# Coefficient Estimates for Initial Taylor-Maclaurin Coefficients for a Subclass of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

Authors: Serap Bulut
Journal: The Scientific World Journal
 (2013)

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## Abstract

We introduce and investigate an interesting subclass𝒩𝒫Σλ,δ(n,β;h) of analytic and bi-univalent functions in the open unit disk 𝕌. For functions belonging to the class 𝒩𝒫Σλ,δ(n,β;h), 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∈ℂand|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 two functionsf and g, analytic in 𝕌, we say that the function f is subordinate to g in 𝕌 and write
(4)f(z)≺g(z)(z∈𝕌),
if there exists a Schwarz function ω, which is analytic in 𝕌 with
(5)ω(0)=0,|ω(z)|<1,(z∈𝕌)
such that
(6)f(z)=g(ω(z)),(z∈𝕌).
Indeed, it is known that
(7)f(z)≺g(z),(z∈𝕌)⟹f(0)=g(0),f(𝕌)⊂g(𝕌).
Furthermore, if the function g is univalent in 𝕌, then we have the following equivalence:
(8)f(z)≺g(z),(z∈𝕌)⟺f(0)=g(0),f(𝕌)⊂g(𝕌).Forf∈𝒜, Al-Oboudi [1] introduced the following operator:
(9)Dδ0f(z)=f(z),(10)Dδ1f(z)=(1-δ)f(z)+δzf′(z)=:Dδf(z),(δ≥0),(11)Dδnf(z)=Dδ(Dδn-1f(z)),(n∈ℕ).
If f is given by (2), then from (10) and (11) we see that
(12)Dδnf(z)=z+∑k=2∞‍[1+(k-1)δ]nakzk,(n∈ℕ0),
with Dδnf(0)=0. When δ=1, we get Sǎlǎgean’s differential operator D1n=Dn, [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
(13)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
(14)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], Porwal and Darus [8], and others (see, e.g., [9–17]).Motivated by the abovementioned works, we define the following subclass of function classΣ.Definition 1.
Leth:𝕌→ℂ be a convex univalent function such that
(15)h(0)=1,h(z¯)=h(z)¯(z∈𝕌;ℜ(h(z))>0).
A function f, defined by (2), is said to be in the class 𝒩𝒫Σλ,δ(n,β;h) if the following conditions are satisfied:
(16)f∈Σ,eiβ((1-λ)Dδnf(z)z+λ(Dδnf(z))′)≺h(z)cosβ+isinβeiβ((1-λ)Dδnf(z)z+λ(Dδnf(z))′)55555≺h(z)(z∈𝕌),eiβ((1-λ)Dδng(w)w+λ(Dδng(w))′)≺h(w)cosβ+isinβ(w∈𝕌),
where β∈(-π/2,π/2),λ≥1, the function g is given by
(17)g(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+⋯,
and Dδn is the Al-Oboudi differential operator.Remark 2.
If we set(18)h(z)=1+Az1+Bz(-1≤B<A≤1)
in Definition 1, then the class 𝒩𝒫Σλ,δ(n,β;h) reduces to the class denoted by 𝒩𝒫Σλ,δ(n,β;A,B) which is the subclass of the functions f∈Σ satisfying
(19)eiβ((1-λ)Dδnf(z)z+λ(Dδnf(z))′)≺1+Az1+Bzcosβ+isinβ(z∈𝕌),eiβ((1-λ)Dδng(w)w+λ(Dδng(w))′)≺1+Aw1+Bwcosβ+isinβ(w∈𝕌),
where β∈(-π/2,π/2),λ≥1, the function g is defined by (17), and Dδn is the Al-Oboudi differential operator.Remark 3.
If we set(20)h(z)=1+(1-2α)z1-z(0≤α<1)
in Definition 1, then the class 𝒩𝒫Σλ,δ(n,β;h) reduces to the class denoted by 𝒩𝒫Σλ,δ(n,β,α) which is the subclass of the functions f∈Σ satisfying
(21)ℜ{eiβ((1-λ)Dδnf(z)z+λ(Dδnf(z))′)}>αcosβℜ{eiβ((1-λ)Dδnf(z)z+λ(Dδnf(z))′55)}(z∈𝕌),ℜ{eiβ((1-λ)Dδng(w)w+λ(Dδng(w))′)}>αcosβℜ{eiβ((1-λ)Dδng(w)w+λ(Dδng(w))′)}>(w∈𝕌),
where β∈(-π/2,π/2),λ≥1, the function g is defined by (17), and Dδn is the Al-Oboudi differential operator.Remark 4.
If we set(22)δ=1,h(z)=1+(1-2α)z1-z(0≤α<1)
in Definition 1, then the class 𝒩𝒫Σλ,δ(n,β;h) reduces to the class denoted by 𝒩𝒫Σλ(n,β,α) which is the subclass of the functions f∈Σ satisfying
(23)ℜ{eiβ((1-λ)Dnf(z)z+λ(Dnf(z))′)}>αcosβℜ{eiβ((1-λ)Dng(w)w+λ(Dng(w))′)}>(z∈𝕌),ℜ{eiβ((1-λ)Dng(w)w+λ(Dng(w))′)}>αcosβℜ{eiβ((1-λ)Dng(w)w+λ(Dng(w))′)}>(w∈𝕌),
where β∈(-π/2,π/2),λ≥1, the function g is defined by (17), and Dn is the Sǎlǎgean differential operator.Remark 5.
If we set(24)n=0,h(z)=1+(1-2α)z1-z(0≤α<1)
in Definition 1, then the class 𝒩𝒫Σλ,δ(n,β;h) reduces to the class denoted by 𝒩𝒫Σλ(β,α) which is the subclass of the functions f∈Σ satisfying
(25)ℜ{eiβ((1-λ)f(z)z+λf′(z))}>αcosβ(z∈𝕌),ℜ{eiβ((1-λ)g(w)w+λg′(w))}>αcosβ(w∈𝕌),
where β∈(-π/2,π/2),λ≥1, and the function g is defined by (17).Remark 6.
If we set(26)n=0,λ=1,h(z)=1+(1-2α)z1-z(0≤α<1)
in Definition 1, then the class 𝒩𝒫Σλ,δ(n,β;h) reduces to the class denoted by 𝒩𝒫Σ(β,α) which is the subclass of the functions f∈Σ satisfying
(27)ℜ{eiβf′(z)}>αcosβ(z∈𝕌),ℜ{eiβg′(w)}>αcosβ(w∈𝕌),
where β∈(-π/2,π/2) and the function g is defined by (17).We note that(28)𝒩𝒫Σλ(n,0,α)=ℋΣ(n,α,λ)(see[8]),𝒩𝒫Σλ(0,α)=ℬΣ(α,λ)(see[7]),𝒩𝒫Σ(0,α)=ℋΣ(α)(see[4]).Firstly, in order to derive our main results, we need the following lemma.Lemma 7 (see [18]).
Let the functionh(z) given by
(29)h(z)=∑n=1∞‍Bnzn
be convex in 𝕌. Suppose also that the function φ(z) given by
(30)φ(z)=∑n=1∞‍cnzn
is holomorphic in 𝕌. If φ(z)≺h(z)(z∈𝕌), then
(31)|cn|≤|B1|(n∈ℕ).The object of the present paper is to find estimates on the Taylor-Maclaurin coefficients|a2| and |a3| for functions in this new subclass 𝒩𝒫Σλ,δ(n,β;h) of the function class Σ.


## 2. A Set of General Coefficient Estimates

In this section, we state and prove our general results involving the bi-univalent function class𝒩𝒫Σλ,δ(n,β;h) given by Definition 1.Theorem 8.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(32)𝒩𝒫Σλ,δ(n,β;h)(β∈(-π/2,π/2),λ≥1,δ≥0,n∈ℕ0)
with
(33)h(z)=1+B1z+B2z2+⋯.
Then
(34)|a2|≤min{|B1|cosβ(1+δ)n(1+λ),|B1|cosβ(1+2δ)n(1+2λ)},(35)|a3|≤|B1|cosβ(1+2δ)n(1+2λ).Proof.
It follows from (16) that
(36)eiβ((1-λ)Dδnf(z)z+λ(Dδnf(z))′)=p(z)cosβ+isinβ(z∈𝕌),(37)eiβ((1-λ)Dδng(w)w+λ(Dδng(w))′)=q(w)cosβ+isinβ(w∈𝕌),
where p(z)≺h(z) and q(w)≺h(w) have the following Taylor-Maclaurin series expansions:
(38)p(z)=1+p1z+p2z2+⋯,(39)q(w)=1+q1w+q2w2+⋯,
respectively. Now, upon equating the coefficients in (36) and (37), we get
(40)eiβ(1+δ)n(1+λ)a2=p1cosβ,(41)eiβ(1+2δ)n(1+2λ)a3=p2cosβ,(42)-eiβ(1+δ)n(1+λ)a2=q1cosβ,(43)eiβ[-(1+2δ)n(1+2λ)a3+2(1+2δ)n(1+2λ)a22]=q2cosβ.
From (40) and (42), we obtain
(44)p1=-q1,(45)2e2iβ(1+δ)2n(1+λ)2a22=(p12+q12)cos2β.
Also, from (41) and (43), we find that
(46)a22=e-iβ(p2+q2)cosβ2(1+2δ)n(1+2λ).
Since p,q∈h(𝕌), according to Lemma 7, we immediately have
(47)|pk|=|p(k)(0)k!|≤|B1|(k∈ℕ),|qk|=|q(k)(0)k!|≤|B1|(k∈ℕ).
Applying (47) and Lemma 7 for the coefficients p1, p2, q1, and q2, from the equalities (45) and (46), we obtain
(48)|a2|2≤|B1|2cos2β(1+δ)2n(1+λ)2,(49)|a2|2≤|B1|cosβ(1+2δ)n(1+2λ),
respectively. So we get the desired estimate on the coefficient |a2| as asserted in (34).
Next, in order to find the bound on the coefficient|a3|, we subtract (43) from (41). We thus get
(50)2(1+2δ)n(1+2λ)a3-2(1+2δ)n(1+2λ)a22=e-iβ(p2-q2)cosβ.
Upon substituting the value of a22 from (45) into (50), it follows that
(51)a3=e-2iβ(p12+q12)cos2β2(1+δ)2n(1+λ)2+e-iβ(p2-q2)cosβ2(1+2δ)n(1+2λ).
So we get
(52)|a3|≤|B1|2cos2β(1+δ)2n(1+λ)2+|B1|cosβ(1+2δ)n(1+2λ).
On the other hand, upon substituting the value of a22 from (46) into (50), it follows that
(53)a3=e-iβ(p2+q2)cosβ2(1+2δ)n(1+2λ)+e-iβ(p2-q2)cosβ2(1+2δ)n(1+2λ).
And we get
(54)|a3|≤|B1|cosβ(1+2δ)n(1+2λ).
Comparing the inequalities in (52) and (54) completes the proof of Theorem 8.


## 3. Corollaries and Consequences

By setting(55)h(z)=1+Az1+Bz(-1≤B<A≤1)
in Theorem 8, we have the following corollary.Corollary 9.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(56)5555555555555555555555555555555𝒩𝒫Σλ,δ(n,β;A,B)(β∈(-π/2,π/2),λ≥1,δ≥0,-1≤B<A≤1,n∈ℕ0).
Then
(57)|a2|≤min{(A-B)cosβ(1+δ)n(1+λ),(A-B)cosβ(1+2δ)n(1+2λ)},|a3|≤(A-B)cosβ(1+2δ)n(1+2λ).By setting(58)h(z)=1+(1-2α)z1-z(0≤α<1)
in Theorem 8, we have the following corollary.Corollary 10.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(59)5555555555555555555555555555𝒩𝒫Σλ,δ(n,β,α)(β∈(-π/2,π/2),λ≥1,δ≥0,0≤α<1,n∈ℕ0).
Then
(60)|a2|≤min{2(1-α)cosβ(1+δ)n(1+λ),2(1-α)cosβ(1+2δ)n(1+2λ)},|a3|≤2(1-α)cosβ(1+2δ)n(1+2λ).By setting(61)δ=1,h(z)=1+(1-2α)z1-z(0≤α<1)
in Theorem 8, we have the following corollary.Corollary 11.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(62)555555555555555555555555𝒩𝒫Σλ(n,β,α)(β∈(-π/2,π/2),λ≥1,0≤α<1,n∈ℕ0).
Then
(63)|a2|≤min{2(1-α)cosβ2n(1+λ),2(1-α)cosβ3n(1+2λ)},|a3|≤2(1-α)cosβ3n(1+2λ).Remark 12.
Whenβ=0, Corollary 11 is an improvement of the following estimates obtained by Porwal and Darus [8].Corollary 13 (see [8]).
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(64)ℋΣ(n,α,λ)(λ≥1,0≤α<1,n∈ℕ0).
Then
(65)|a2|≤2(1-α)3n(1+2λ),|a3|≤4(1-α)222n(1+λ)2+2(1-α)3n(1+2λ).By setting(66)n=0,h(z)=1+(1-2α)z1-z(0≤α<1)
in Theorem 8, we have the following corollary.Corollary 14.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(67)𝒩𝒫Σλ(β,α)(β∈(-π/2,π/2),λ≥1,0≤α<1).
Then
(68)|a2|≤min{2(1-α)cosβ1+λ,2(1-α)cosβ1+2λ},|a3|≤2(1-α)cosβ1+2λ.Remark 15.
Whenβ=0, Corollary 14 is 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 function class
(69)ℬΣ(α,λ)(λ≥1,0≤α<1).
Then
(70)|a2|≤2(1-α)1+2λ,|a3|≤4(1-α)2(1+λ)2+2(1-α)1+2λ.By setting(71)n=0,λ=1,h(z)=1+(1-2α)z1-z(0≤α<1)
in Theorem 8, we have the following corollary.Corollary 17.
Let the functionf(z) given by the Taylor-Maclaurin series expansion (2) be in the function class
(72)𝒩𝒫Σ(β,α)(β∈(-π/2,π/2),0≤α<1).
Then
(73)|a2|≤min{(1-α)cosβ,2(1-α)cosβ3},|a3|≤2(1-α)cosβ3.Remark 18.
Whenβ=0, Corollary 17 is 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 function class
(74)ℋΣ(α)(0≤α<1).
Then
(75)|a2|≤2(1-α)3,|a3|≤(1-α)(5-3α)3.



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