JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi 10.1155/2017/4150210 4150210 Research Article Coefficient Bounds for Subclasses of Biunivalent Functions Associated with the Chebyshev Polynomials http://orcid.org/0000-0002-3010-7795 Güney Hatun Özlem 1 http://orcid.org/0000-0001-8285-6619 Murugusundaramoorthy G. 2 http://orcid.org/0000-0002-3216-7038 Vijaya K. 2 Xu Yan 1 Faculty of Science Department of Mathematics Dicle University 21280 Diyarbakir Turkey dicle.edu.tr 2 School of Advanced Sciences VIT University Vellore 632014 India vit.ac.in 2017 26112017 2017 28 08 2017 26 10 2017 26112017 2017 Copyright © 2017 Hatun Özlem Güney et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let A denote the class of analytic functions of the form (1)fz=z+n=2anznnormalized by the conditions f(0)=0=f(0)-1 defined in the open unit disk (2)=zC: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(0α<1), respectively.

The Koebe one-quarter theorem  ensures that the image of under every univalent function fA 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,r0f14.A function fA 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 fΣ 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 function f 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 kinds Tn(x) and Un(x) and their numerous uses in different applications; see Doha  and Mason .

The well-known kinds of the Chebyshev polynomials are the first and second kinds. In the case of real variable x on (-1,1), the first and second kinds are defined by (5)Tnx=cosnθ,Unx=sinn+1θsinθ,where the subscript n denotes the polynomial degree and x=cosθ. We consider the function (6)Φz,t=11-2tz+z2.We note that if t=cosα, α-π/3,π/3, then for all z(7)Φz,t=11-2tz+z2=1+n=1sinn+1αsinαzn=1+2cosαz+3cos2α-sin2αz2+.Thus, we write (8)Φz,t=1+U1tz+U2tz2+z,t-1,1,where Un-1=sinnarccost/1-t2, for nN, 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=0Tntzn=1-tz1-2tz+z2z.

All the same, the Chebyshev polynomials of the first kind Tn(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 ). In , making use of the Sălăgean  differential operator, (13)Dk:AAdefined by (14)D0fz=fz,D1fz=Dfz=zfz,Dkfz=DDk-1fz=zDk-1fz,kN=1,2,3,,(15)Dkfz=z+n=2nkanzn,kN0=N0,and further for functions g of the form (4) Vijaya et al.  (also see ) 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|.

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Motivated by recent works of Altinkaya and Yalcin  (also see ) 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.

For 0λ1 and t(-1,1) a function fΣ 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+1fzΦz,t,1-λDk+1gwDkgw+λDk+2gwDk+1gwΦw,t,where z,w and g is given by (4).

We note that by specializing the parameters λ 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.

Supposing f(z)Σ and t(-1,1), then we denote

MΣk0,Φz,tSΣkΦz,t,

MΣk1,Φz,tKΣkΦz,t,

MΣ00,Φz,t=SΣΦz,t,

MΣ01,Φz,t=KΣΦz,t.

Due to Frasin and Aouf  and Panigarhi and Murugusundaramoorthy  (also see [11, 13]) we define the following new subclass involving the Sălăgean operator .

Definition 3.

For 0β1 and t(-1,1) a function fΣ of form (1) is said to be in the class FΣk(β,Φ(z,t)) if the following subordination holds: (18)1-βDkfzz+βDkfzΦz,t,1-βDkgww+βDkgwΦw,t,where z,w, g=f-1, Dkf(z) and Dkg(w) are given by (4), (15), and (16), respectively.

In Definition 3, 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.

Supposing f(z)Σ and t(-1,1), then we denote

FΣk(0,Φ(z,t))RΣk(Φ(z,t)),

FΣk(1,Φ(z,t))HΣk(Φ(z,t)),

FΣ0(β,Φ(z,t))FΣ(β,Φ(z,t)),

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 fMΣk(λ,Φ(z,t)) and fFΣk(β,Φ(z,t)).

Theorem 5.

Let f given by (1) be in the class MΣk(λ,Φ(z,t)) and t(0,1). Then (19)a22t2t21+2λ3k-λ2+5λ+222k4t2+1+λ222k,a34t21+λ222k+t1+2λ3k,where 0λ1 and t1/2.

Proof.

Let fMΣk(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)cj1,dj1jN.

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)a22t2t21+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)a34t21+λ222k+t1+2λ3k.

By taking λ=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.

Let f given by (1) be in the class SΣk(Φ(z,t)). Then (38)a22t2t3k-22k8t2+22k,a34t222k+t3k.

Remark 7.

Let f given by (1) be in the class KΣk(Φ(z,t)). Then (39)a22t2t3k+1-22k+18t2+22k+1,(40)a3t222k+t3k+1.

For k=0, Theorem 5 yields the following corollary.

Corollary 8.

Let f given by (1) be in the class MΣ0(λ,Φ(z,t)). Then (41)a22t2t1+λ2-λ2+λ4t2,a34t21+λ2+t1+2λ,where 0λ1 and t1/2.

By taking k=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.

Let f given by (1) be in the class SΣk(Φ(z,t)). Then (42)a22t2t,a34t2+t.

Remark 10.

Let f given by (1) be in the class KΣk(Φ(z,t)). Then, for t1/2, (43)a22t2t4-8t2,a3t2+t3.

Theorem 11.

Let f given by (1) be in the class FΣk(β,Φ(z,t)) and t(0,1). Then (44)a22t2t1+2β3k-1+β222k4t2+1+β222k,(45)a34t21+β222k+2t1+2β3k.

Proof.

Proceeding as in the proof of Theorem 5 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.

Let f given by (1) be in the class RΣk(Φ(z,t)). Then (54)a22t2t3k-22k4t2+22k,a34t222k+2t3k.

Remark 13.

Let f given by (1) be in the class HΣk(Φ(z,t)). Then (55)a22t2t3k+1-22k+14t2+22k+1,a34t222k+1+2t3k+1.

By taking k=0 we deduce the following results.

Remark 14.

Let f given by (1) be in the class FΣ(β,Φ(z,t)). Then (56)a22t2t1+β2-4t2β2,a34t21+β2+2t1+2β.

Remark 15.

Let f given by (1) be in the class FΣ0(1,Φ(z,t))HΣ(Φ(z,t)). Then (57)a2t2t1-t2,a3t2+2t3.

Remark 16.

Let f given by (1) be in the class FΣ0(0,Φ(z,t))RΣ(Φ(z,t)). Then (58)a22t2t,a34t2+2t.

3. Fekete-Szegö Inequality for the Function Classes <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M202"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>Φ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M203"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">F</mml:mi></mml:mrow><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>Φ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>

Due to Zaprawa , 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.

Let f given by (1) be in the class MΣk(λ,Φ(z,t)) and μR. Then one has (59)a3-μa22t1+2λ3k,μ-11+λ222k/4t2+21+2λ3k-λ2+5λ+222k21+2λ3k81-μt321+2λ3k-λ2+5λ+222k4t2+1+λ222k,μ-11+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-μa22t1+2λ3k,0hμ141+2λ3k4thμ,hμ141+2λ3k.

Taking μ=1, we have the following corollary.

Corollary 18.

If fMΣk(λ,Φ(z,t)), then(63)a3-a22t1+2λ3k.

Corollary 19.

Let f given by (1) be in the class SΣk(Φ(z,t)) and μR. Then one has (64)a3-μa22t3k,μ-122k/8t2+3k-22k3k81-μt33k-22k8t2+22k,μ-122k/8t2+3k-22k3k.Particularly, for μ=1 if fSΣ(Φ(z,t)) one obtains (65)a3-a22t.

Corollary 20.

Let f given by (1) be in the class KΣk(Φ(z,t)) and μR. Then one has(66)a3-μa22t3k+1,μ-122k/2t2+3k+1-22k+23k+121-μt33k+1-22k+22t2+22k,μ-122k/2t2+3k+1-22k+23k+1.Particularly, for μ=1 if fKΣ0(Φ(z,t)) one obtains (67)a3-a22t3.

Theorem 21.

Let f given by (1) be in the class FΣk(β,Φ(z,t)) and μR. Then one has (68)a3-μa222t1+2β3k,μ-11+β222k/4t2+1+2β3k-1+β222k1+2β3k81-μt31+2β3k-1+β222k4t2+1+β222k,μ-11+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-μa222t1+2β3k,0hμ121+2β3k4thμ,hμ121+2β3k.

Taking μ=1, we have the following corollary.

Corollary 22.

If fFΣk(β,Φ(z,t)), then (72)a3-a222t1+2β3k.

Corollary 23.

Let f given by (1) be in the class RΣk(Φ(z,t)) and μR. Then one has (73)a3-μa222t3k,μ-122k/4t2+3k-22k3k81-μt33k-22k4t2+22k,μ-122k/4t2+3k-22k3k.Particularly, for μ=1 if fRΣ0(Φ(z,t)) one obtains (74)a3-a222t.

Corollary 24.

Let f given by (1) be in the class HΣk(Φ(z,t)) and μR. Then one has (75)a3-μa222t3k+1,μ-122k/t2+3k+1-22k+23k+121-μt33k+1-22k+2t2+22k,μ-122k/t2+3k+1-22k+23k+1.Particularly, for μ=1 if fHΣ0(Φ(z,t)) one obtains (76)a3-a222t3.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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