(α,β,γ,λ) if and only if (19)1-βRqλfz+βz∂qRqλfz+γz2∂q2Rqλfzz≺1+Az1+Bz,z∈E,1-βRqλgw+βw∂qRqλgw+γw2∂q2Rqλgww≺1+Aw1+Bw,w∈E,where(20)A=1-α1+q,B=-qand g(w)=f-1(w) is defined by (3).
On specializing the parametersα, β, and γ, one can state the various new subclasses as illustrated in the following definition.Definition 2.
Forβ=1, -1<λ, 0≤γ, and 0≤α<1. A function, f∈Σ, is in the class NΣq(α,γ,λ) if the following conditions are satisfied: (21)∂qRqλfz+γz∂q2Rqλfz≺1+Az1+Bz,z∈E,∂qRqλgw+γw∂q2Rqλgw≺1+Aw1+Bw,w∈E, where A, B are given by (20), and g(w)=f-1(w) is defined by (3).Definition 3.
Forβ=1, γ=0, -1<λ, and 0≤α<1. A function, f∈Σ, is in the class NΣq(α,λ) if the following conditions are satisfied: (22)∂qRqλfz≺1+Az1+Bz,z∈E,∂qRqλgw≺1+Aw1+Bw,w∈E, where A, B are given by (20) and g(w)=f-1(w) is defined by (3).Definition 4.
Forγ=0, -1<λ and 0≤α<1. A function, f∈Σ, is in the class NΣq(α,β,λ) if the following conditions are satisfied: (23)1-βRqλfzz+β∂qRqλfz≺1+Az1+Bz,z∈E,1-βRqλgww+β∂qRqλgw≺1+Aw1+Bw,w∈E, where A, B are given by (20) and g(w)=f-1(w) is defined by (3).
It is well known that(24)fz≺1+1-2αz1-ziffRefz>α.
Special Cases (i)
Forλ=0 and q→1, the class NΣq(α,β,γ,λ)=NΣ(α,β,γ); see [21]. (ii)
Forβ=1, λ=0, and q→1, the class NΣq(α,β,γ,λ)=BΣ(α,γ); see [25]. (iii)
Forβ=1, λ=0, γ=0, and q→1, the class NΣq(α,β,γ,λ)=HΣα; see [2]. (iv)
Forλ=0, γ=0, and q→1, the class NΣq(α,β,γ,λ)=NΣ(α,β); see [26].Lemma 5 (see [27]).
Let the Schwarz functionw(z) be given by (25)wz=w1z+w2z2+w3z3+⋯,z∈E; then (26)w1≤1,w2≤1-w12,w2-tw12≤1+t-1w12.


## 2. Main Results

Using the Faber polynomial expansion of functionsf∈A of the form (1), the coefficients of its inverse map g=f-1 may be expressed as [16] given by (27)gw=f-1w=w+∑n=2∞1nKn-1-na2,a3,…wn,where(28)Kn-1-n=-n!-2n+1!n-5!a2n-1+-n!2-n+1!n-3!a2n-3a3+-n!-2n+3!n-4!a2n-4a4+-n!2-n+2!n-5!a2n-5a5+-n+2a32+-n!-2n+5!n-6!a2n-6a6+-2n+5a3a4+∑j≥7a2n-jVj,such that Vj with 7≤j≤n is a homogeneous polynomial in the variables a2,a3,…,an; see [28]. In particular, the first three terms of Kn-1-n are in general, for any p∈N and n≥2, an expansion of Kn-1p (for details we refer [16, 29]), which is as follows: (29)Kn-1p=pan+pp-12En-12+p!p-3!3!En-13+⋯+p!p-n+1!n-1!En-1n-1,where En-1p=En-1p(a2,a3,…) and, by [28], (30)En-1ma2,…,an=∑n=2∞m!a2μ1⋯anμn-1μ1!,…,μn-1!,form≤n, while a1=1, and the sum is taken over all nonnegative integer μ1,…,μn satisfying (31)μ1+μ2+⋯+μn=m,μ1+2μ2+⋯+n-1μn-1=n-1. Evidently, En-1n-1(a2,…,an)=a2n-1 [15], or, equivalently, (32)Enma1,a2,…,an=∑n=1∞m!a1μ1⋯anμnμ1!,…,μn!,form≤n, while a1=1, and the sum is taken over all nonnegative integer μ1,…,μn satisfying (33)μ1+μ2+⋯+μn=m,μ1+2μ2+⋯+nμn=n.It is clear that Enn(a1,…,an)=E1n; the first and last polynomials are Enn=a1n, En1=an.Theorem 6.
For0≤γ, 1≤β, -1<λ, and 0≤α<1. If f∈NΣq(α,β,γ,λ), if am=0, 2≤m≤n-1, then(34)an≤1-α1+qn-1!Γq1+λ1-β+βn+γnn-1Γqn+λ,forn≥3.Proof.
For the functionf∈NΣq(α,β,γ,λ) of the form (1), we have(35)1-βRqλfz+βz∂qRqλfz+γz2∂q2Rqλfzz=1+∑n=2∞1-β+βn+γnn-1×Ψn-1anzn-1,and, for its inverse map g=f-1, we have(36)1-βRqλgw+βw∂qRqλgw+γw2∂q2Rqλgww=1+∑n=2∞1-β+βn+γnn-1×Ψn-1bnwn-1,where Ψn-1 is given by (16) and bn=1/nKn-1-n(a2,a3,…,an).
Since both functionf and its inverse map g=f-1 are in NΣq(α,β,γ,λ), by the definition of subordination there exist two Schwarz functions p(z)=∑n=1∞cnzn and q(w)=∑n=1∞dnwn, where z, w∈E. We have (37)1-βRqλfz+βz∂qRqλfz+γz2∂q2Rqλfzz=1+Apz1+Bpz,where(38)1+Apz1+Bpz=1-∑n=1∞A-BKn-1c1,c2,…,cn,Bzn,(39)1-βRqλgw+βw∂qRqλgw+γw2∂q2Rqλgww=1+Aqw1+Bqw,where(40)1+Aqw1+Bqw=1-∑n=1∞A-BKn-1d1,d2,…,dn,Bwn.In general [15, 16] for any p∈N and n≥2, an expansion of Knp(k1,k2,…,kn,B),(41)Knpk1,k2,…,kn,B=p!p-n!n!k1nBn-1+p!p-n+1!n-2!k1n-2k2Bn-2+p!p-n+2!n-3!×k1n-3k3Bn-3+p!p-n+3!n-4!k1n-4k4Bn-4+p-n+32k32B+p!p-n+4!n-5!k1n-5k5Bn-5+p-n+4k3k4B+∑j≥6k1n-1Xj,where Xj is a homogeneous polynomial of degree j in the variables k1,k2,…,kn.
For the coefficients of the Schwarz functionsp(z) and q(w), cn≤1 and dn≤1 [9].
Comparing the corresponding coefficients of (35) and (37), we have (42)1-β+βn+γnn-1Ψn-1an=-A-BKn-1-1c1,c2,…,cn-1,B.Similarly, corresponding to coefficients of (36) and (39), we have(43)1-β+βn+γnn-1Ψn-1bn=-A-BKn-1-1d1,d2,…,dn-1,B.Note that, for am=0, 2≤m≤n-1, we have bn=-an(44)1-β+βn+βγnn-1Ψn-1an=-A-Bcn-1,(45)-1-β+βn+βγnn-1Ψn-1an=-A-Bdn-1.Taking the absolute values of (44) and (45), we have(46)an≤A-B1-β+βn+γnn-1Ψn-1cn-1=A-B1-β+βn+γnn-1Ψn-1dn-1,(47)an≤A-B1-β+βn+γnn-1Ψn-1,forn≥3.By using A=1-α(1+q), B=-q, from (20) and Ψn-1=Γq(n+λ)/[n-1]!Γq(1+λ) from (16) in (47), we have(48)an≤1-α1+qn-1!Γq1+λ1-β+βn+γnn-1Γqn+λ,forn≥3,which completes the proof of theorem.By puttingβ=1, in Theorem 6, we have the following corollary.Corollary 7.
For0≤γ, -1<λ, and 0≤α<1, if f∈NΣq(α,γ,λ) and if am=0, 2≤m≤n-1, then(49)an≤1-α1+qn-1!Γq1+λn1+γn-1Γqn+λ,forn≥3.For β=1, γ=0, in Theorem 6, we have the following corollary.Corollary 8.
For0≤α<1, -1<λ, if f∈NΣq(α,λ) and if am=0, 2≤m≤n-1, then(50)an≤1-α1+qn-1!Γq1+λnΓqn+λ,forn≥3.For λ=0, q→1 in Theorem 6, we obtain the following corollary.Corollary 9 (see [21]).
For1≤β and 0≤γ, 0≤α<1, if f∈NΣ(α,β,γ) and if am=0, 2≤m≤n-1, then(51)an≤21-α1+βn-1+γnn-1,forn≥3.For β=1, λ=0, and q→1 in Theorem 6, we obtain the following corollary.Corollary 10 (see [25]).
For0≤α<1 and 0≤γ, if f∈BΣ(α,γ) and if am=0, 2≤m≤n-1, then(52)an≤21-αn1+γn-1,forn≥3.For β=1, γ=0, λ=0, q→1 in Theorem 6, we obtain the following corollary.Corollary 11.
For0≤α<1, if f∈HΣ(α) and if am=0, 2≤m≤n-1, then(53)an≤21-αn,forn≥3.Forγ=0, λ=0, q→1 in Theorem 6, we obtain the following corollary.Corollary 12 (see [30]).
For1≤β and 0≤α<1, if f∈NΣ(α,β) and if am=0, 2≤m≤n-1, then(54)an≤21-α1+n-1β,forn≥3.Theorem 13.
For0≤α<1, -1<λ, 1≤β, and 0≤γ, if f∈NΣq(α,β,γ,λ), then (55)a2≤min⁡1-α1+q1!Γq1+λ1-β+β2+γ21Γq2+λ,2q1-α1+q2!Γq1+λ21-β+β3+γ32Γq3+λ,a3≤min⁡21+11-α1+q1!Γq1+λ1-β+β2+γ21Γq2+λ2,21+12q1-α1+q2!Γq1+λ21-β+β3+γ32Γq3+λ,a3-2a22≤q1-α1+q2!Γq1+λ1-β+β3+γ32Γq3+λ.Proof.
Replacingn by 2 and 3 in (44) and (45), respectively, we have (56)1-β+β2+γ21Ψ1a2=-A-Bc1,(57)1-β+β3+γ32Ψ2a3=-A-BBc12-c2,(58)-1-β+β2+γ21Ψ1a2=-A-Bd1,(59)1-β+β3+γ32Ψ22a22-a3=-A-BBd12-d2.From (56) and (58) we have(60)c1=-d1,(61)a2≤A-B1-β+β2+γ21Ψ1c1=A-B1-β+β2+γ21Ψ1d1,≤A-B1-β+β2+γ21Ψ1.Using (20) and (16) in (61), we have(62)a2≤1-α1+q1!Γq1+λ1-β+β2+γ21Γq2+λ.Adding (57) and (59) we have (63)21-β+β3+γ32Ψ2a22=-A-BBc12+d12-c2+d2.Taking absolute values of both sides of (63) and applying the estimates c2≤1-c12, d2≤1-d12 of Lemma 5 and c1≤1, d1≤1, we have(64)a22≤2A-BB21-β+β3+γ32Ψ2,a2≤2A-BB21-β+β3+γ32Ψ2;using (20) and (16) in (64), we have(65)a2≤2q1-α1+q2!Γq1+λ21-β+β3+γ32Γq3+λ.Now, in order to find a3, we subtract (59) from (57) and we have (66)a3=A-BBd12-c12-c2-d21+11-β+β3+γ32Ψ2+21+1a22.Using (60) in (66), we have(67)a3=-A-Bc2-d21+11-β+β3+γ32Ψ2+21+1a22.Taking the modulus of (67), we have(68)a3≤A-Bc2+d21+11-β+β3+γ32Ψ2+21+1a22.By using the estimates c2≤1-c12, d2≤1-d12, of Lemma 5, and c1≤1, d1≤1, in (68), we have (69)a3≤21+1a22,and using (56) in (69) we have(70)a3≤21+11-α1+q1-β+β2+γ21Ψ12.Using (20) and (16) in (70), we have(71)a3≤21+11-α1+q1!Γq1+λ1-β+β2+γ21Γq2+λ2.Again using (64) in (69) we have(72)a3≤21+12q1-α1+q2!Γq1+λ21-β+β3+γ32Γq3+λ.From (59) we have (73)a3-2a22≤A-BBd12+A-Bd21-β+β3+γ32Ψ2.Using d2≤1-d12, d1 of Lemma 5 and (20), (16), on (73), we have (74)a3-2a22≤q1-α1+q2!Γq1+λ1-β+β3+γ32Γq3+λ.Forβ=1, in Theorem 13, we obtain the following corollary.Corollary 14.
For-1<λ, 0≤γ, and 0≤α<1, if f∈NΣq(α,γ,λ), then(75)a2≤min⁡1-α1+q1!Γq1+λ21+γΓq2+λ,2q1-α1+q2!Γq1+λ231+γ2Γq3+λ,a3≤min⁡21+11-α1+q1!Γq1+λ21+γ1Γq2+λ2,21+12q1-α1+q2!Γq1+λ231+γ2Γq3+λ,a3-2a22≤q1-α1+q2!Γq1+λ31+γ2Γq3+λ.Forβ=1 and γ=0, in Theorem 13, we obtain the following corollary.Corollary 15.
For-1<λ, 0≤α<1, if f∈NΣq(α,γ,λ), then (76)a2≤min⁡1-α1+q1!Γq1+λ2Γq2+λ,2q1-α1+q2!Γq1+λ23Γq3+λ,a3≤min⁡21+11-α1+q1!Γq1+λ2Γq2+λ2,21+12q1-α1+q2!Γq1+λ23Γq3+λ,a3-2a22≤q1-α1+q2!Γq1+λ3Γq3+λ.Forλ=0, q→1, in Theorem 13, we obtain the following corollary.Corollary 16 (see [21]).
For0≤α<1, 1≤β, and 0≤γ, if f∈NΣ(α,β,γ), then (77)a2≤21-α1+2β+6γ,0≤α<1-1+β+2γ221+2β+6γ,21-α1+β+2γ,1-1+β+2γ221+2β+6γ≤α<1,a3≤21-α1+2β+6γ,a3-2a22≤21-α1+2β+6γ.Forλ=0, γ=0, q→1, in Theorem 13, we obtain the following corollary.Corollary 17 (see [30]).
For0≤α<1 and 1≤β, if f∈NΣ(α,β), then (78)a2≤21-α1+2β,0≤α<1-1+β221+2β,21-α1+β,1-1+β221+2β≤α<1,a3≤21-α1+2β,a3-2a22≤21-α1+2β.Forλ=0, β=1, γ=0, q→1, in Theorem 13, we obtain the following [21].Corollary 18 (see [21]).
For0≤α<1, if f∈HΣ(α), then(79)a2≤21-α3,0≤α<13,1-α,13≤α<1.a3≤21-α3,a3-2a22≤21-α3.Forλ=0, q→1, β=1, in Theorem 13, we obtain the following corollary.Corollary 19 (see [25]).
For0≤α<1, 1≤β, and 0≤γ, if f∈BΣ(α,γ), then (80)a2≤21-α31+2γ,0≤α<1+2γ-2γ231+2γ,21-α1+2γ,1+2γ-2γ231+2γ,≤α<1.a3≤21-α31+2γ.



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