JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 181932 10.1155/2013/181932 181932 Research Article Coefficient Estimates for New Subclasses of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator Bulut Serap Hencl Stanislav Civil Aviation College Kocaeli University Arslanbey Campus, İzmit 41285 Kocaeli Turkey kocaeli.edu.tr 2013 25 11 2013 2013 07 05 2013 04 10 2013 2013 Copyright © 2013 Serap Bulut. 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 two new subclasses 𝒩Σδ,μn,α,λ and 𝒩Σδ,μn,β,λ of analytic and bi-univalent functions in the open unit disk 𝕌. For functions belonging to these classes, we obtain estimates on the first two Taylor-Maclaurin coefficients a2 and a3.

1. Introduction

Let =(-,) be the set of real numbers, the set of complex numbers, and (1)  :={1,2,3,}=0{0} the set of positive integers.

Let 𝒜 denote the class of all functions of the form (2)f(z)=z+k=2akzk which are analytic in the open unit disk (3)𝕌={z:z,|z|<1}. We also denote by 𝒮 the class of all functions in the normalized analytic function class 𝒜 which are univalent in 𝕌.

For f𝒜, Al-Oboudi  introduced the following operator: (4)Dδ0f(z)=f(z),(5)Dδ1f(z)=(1-δ)f(z)+δzf(z)=:Dδf(z)  (δ0)(6)Dδnf(z)=Dδ(Dδn-1f(z))  (n). If f is given by (2), then from (5) and (6) we see that (7)Dδnf(z)=z+k=2[1+(k-1)δ]nakzk  (n0), with Dδnf(0)=0. When δ=1, we get Sălăgean’s differential operator .

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk 𝕌. In fact, the Koebe one-quarter theorem  ensures that the image of 𝕌 under every univalent function f𝒮 contains a disk of radius 1/4. Thus, every function f𝒜 has an inverse f-1, which is defined by (8)f-1(f(z))=z(z𝕌),f(f-1(w))=w(|w|<r0(f);  r0(f)14). In fact, the inverse function f-1 is given by (9)f-1(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+.

A function f𝒜 is said to be bi-univalent in 𝕌 if both f and f-1 are univalent in 𝕌. Let Σ denote the class of bi-univalent functions in 𝕌 given by (2). For a brief history and interesting examples of functions in the class Σ, see  (see also [5, 6]). In fact, the aforecited work of Srivastava et al.  essentially revived the investigation of various subclasses of the bi-univalent function class Σ in recent years; it was followed by such works as those by Frasin and Aouf , Çağlar et al. , Porwal and Darus , and others (see, for example, ).

The object of the present paper is to introduce two new subclasses of the function class Σ and find estimates on the coefficients |a2| and |a3| for functions in these new subclasses of the function class Σ.

Firstly, in order to derive our main results, we need the following lemma.

Lemma 1 (see [<xref ref-type="bibr" rid="B8">14</xref>]).

If p𝒫, then |ck|2 for each k, where 𝒫 is the family of all functions p analytic in 𝕌 for which (10)(p(z))>0,p(z)=1+c1z+c2z2+ for z𝕌.

2. Coefficient Bounds for the Function Class <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M51"><mml:msubsup><mml:mrow><mml:mi>𝒩</mml:mi></mml:mrow><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mi>δ</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>α</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>λ</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> Definition 2.

A function f(z) given by (2) is said to be in the class 𝒩Σδ,μ(n,α,λ) if the following conditions are satisfied: (11)fΣ,|arg((1-λ)(Dδnf(z)z)μ+λ(Dδnf(z))(Dδnf(z)z)μ-1)|<απ2(0<α1,λ1,μ0,δ0,n0,z𝕌),|arg((1-λ)(Dδng(w)w)μ+λ(Dδng(w))(Dδng(w)w)μ-1)|<απ2(0<α1,λ1,μ0,δ0,n0,w𝕌), where the function g is given by (12)g(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+ and Dδn is the Al-Oboudi differential operator.

Remark 3.

In Definition 2, if we choose

n=0, then we have the class (13)𝒩Σδ,μ(0,α,λ)=𝒩Σμ(α,λ) introduced by Çag˘lar et al. ;

n=0 and μ=1, then we have the class (14)𝒩Σδ,1(0,α,λ)=Σ(α,λ) introduced by Frasin and Aouf ;

n=0,  μ=1, and λ=1, then we have the class (15)𝒩Σδ,1(0,α,1)=Σα introduced by Srivastava et al. ;

n=0,  μ=0, and λ=1, then we have the class (16)𝒩Σδ,0(0,α,1)=𝒮Σ*[α] of strongly bi-starlike functions of order α, introduced by Brannan and Taha [5, 6];

δ=1 and μ=1, then we have the class (17)𝒩Σ1,1(n,α,λ)=Σ(n,α,λ) introduced by Porwal and Darus .

Theorem 4.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the function class (18)𝒩Σδ,μ(n,α,λ)(0<α1,λ1,μ0,δ0,n0). Then, (19)|a2|(2α)×(-(1+δ)2n(λ2+2λ+μ)])1/2((1+δ)2n(λ+μ)2+α[(1+δ)2n(λ2+2λ+μ)2(1+2δ)n(2λ+μ)-(1+δ)2n(λ2+2λ+μ)])1/2)-1(20)|a3|4α2(1+δ)2n(λ+μ)2+2α(1+2δ)n(2λ+μ).

Proof.

First of all, it follows from conditions (11) that (21)(1-λ)(Dδnf(z)z)μ+λ(Dδnf(z))(Dδnf(z)z)μ-1=[p(z)]α(z𝕌),(1-λ)(Dδng(w)w)μ+λ(Dδng(w))(Dδng(w)w)μ-1=[q(w)]α(w𝕌), respectively, where (22)p(z)=1+p1z+p2z2+,q(w)=1+q1w+q2w2+ in 𝒫. Now, upon equating the coefficients in (21), we get (23)(1+δ)n(λ+μ)a2=αp1,(24)(1+2δ)n(2λ+μ)a3+(1+δ)2n(μ-1)(λ+μ2)a22=αp2+α(α-1)2p12,(25)-(1+δ)n(λ+μ)a2=αq1,(26)-(1+2δ)n(2λ+μ)a3+[4(1+2δ)n+(1+δ)2n(μ-1)]×(λ+μ2)a22=αq2+α(α-1)2q12. From (23) and (25), we obtain (27)p1=-q1,(28)2(1+δ)2n(λ+μ)2a22=α2(p12+q12). Also, from (24), (26), and (28), we find that (29)[2(1+2δ)n+(1+δ)2n(μ-1)](2λ+μ)a22=α(p2+q2)+α(α-1)2(p12+q12)=α(p2+q2)+α-1α(1+δ)2n(λ+μ)2a22. Therefore, we obtain (30)a22=(α2(p2+q2))×(+(1+δ)2n[(λ+μ)2-α(λ2+2λ+μ)]2α(1+2δ)n(2λ+μ)+(1+δ)2n[(λ+μ)2-α(λ2+2λ+μ)])-1. Applying Lemma 1 for the aforementioned equality, we get the desired estimate on the coefficient |a2| as asserted in (19).

Next, in order to find the bound on the coefficient |a3|, we subtract (26) from (24). We thus get (31)2(1+2δ)n(2λ+μ)a3-2(1+2δ)n(2λ+μ)a22=α(p2-q2)+α(α-1)2(p12-q12). It follows from (27), (28), and (31) that (32)a3=α2(p12+q12)2(1+δ)2n(λ+μ)2+α(p2-q2)2(1+2δ)n(2λ+μ). Applying Lemma 1 for the previous equality, we get the desired estimate on the coefficient |a3| as asserted in (20).

If we take n=0 in Theorem 4, then we have the following corollary.

Corollary 5 (see [<xref ref-type="bibr" rid="B4">8</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒩Σμ(α,λ)  (0<α1,λ1,μ0). Then, (33)|a2|2α(λ+μ)2+α(μ+2λ-λ2),|a3|4α2(λ+μ)2+2α2λ+μ.

If we take n=0 and μ=1 in Theorem 4, then we have the following corollary.

Corollary 6 (see [<xref ref-type="bibr" rid="B6">7</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class Σ(α,λ)  (0<α1,λ1). Then (34)|a2|2α(λ+1)2+α(1+2λ-λ2),|a3|4α2(λ+1)2+2α2λ+1.

If we take n=0,μ=1 and λ=1 in Theorem 4, then we have the following corollary.

Corollary 7 (see [<xref ref-type="bibr" rid="B12">4</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class Σα  (0<α1). Then, (35)|a2|α2α+2,|a3|α(3α+2)3.

If we take n=0,μ=0, and λ=1 in Theorem 4, then we have the following corollary.

Corollary 8.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮Σ*[α]  (0<α1). Then, (36)|a2|2α1+α,|a3|4α2+α.

If we take δ=1 and μ=1 in Theorem 4, then we have the following corollary.

Corollary 9 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the function class Σ(n,α,λ)  (0<α1,λ1,n0). Then, (37)|a2|2α4n(λ+1)2+α[2.3n(2λ+1)-4n(λ+1)2],|a3|4α24n(λ+1)2+2α3n(2λ+1).

3. Coefficient Bounds for the Function Class <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M113"><mml:msubsup><mml:mrow><mml:mi>𝒩</mml:mi></mml:mrow><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mi>δ</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>β</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>λ</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> Definition 10.

A function f(z) given by (2) is said to be in the class 𝒩Σδ,μ(n,β,λ) if the following conditions are satisfied: (38)fΣ,{(1-λ)(Dδnf(z)z)μ+λ(Dδnf(z))(Dδnf(z)z)μ-1}>β(0β<1,λ1,μ0,δ0,n0,z𝕌),{(1-λ)(Dδng(w)w)μ+λ(Dδng(w))(Dδng(w)w)μ-1}>β(0β<1,λ1,μ0,δ0,n0,w𝕌), where the function g is defined by (12) and Dδn is the Al-Oboudi differential operator.

Remark 11.

In Definition 10, if we choose

n=0, then we have the class (39)𝒩Σδ,μ(0,β,λ)=𝒩Σμ(β,λ) introduced by Çag˘lar et al. ;

n=0 and μ=1, then we have the class (40)𝒩Σδ,1(0,β,λ)=Σ(β,λ) introduced by Frasin and Aouf ;

n=0,μ=1, and λ=1, then we have the class (41)𝒩Σδ,1(0,β,1)=Σ(β) introduced by Srivastava et al. ;

n=0,μ=0, and λ=1, then we have the class (42)𝒩Σδ,0(0,β,1)=𝒮Σ*(β) of bi-starlike functions of order β, introduced by Brannan and Taha [5, 6];

δ=1 and μ=1, then we have the class (43)𝒩Σ1,1(n,β,λ)=Σ(n,β,λ) introduced by Porwal and Darus .

Theorem 12.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the function class (44)𝒩Σδ,μ(n,β,λ)(0<α1,λ1,μ0,δ0,n0). Then, (45)|a2|min{2(1-β)(1+δ)n(λ+μ),(×(2λ+μ))-1(4(1-β))×(×(2λ+μ)|2(1+2δ)n+(1+δ)2n(μ-1)|×(2λ+μ))-1)1/22(1-β)(1+δ)n(λ+μ)}(46)|a3|min{4(1-β)2(1+δ)2n(λ+μ)2+2(1-β)(1+2δ)n(2λ+μ),((1-β)[|4(1+2δ)n+(1+δ)2n(μ-1)|+(1+δ)2n|μ-1|])×(|2(1+2δ)n+(1+δ)2n(μ-1)|(2λ+μ))-1}(1+2δ)n|2(1+2δ)n+(1+δ)2n(μ-1)|×(2λ+μ)2(1+2δ)n(1+δ)2n)-14(1-β)2(1+δ)2n(λ+μ)2}.

Proof.

First of all, it follows from conditions (38) that (47)(1-λ)(Dδnf(z)z)μ+λ(Dδnf(z))(Dδnf(z)z)μ-1=β+(1-β)p(z)(z𝕌),(1-λ)(Dδng(w)w)μ+λ(Dδng(w))(Dδng(w)w)μ-1=β+(1-β)q(w)(w𝕌), respectively, where (48)p(z)=1+p1z+p2z2+,q(w)=1+q1w+q2w2+ in 𝒫. Now, upon equating the coefficients in (47), we get (49)(1+δ)n(λ+μ)a2=(1-β)p1,(50)(1+2δ)n(2λ+μ)a3+(1+δ)2n(μ-1)(λ+μ2)a22=(1-β)p2,(51)-(1+δ)n(λ+μ)a2=(1-β)q1,(52)-(1+2δ)n(2λ+μ)a3+[4(1+2δ)n+(1+δ)2n(μ-1)]×(λ+μ2)a22=(1-β)q2. From (49) and (51), we obtain (53)p1=-q1,(54)2(1+δ)2n(λ+μ)2a22=(1-β)2(p12+q12). Also, from (50) and (52), we have (55)[2(1+2δ)n+(1+δ)2n(μ-1)]×(2λ+μ)a22=(1-β)(p2+q2). Therefore, from equalities (54) and (55) we find that (56)|a2|2(1-β)2(|p1|2+|q1|2)2(1+δ)2n(λ+μ)2,|a2|2(1-β)(|p2|+|q2|)|2(1+2δ)n+(1+δ)2n(μ-1)|(2λ+μ), respectively, and applying Lemma 1, we get the desired estimate on the coefficient |a2| as asserted in (45).

Next, in order to find the bound on the coefficient |a3|, we subtract (52) from (50). We thus get (57)2(1+2δ)n(2λ+μ)a3-2(1+2δ)n(2λ+μ)a22=(1-β)(p2-q2), which, upon substitution of the value of a22 from (54), yields (58)a3=(1-β)2(p12+q12)2(1+δ)2n(λ+μ)2+(1-β)(p2-q2)2(1+2δ)n(2λ+μ). On the other hand, by using the (55) into (57), it follows that (59)a3=(1-β)×(2(1+2δ)n[2(1+2δ)n+(1+δ)2n(μ-1)]×(2λ+μ))-1×{[4(1+2δ)n+(1+δ)2n(μ-1)]p2-(1+δ)2n(μ-1)q2}. Applying Lemma 1 for (58) and (59), we get the desired estimate on the coefficient |a3| as asserted in (46).

If we take n=0 in Theorem 12, then we have the following corollary.

Corollary 13 (see [<xref ref-type="bibr" rid="B4">8</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒩Σμ(β,λ)  (0β<1,λ1,μ0). Then, (60)|a2|min{4(1-β)(μ+1)(2λ+μ),2(1-β)λ+μ},|a3|{min{4(1-β)(μ+1)(2λ+μ),4(1-β)2(λ+μ)2+2(1-β)2λ+μ},0μ<12(1-β)2λ+μ,μ1.

If we take n=0 and μ=1 in Theorem 12, then we have the following corollary.

Corollary 14.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class Σ(β,λ)  (0β<1,λ1). Then, (61)|a2|min{2(1-β)2λ+1,2(1-β)λ+1},|a3|2(1-β)2λ+1.

Remark 15.

Corollary 14 provides an improvement of the following estimates obtained by Frasin and Aouf .

Corollary 16 (see [<xref ref-type="bibr" rid="B6">7</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class Σ(β,λ)  (0β<1,λ1). Then, (62)|a2|2(1-β)2λ+1,|a3|4(1-β)2(λ+1)2+2(1-β)2λ+1.

If we take n=0,μ=1, and λ=1 in Theorem 12, then we have the following corollary.

Corollary 17.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class Σ(β)  (0β<1). Then, (63)|a2|{2(1-β)3,0β131-β,13β<1,|a3|2(1-β)3.

Remark 18.

Corollary 17 provides an improvement of the following estimates obtained by Srivastava et al. .

Corollary 19 (see [<xref ref-type="bibr" rid="B12">4</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class Σ(β)  (0β<1). Then, (64)|a2|2(1-β)3,(65)|a3|(1-β)(5-3β)3.

If we take n=0,μ=0, and λ=1 in Theorem 12, then we have the following corollary.

Corollary 20.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮Σ*(β)  (0β<1). Then, (66)|a2|2(1-β),|a3|{2(1-β),0β34(1-β)(5-4β),34β<1.

Remark 21.

Corollary 20 provides an improvement of the following estimates obtained by Brannan and Taha [5, 6] (see also [10, Corollary 3.2]).

Corollary 22 (see [<xref ref-type="bibr" rid="B2a">5</xref>, <xref ref-type="bibr" rid="B2b">6</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮Σ*(β)  (0β<1). Then, (67)|a2|2(1-β),|a3|2(1-β).

If we take δ=1 and μ=1 in Theorem 12, then we have the following corollary.

Corollary 23.

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the function class Σ(n,β,λ)  (0<α1,λ1,n0). Then, (68)|a2|min{2(1-β)2n(λ+1),2(1-β)3n(2λ+1)},|a3|2(1-β)3n(2λ+1).

Remark 24.

Corollary 23 provides an improvement of the following estimates obtained by Porwal and Darus .

Corollary 25 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let the function f(z) given by the Taylor-Maclaurin series expansion (2) be in the function class Σ(n,β,λ)  (0<α1,λ1,n0). Then, (69)|a2|2(1-β)3n(2λ+1),|a3|4(1-β)24n(λ+1)2+2(1-β)3n(2λ+1).

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