# Coefficient Bounds for Subclasses of Biunivalent Functions Associated with the Chebyshev Polynomials

Authors: Hatun Özlem Güney; G. Murugusundaramoorthy; K. Vijaya
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

We introduce and investigate new subclasses of biunivalent functions defined in the open unit disk, involving Sălăgean operator associated with Chebyshev polynomials. Furthermore, we find estimates of the first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we give Fekete-Szegö inequalities for these function classes. Several consequences of the results are also pointed out.


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## Body

## 1. Introduction

LetA denote the class of analytic functions of the form (1)fz=z+∑n=2∞anznnormalized by the conditions f(0)=0=f′(0)-1 defined in the open unit disk (2)∆=z∈C:z<1.Let S be the subclass of A consisting of functions of form (1) which are also univalent in ∆. Let S∗(α) and K(α) denote the well-known subclasses of S, consisting of starlike and convex functions of order α(0≤α<1), respectively.The Koebe one-quarter theorem [1] ensures that the image of ∆ under every univalent function f∈A contains a disk of radius 1/4. Thus every univalent function f has an inverse f-1 satisfying (3)f-1fz=z,z∈∆,ff-1w=ww<r0f,r0f≥14.A function f∈A is said to be biunivalent in ∆ if both f and f-1 are univalent in ∆. Let Σ denote the class of biunivalent functions defined in the unit disk ∆. Since f∈Σ has the Maclaurin series given by (1), a computation shows that its inverse g=f-1 has the expansion (4)gw=f-1w=w-a2w2+2a22-a3w3+⋯.An analytic functionf is subordinate to an analytic function g, written as f(z)≺g(z), provided there is an analytic function w defined on ∆ with w(0)=0 and |w(z)|<1 satisfying f(z)=g(w(z)).Chebyshev polynomials, which are used by us in this paper, play a considerable role in numerical analysis. We know that the Chebyshev polynomials are four kinds. The most of books and research articles related to specific orthogonal polynomials of Chebyshev family contain essentially results of Chebyshev polynomials of first and second kindsTn(x) and Un(x) and their numerous uses in different applications; see Doha [2] and Mason [3].The well-known kinds of the Chebyshev polynomials are the first and second kinds. In the case of real variablex on (-1,1), the first and second kinds are defined by (5)Tnx=cos⁡nθ,Unx=sin⁡n+1θsin⁡θ,where the subscript n denotes the polynomial degree and x=cos⁡θ. We consider the function (6)Φz,t=11-2tz+z2.We note that if t=cos⁡α, α∈-π/3,π/3, then for all z∈∆(7)Φz,t=11-2tz+z2=1+∑n=1∞sin⁡n+1αsin⁡αzn=1+2cos⁡αz+3cos2α-sin2αz2+⋯.Thus, we write (8)Φz,t=1+U1tz+U2tz2+⋯z∈∆,t∈-1,1,where Un-1=sin⁡narccos⁡t/1-t2, for n∈N, are the second kind of the Chebyshev polynomials. Also, it is known that (9)Unt=2tUn-1t-Un-2t,(10)U1t=2t;U2t=4t2-1,U3t=8t3-4t,….The Chebyshev polynomials Tn(t),  t∈[-1,1], of the first kind have the generating function of the form (11)∑n=0∞Tntzn=1-tz1-2tz+z2z∈∆.All the same, the Chebyshev polynomials of the first kindTn(t) and the second kind Un(t) are well connected by the following relationship:(12)dTntdt=nUn-1t,Tnt=Unt-tUn-1t,2Tnt=Unt-Un-2t.Several authors have introduced and investigated subclasses of biunivalent functions and obtained bounds for the initial coefficients (see [4–10]). In [11], making use of the Sălăgean [12] differential operator, (13)Dk:A⟶Adefined by (14)D0fz=fz,D1fz=Dfz=zf′z,Dkfz=DDk-1fz=zDk-1fz′,k∈N=1,2,3,…,(15)Dkfz=z+∑n=2∞nkanzn,k∈N0=N∪0,and further for functions g of the form (4) Vijaya et al. [11] (also see [13]) defined (16)Dkgw=w-a22kw2+2a22-a33kw3+⋯and introduced two new subclasses of biunivalent functions. In this paper, we use Chebyshev polynomials to obtain the estimates on the coefficients |a2| and |a3|.


## 2. Biunivalent Function ClassesMΣk(λ,Φ(z,t)) and FΣk(β,Φ(z,t))

Motivated by recent works of Altinkaya and Yalcin [14] (also see [15]) and recent studies on biunivalent functions involving Sălăgean operator [11, 13], in this section, we introduce two new subclasses of Σ associated with Chebyshev polynomials and obtain the initial Taylor coefficients |a2| and |a3| for the function classes by subordination.Definition 1.
For0≤λ≤1 and t∈(-1,1) a function f∈Σ of form (1) is said to be in the class MΣk(λ,Φ(z,t)) if the following subordination holds: (17)1-λDk+1fzDkfz+λDk+2fzDk+1fz≺Φz,t,1-λDk+1gwDkgw+λDk+2gwDk+1gw≺Φw,t,where z,w∈∆ and g is given by (4).We note that by specializing the parametersλ and suitably fixing the values for k in Definition 1, we introduce (had not been studied so far) the following new subclasses of Σ as listed below.Remark 2.
Supposingf(z)∈Σ and t∈(-1,1), then we denote(1)
MΣk0,Φz,t≡SΣkΦz,t,(2)
MΣk1,Φz,t≡KΣkΦz,t,(3)
MΣ00,Φz,t=SΣ∗Φz,t,(4)
MΣ01,Φz,t=KΣΦz,t.Due to Frasin and Aouf [16] and Panigarhi and Murugusundaramoorthy [17] (also see [11, 13]) we define the following new subclass involving the Sălăgean operator [12].Definition 3.
For0≤β≤1 and t∈(-1,1) a function f∈Σ of form (1) is said to be in the class FΣk(β,Φ(z,t)) if the following subordination holds: (18)1-βDkfzz+βDkfz′≺Φz,t,1-βDkgww+βDkgw′≺Φw,t,where z,w∈∆, g=f-1, Dkf(z) and Dkg(w) are given by (4), (15), and (16), respectively.In Definition3, by specializing the parameters β and suitably fixing the values for k (had not been studied so far) the following new subclasses of Σ are as listed below.Remark 4.
Supposingf(z)∈Σ and t∈(-1,1), then we denote (1)
FΣk(0,Φ(z,t))≡RΣk(Φ(z,t)),(2)
FΣk(1,Φ(z,t))≡HΣk(Φ(z,t)),(3)
FΣ0(β,Φ(z,t))≡FΣ(β,Φ(z,t)),(4)
FΣ0(1,Φ(z,t))≡HΣ0(Φ(z,t)).In the following theorems we determine the initial Taylor coefficients|a2| and |a3| for the function classes f∈MΣk(λ,Φ(z,t)) and f∈FΣk(β,Φ(z,t)).Theorem 5.
Letf given by (1) be in the class MΣk(λ,Φ(z,t)) and t∈(0,1). Then (19)a2≤2t2t21+2λ3k-λ2+5λ+222k4t2+1+λ222k,a3≤4t21+λ222k+t1+2λ3k,where 0≤λ≤1 and t≠1/2.Proof.
Letf∈MΣk(λ,Φ(z,t)) and g=f-1. Considering (17), we have (20)1-λDk+1fzDkfz+λDk+2fzDk+1fz=Φz,t,(21)1-λDk+1gwDkgw+λDk+2gwDk+1gw=Φw,t.Define the functions u(z) and v(w) by (22)uz=c1z+c2z2+⋯,(23)vw=d1w+d2w2+⋯which are analytic in ∆ with u(0)=0=v(0) and |u(z)|<1, |v(w)|<1, for all z∈∆. It is well known that (24)uz=c1z+c2z2+⋯<1,vw=d1w+d2w2+⋯<1,z,w∈∆,and then (25)cj≤1,dj≤1∀j∈N.
Using (22) and (23) in (20) and (21), respectively, we have (26)1-λDk+1fzDkfz+λDk+2fzDk+1fz=1+U1tuz+U2tu2z+⋯,1-λDk+1gwDkgw+λDk+2gwDk+1gw=1+U1tvw+U2tv2w+⋯.In light of (1), (4), (10), (15), and (16) and from (26), we have (27)1+1+λ2ka2z+21+2λ3ka3-1+3λ22ka22z2+⋯=1+U1tc1z+U1tc2+U2tc12z2+⋯,1-1+λ2ka2w+8λ+43k-3λ+122ka22-21+2λ3ka3w2+⋯=1+U1td1w+U1td2+U2td12w2+⋯.This yields the following relations:(28)1+λ2ka2=U1tc1,(29)-1+3λ22ka22+21+2λ3ka3=U1tc2+U2tc12,(30)-1+λ2ka2=U1td1,(31)41+2λ3k-1+3λ22ka22-21+2λ3ka3=U1td2+U2td12.From (28) and (30) it follows that (32)c1=-d1,(33)21+λ222ka22=U12tc12+d12.Adding (29) to (31) and using (33), we obtain (34)a22=U13tc2+d2221+2λ3k-1+3λ22kU12t-1+λ222kU2t.Applying (25) to the coefficients c2 and d2 and using (10) we have (35)a2≤2t2t21+2λ3k-λ2+5λ+222k4t2+1+λ222k.By subtracting (31) from (29) and using (32) and (33), we get (36)a3=U12tc12+d1221+λ222k+U1c2-d241+2λ3k.Using (10), once again applying (25) to the coefficients c1, c2, d1, and d2, we get (37)a3≤4t21+λ222k+t1+2λ3k.By takingλ=0 or λ=1 and t∈(0,1), one can easily state the estimates |a2| and |a3| for the function classes MΣk(0,Φ(z,t))=SΣk(Φ(z,t)) and MΣk(1,Φ(z,t))=KΣk(Φ(z,t)), respectively.Remark 6.
Letf given by (1) be in the class SΣk(Φ(z,t)). Then (38)a2≤2t2t3k-22k8t2+22k,a3≤4t222k+t3k.Remark 7.
Letf given by (1) be in the class KΣk(Φ(z,t)). Then (39)a2≤2t2t3k+1-22k+18t2+22k+1,(40)a3≤t222k+t3k+1.Fork=0, Theorem 5 yields the following corollary.Corollary 8.
Letf given by (1) be in the class MΣ0(λ,Φ(z,t)). Then (41)a2≤2t2t1+λ2-λ2+λ4t2,a3≤4t21+λ2+t1+2λ,where 0≤λ≤1 and t≠1/2.By takingk=0 in the above remarks we get the estimates a2 and |a3| for the function classes SΣ∗(Φ(z,t)) and KΣ(Φ(z,t)).Remark 9.
Letf given by (1) be in the class SΣk(Φ(z,t)). Then (42)a2≤2t2t,a3≤4t2+t.Remark 10.
Letf given by (1) be in the class KΣk(Φ(z,t)). Then, for t≠1/2, (43)a2≤2t2t4-8t2,a3≤t2+t3.Theorem 11.
Letf given by (1) be in the class FΣk(β,Φ(z,t)) and t∈(0,1). Then (44)a2≤2t2t1+2β3k-1+β222k4t2+1+β222k,(45)a3≤4t21+β222k+2t1+2β3k.Proof.
Proceeding as in the proof of Theorem5 we can arrive at the following relations:(46)1+β2ka2=U1tc1,(47)1+2β3ka3=U1tc2+U2tc12,(48)-1+β2ka2=U1td1,(49)21+2β3ka22-1+2β3ka3=U1td2+U2td12.From (46) and (48) it follows that (50)c1=-d1,(51)21+β222ka22=U12tc12+d12.From (47), (49), and (51), we obtain (52)a22=U13tc2+d221+2β3kU12t-1+β222kU2t.Using (10) and (25) for the coefficients c2 and d2, we immediately get the desired estimate on |a2| as asserted in (44).
By subtracting (49) from (47) and using (50) and (51), we get (53)a3=U12tc12+d1221+β222k+U1tc2-d221+2β3k.Again using (10) and (25) for the coefficients c1, c2, d1, and d2, we get the desired estimate on |a3| as asserted in (45).Remark 12.
Letf given by (1) be in the class RΣk(Φ(z,t)). Then (54)a2≤2t2t3k-22k4t2+22k,a3≤4t222k+2t3k.Remark 13.
Letf given by (1) be in the class HΣk(Φ(z,t)). Then (55)a2≤2t2t3k+1-22k+14t2+22k+1,a3≤4t222k+1+2t3k+1.By takingk=0 we deduce the following results.Remark 14.
Letf given by (1) be in the class FΣ(β,Φ(z,t)). Then (56)a2≤2t2t1+β2-4t2β2,a3≤4t21+β2+2t1+2β.Remark 15.
Letf given by (1) be in the class FΣ0(1,Φ(z,t))≡HΣ(Φ(z,t)). Then (57)a2≤t2t1-t2,a3≤t2+2t3.Remark 16.
Letf given by (1) be in the class FΣ0(0,Φ(z,t))≡RΣ(Φ(z,t)). Then (58)a2≤2t2t,a3≤4t2+2t.


## 3. Fekete-Szegö Inequality for the Function ClassesMΣk(λ,Φ(z,t)) and FΣk(λ,Φ(z,t))

Due to Zaprawa [18], in this section we obtain the Fekete-Szegö inequality for the function classes MΣk(λ,Φ(z,t)) and FΣk(β,Φ(z,t)).Theorem 17.
Letf given by (1) be in the class MΣk(λ,Φ(z,t)) and μ∈R. Then one has (59)a3-μa22≤t1+2λ3k,μ-1≤1+λ222k/4t2+21+2λ3k-λ2+5λ+222k21+2λ3k81-μt321+2λ3k-λ2+5λ+222k4t2+1+λ222k,μ-1≥1+λ222k/4t2+21+2λ3k-λ2+5λ+222k21+2λ3k.Proof.
From (29) and (31) (60)a3-μa22=1-μU13tc2+d241+2λ3k-21+3λ22kU12t-2U2t1+λ222k+U1tc2-d241+2λ3k=U1thμ+141+2λ3kc2+hμ-141+2λ3kd2,where (61)hμ=1-μU12t221+2λ3k-1+3λ22kU12t-1+λ222kU2t.
Then, in view of (10), we conclude that(62)a3-μa22≤t1+2λ3k,0≤hμ≤141+2λ3k4thμ,hμ≥141+2λ3k.
Takingμ=1, we have the following corollary.Corollary 18.
Iff∈MΣk(λ,Φ(z,t)), then(63)a3-a22≤t1+2λ3k.Corollary 19.
Letf given by (1) be in the class SΣk(Φ(z,t)) and μ∈R. Then one has (64)a3-μa22≤t3k,μ-1≤22k/8t2+3k-22k3k81-μt33k-22k8t2+22k,μ-1≥22k/8t2+3k-22k3k.Particularly, for μ=1 if f∈SΣ∗(Φ(z,t)) one obtains (65)a3-a22≤t.Corollary 20.
Letf given by (1) be in the class KΣk(Φ(z,t)) and μ∈R. Then one has(66)a3-μa22≤t3k+1,μ-1≤22k/2t2+3k+1-22k+23k+121-μt33k+1-22k+22t2+22k,μ-1≥22k/2t2+3k+1-22k+23k+1.Particularly, for μ=1 if f∈KΣ0(Φ(z,t)) one obtains (67)a3-a22≤t3.Theorem 21.
Letf given by (1) be in the class FΣk(β,Φ(z,t)) and μ∈R. Then one has (68)a3-μa22≤2t1+2β3k,μ-1≤1+β222k/4t2+1+2β3k-1+β222k1+2β3k81-μt31+2β3k-1+β222k4t2+1+β222k,μ-1≥1+β222k/4t2+1+2β3k-1+β222k1+2β3k.Proof.
From (29) and (31) (69)a3-μa22=1-μU13tc2+d221+2β3kU12t-1+β222kU2t+U1tc2-d221+2β3k=U1thμ+121+2β3kc2+hμ-121+2β3kd2,where (70)hμ=1-μU12t21+2β3kU12t-1+β222kU2t.
Then, in view of (10), we conclude that(71)a3-μa22≤2t1+2β3k,0≤hμ≤121+2β3k4thμ,hμ≥121+2β3k.
Takingμ=1, we have the following corollary.Corollary 22.
Iff∈FΣk(β,Φ(z,t)), then (72)a3-a22≤2t1+2β3k.Corollary 23.
Letf given by (1) be in the class RΣk(Φ(z,t)) and μ∈R. Then one has (73)a3-μa22≤2t3k,μ-1≤22k/4t2+3k-22k3k81-μt33k-22k4t2+22k,μ-1≥22k/4t2+3k-22k3k.Particularly, for μ=1 if f∈RΣ0(Φ(z,t)) one obtains (74)a3-a22≤2t.Corollary 24.
Letf given by (1) be in the class HΣk(Φ(z,t)) and μ∈R. Then one has (75)a3-μa22≤2t3k+1,μ-1≤22k/t2+3k+1-22k+23k+121-μt33k+1-22k+2t2+22k,μ-1≥22k/t2+3k+1-22k+23k+1.Particularly, for μ=1 if f∈HΣ0(Φ(z,t)) one obtains (76)a3-a22≤2t3.



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