# Coefficient Estimates for New Subclasses of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

Authors: Serap Bulut
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce and investigate two new subclasses𝒩Σδ,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.


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## Body

## 1. Introduction

Letℝ=(-∞,∞) be the set of real numbers, ℂ the set of complex numbers, and
(1)ℕ:={1,2,3,…}=ℕ0∖{0}
the set of positive integers.Let𝒜 denote the class of all functions of the form
(2)f(z)=z+∑k=2∞‍akzk
which are analytic in the open unit disk
(3)𝕌={z:z∈ℂ,|z|<1}.
We also denote by 𝒮 the class of all functions in the normalized analytic function class 𝒜 which are univalent in 𝕌.Forf∈𝒜, Al-Oboudi [1] introduced the following operator:
(4)Dδ0f(z)=f(z),(5)Dδ1f(z)=(1-δ)f(z)+δzf′(z)=:Dδf(z)(δ≥0)(6)Dδnf(z)=Dδ(Dδn-1f(z))(n∈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(n∈ℕ0),
with Dδnf(0)=0. When δ=1, we get Sălăgean’s differential operator [2].Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk𝕌. In fact, the Koebe one-quarter theorem [3] ensures that the image of 𝕌 under every univalent function f∈𝒮 contains a disk of radius 1/4. Thus, every function f∈𝒜 has an inverse f-1, which is defined by
(8)f-1(f(z))=z(z∈𝕌),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 functionf∈𝒜 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 [4] (see also [5, 6]). In fact, the aforecited work of Srivastava et al. [4] 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 [7], Çağlar et al. [8], Porwal and Darus [9], and others (see, for example, [10–13]).The object of the present paper is to introduce two new subclasses of the function classΣ 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 [14]).
Ifp∈𝒫, then |ck|≤2 for each k, where 𝒫 is the family of all functions p analytic in 𝕌 for which
(10)ℜ(p(z))>0,p(z)=1+c1z+c2z2+⋯
for z∈𝕌.


## 2. Coefficient Bounds for the Function Class𝒩Σδ,μ(n,α,λ)

Definition 2.
A functionf(z) given by (2) is said to be in the class 𝒩Σδ,μ(n,α,λ) if the following conditions are satisfied:
(11)f∈Σ,|arg((1-λ)(Dδnf(z)z)μ+λ(Dδnf(z))′(Dδnf(z)z)μ-1)|<απ2(0<α≤1,λ≥1,μ≥0,δ≥0,n∈ℕ0,z∈𝕌),|arg((1-λ)(Dδng(w)w)μ+λ(Dδng(w))′(Dδng(w)w)μ-1)|<απ2(0<α≤1,λ≥1,μ≥0,δ≥0,n∈ℕ0,w∈𝕌),
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 Definition2, if we choose(i)
n=0, then we have the class
(13)𝒩Σδ,μ(0,α,λ)=𝒩Σμ(α,λ)
introduced by Çag˘lar et al. [8];(ii)
n=0 and μ=1, then we have the class
(14)𝒩Σδ,1(0,α,λ)=ℬΣ(α,λ)
introduced by Frasin and Aouf [7];(iii)
n=0,μ=1, and λ=1, then we have the class
(15)𝒩Σδ,1(0,α,1)=ℋΣα
introduced by Srivastava et al. [4];(iv)
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];(v)
δ=1 and μ=1, then we have the class
(17)𝒩Σ1,1(n,α,λ)=ℬΣ(n,α,λ)
introduced by Porwal and Darus [9].Theorem 4.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(18)𝒩Σδ,μ(n,α,λ)(0<α≤1,λ≥1,μ≥0,δ≥0,n∈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(z∈𝕌),(1-λ)(Dδng(w)w)μ+λ(Dδng(w))′(Dδng(w)w)μ-1=[q(w)]α(w∈𝕌),
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 taken=0 in Theorem 4, then we have the following corollary.Corollary 5 (see [8]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒩Σμ(α,λ)(0<α≤1,λ≥1,μ≥0). Then,
(33)|a2|≤2α(λ+μ)2+α(μ+2λ-λ2),|a3|≤4α2(λ+μ)2+2α2λ+μ.If we taken=0 and μ=1 in Theorem 4, then we have the following corollary.Corollary 6 (see [7]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class ℬΣ(α,λ)(0<α≤1,λ≥1). Then
(34)|a2|≤2α(λ+1)2+α(1+2λ-λ2),|a3|≤4α2(λ+1)2+2α2λ+1.If we taken=0,μ=1 and λ=1 in Theorem 4, then we have the following corollary.Corollary 7 (see [4]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class ℋΣα(0<α≤1). Then,
(35)|a2|≤α2α+2,|a3|≤α(3α+2)3.If we taken=0,μ=0, and λ=1 in Theorem 4, then we have the following corollary.Corollary 8.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮Σ*[α](0<α≤1). Then,
(36)|a2|≤2α1+α,|a3|≤4α2+α.If we takeδ=1 and μ=1 in Theorem 4, then we have the following corollary.Corollary 9 (see [9]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class ℬΣ(n,α,λ)(0<α≤1,λ≥1,n∈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 Class𝒩Σδ,μ(n,β,λ)

Definition 10.
A functionf(z) given by (2) is said to be in the class 𝒩Σδ,μ(n,β,λ) if the following conditions are satisfied:
(38)f∈Σ,ℜ{(1-λ)(Dδnf(z)z)μ+λ(Dδnf(z))′(Dδnf(z)z)μ-1}>β(0≤β<1,λ≥1,μ≥0,δ≥0,n∈ℕ0,z∈𝕌),ℜ{(1-λ)(Dδng(w)w)μ+λ(Dδng(w))′(Dδng(w)w)μ-1}>β(0≤β<1,λ≥1,μ≥0,δ≥0,n∈ℕ0,w∈𝕌),
where the function g is defined by (12) and Dδn is the Al-Oboudi differential operator.Remark 11.
In Definition10, if we choose(i)
n=0, then we have the class
(39)𝒩Σδ,μ(0,β,λ)=𝒩Σμ(β,λ)
introduced by Çag˘lar et al. [8];(ii)
n=0 and μ=1, then we have the class
(40)𝒩Σδ,1(0,β,λ)=ℬΣ(β,λ)
introduced by Frasin and Aouf [7];(iii)
n=0,μ=1, and λ=1, then we have the class
(41)𝒩Σδ,1(0,β,1)=ℋΣ(β)
introduced by Srivastava et al. [4];(iv)
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];(v)
δ=1 and μ=1, then we have the class
(43)𝒩Σ1,1(n,β,λ)=ℋΣ(n,β,λ)
introduced by Porwal and Darus [9].Theorem 12.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(44)𝒩Σδ,μ(n,β,λ)(0<α≤1,λ≥1,μ≥0,δ≥0,n∈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)(z∈𝕌),(1-λ)(Dδng(w)w)μ+λ(Dδng(w))′(Dδng(w)w)μ-1=β+(1-β)q(w)(w∈𝕌),
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 taken=0 in Theorem 12, then we have the following corollary.Corollary 13 (see [8]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒩Σμ(β,λ)(0≤β<1,λ≥1,μ≥0). 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},0≤μ<12(1-β)2λ+μ,μ≥1.If we taken=0 and μ=1 in Theorem 12, then we have the following corollary.Corollary 14.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class ℬΣ(β,λ)(0≤β<1,λ≥1). Then,
(61)|a2|≤min{2(1-β)2λ+1,2(1-β)λ+1},|a3|≤2(1-β)2λ+1.Remark 15.
Corollary14 provides an improvement of the following estimates obtained by Frasin and Aouf [7].Corollary 16 (see [7]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class ℬΣ(β,λ)(0≤β<1,λ≥1). Then,
(62)|a2|≤2(1-β)2λ+1,|a3|≤4(1-β)2(λ+1)2+2(1-β)2λ+1.If we taken=0,μ=1, and λ=1 in Theorem 12, then we have the following corollary.Corollary 17.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class ℋΣ(β)(0≤β<1). Then,
(63)|a2|≤{2(1-β)3,0≤β≤131-β,13≤β<1,|a3|≤2(1-β)3.Remark 18.
Corollary17 provides an improvement of the following estimates obtained by Srivastava et al. [4].Corollary 19 (see [4]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class ℋΣ(β)(0≤β<1). Then,
(64)|a2|≤2(1-β)3,(65)|a3|≤(1-β)(5-3β)3.If we taken=0,μ=0, and λ=1 in Theorem 12, then we have the following corollary.Corollary 20.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮Σ*(β)(0≤β<1). Then,
(66)|a2|≤2(1-β),|a3|≤{2(1-β),0≤β≤34(1-β)(5-4β),34≤β<1.Remark 21.
Corollary20 provides an improvement of the following estimates obtained by Brannan and Taha [5, 6] (see also [10, Corollary 3.2]).Corollary 22 (see [5, 6]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the class 𝒮Σ*(β)(0≤β<1). Then,
(67)|a2|≤2(1-β),|a3|≤2(1-β).If we takeδ=1 and μ=1 in Theorem 12, then we have the following corollary.Corollary 23.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class ℋΣ(n,β,λ)(0<α≤1,λ≥1,n∈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.
Corollary23 provides an improvement of the following estimates obtained by Porwal and Darus [9].Corollary 25 (see [9]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class ℋΣ(n,β,λ)(0<α≤1,λ≥1,n∈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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