# Coefficient Estimates and Other Properties for a Class of

Authors: Halit Orhan; Dorina Răducanu; Murat Çağlar; Mustafa Bayram
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

For0≤η<1, 0≤λ<1, −π/2<γ<π/2, 0≤β≤α, and m∈ℕ∪{0}, a new class Sα,βm(η,γ,λ) of analytic functions defined by means of the differential operator Dα,βm is introduced. Our main object is to provide sharp upper bounds for Fekete-Szegö problem in Sα,βm(η,γ,λ). We also find sufficient conditions for a function to be in this class. Some interesting
consequences of our results are pointed out.


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## Body

## 1. Introduction

Let𝒜 denote the class of functions f of the form
(1)f(z)=z+∑n=2∞‍anzn,
which are analytic in the open unit disk 𝒰={z∈ℂ:|z|<1}.Let𝒮 denote the subclass of 𝒜 consisting of functions that are univalent in 𝒰.A functionf∈𝒜 is said to be in the class of γ-spirallike functions of order λ in 𝒰, denoted by 𝒮*(γ,λ), if
(2)ℜ(eiγzf′(z)f(z))>λcosγ,z∈𝒰,
for 0≤λ<1 and some real γ with |γ|<π/2.The class𝒮*(γ,λ) was studied by Libera [1] and Keogh and Merkes [2].Note that𝒮*(γ,0) is the class of spirallike functions introduced by Špaček [3], 𝒮*(0,λ)=𝒮*(λ) is the class of starlike functions of order λ, and 𝒮*(0,0)=𝒮* is the familiar class of starlike functions.For the constantsλ,γ with 0≤λ<1 and |γ|<π/2, denote
(3)pλ,γ(z)=1+e-iγ(e-iγ-2λcosγ)z1-z,z∈𝒰.
The function pλ,γ(z) maps the open unit disk onto the half-plane Hλ,γ={z∈ℂ:ℜ(eiγz)>λcosγ}. If
(4)pλ,γ(z)=1+∑n=1∞‍pnzn,
then it is easy to check that
(5)pn=2e-iγ(1-λ)cosγ,∀n≥1.Forf∈𝒜 given by (1) and g∈𝒜 given by
(6)g(z)=z+∑n=2∞‍bnzn,
the Hadamard product (or convolution), denoted by f*g, is defined by
(7)(f*g)(z)=z+∑n=2∞‍anbnzn,z∈𝒰.Denote byℬ the family of all analytic functions w(z) that satisfy the conditions w(0)=0 and |w(z)|<1, z∈𝒰.A functionf∈𝒜 is said to be subordinate to a function g∈𝒜, written f≺g, if there exists a function w∈ℬ such that f(z)=g(w(z)), z∈𝒰.A classical theorem of Fekete and Szegö (see [4]) states that if f∈𝒮 is given by (1), then
(8)|a3-μa22|≤{3-4μ,ifμ≤0,1+2exp(-2μ1-μ),if0≤μ≤1,4μ-3,ifμ≥1.
This inequality is sharp in the sense that for each μ there exists a function in 𝒮 such that the equality holds. Later Pfluger (see [5]) has considered the same problem but for complex values of μ. The problem of finding sharp upper bounds for the functional |a3-μa22| for different subclasses of 𝒜 is known as the Fekete-Szegö problem. Over the years, this problem has been investigated by many authors including [6–12].For a functionf∈𝒜, we consider the following differential operator introduced by Răducanu and Orhan [13]:(i)
Dα,β0f(z)=f(z),(ii)
Dα,β1f(z)=Dα,βf(z)=αβz2f′′(z)+(α-β)zf′(z)+(1-α+β)f(z),(iii)
Dα,βmf(z)=Dα,β(Dα,βm-1f(z)),z∈𝒰,where 0≤β≤α and m∈ℕ0={0,1,…}.If the functionf is given by (1), then, from the definition of the operator Dα,βmf, it is easy to observe that
(9)Dα,βmf(z)=z+∑n=2∞‍Φn(α,β,m)anzn,
where
(10)Φn(α,β,m)=[1+(αβn+α-β)(n-1)]m,n≥2.It should be remarked that the operatorDα,βmf generalizes other differential operators considered earlier. For f∈𝒜, we have(i)
D1,0mf(z)=Dmf(z), the operator introduced by Sălăgean [14];(ii)
Dα,0mf(z)=Dαmf(z), the operator studied by Al-Oboudi [15].In view of (9), Dα,βmf(z) can be written in terms of convolution as
(11)Dα,βmf(z)=(gα,β*f)(z),z∈𝒰,
where
(12)gα,β(z)=z+∑n=2∞‍Φn(α,β,m)zn,z∈𝒰.Define the functiongα,β(-1) such that
(13)gα,β(-1)(z)*gα,β(z)=z1-z,z∈𝒰.It is easy to observe that(14)f(z)=gα,β(-1)(z)*Dα,βmf(z).Making use of the differential operatorDα,βmf, we define the following class of functions.Definition 1.
For0≤η<1, 0≤λ<1, and |γ|<π/2, denote by 𝒮α,βm(η,γ,λ) the class of functions f∈𝒜 which satisfy the condition
(15)ℜ(eiγz(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′)>λcosγ,z∈𝒰.
The class𝒮α,βm(η,γ,λ) contains as particular cases the following classes of functions:
(16)𝒮α,β0(0,γ,λ)=𝒮*(γ,λ),𝒮α,β0(0,γ,0)=𝒮*(γ),𝒮α,β0(0,0,0)=𝒮*.
Also, the class 𝒮α,β0(η,γ,λ) consists of functions f∈𝒜 satisfying the inequality
(17)ℜ(eiγzf′(z)(1-η)f(z)+ηzf′(z))>λcosγ,z∈𝒰.
An analogous of the class 𝒮α,β0(η,γ,λ) has been recently studied by Murugusundaramoorthy [16].
The main object of this paper is to obtain sharp upper bounds for the Fekete-Szegö problem for the class𝒮α,βm(η,γ,λ). We also find sufficient conditions for a function to be in this class.


## 2. Membership Characterizations

In this section, we obtain several sufficient conditions for a functionf∈𝒜 to be in the class 𝒮α,βm(η,γ,λ).Theorem 2.
Letf∈𝒜, and let δ be a real number with 0≤δ<1. If
(18)|z(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′-1|≤1-δ,z∈𝒰,
then f∈𝒮α,βm(η,γ,λ) provided that
(19)|γ|≤cos-1(1-δ1-λ).Proof.
From (18), it follows that
(20)z(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′=1+(1-δ)w(z),
where w(z)∈ℬ. We have
(21)ℜ(eiγz(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′)=ℜ[eiγ(1+(1-δ)w(z))]=cosγ+(1-δ)ℜ(eiγw(z))≥cosγ-(1-δ)|eiγw(z)|>cosγ-(1-δ)≥λcosγ,
provided that |γ|≤cos-1((1-δ)/(1-λ)). Thus, the proof is completed.If in Theorem2 we take δ=1-(1-λ)cosγ, we will obtain the following result.Corollary 3.
Letf∈𝒜. If
(22)|z(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′-1|≤(1-λ)cosγ,z∈𝒰
then f∈𝒮α,βm(η,γ,λ).A sufficient condition for a functionf∈𝒜 to be in the class 𝒮α,βm(η,γ,λ), in terms of coefficients inequality, is obtained in the next theorem.Theorem 4.
If a functionf∈𝒜 given by (1) satisfies the inequality
(23)∑n=2∞‍[(1-η)(n-1)secγ+(1-λ)(1+η(n-1))]∑n=2∞‍×Φn(α,β,m)|an|≤1-λ,
where 0≤η<1, 0≤λ<1, |γ|<π/2, and Φn(α,β,m) is defined by (10), then it belongs to the class 𝒮α,βm(η,γ,λ).Proof.
In virtue of Corollary3, it suffices to show that the condition (22) is satisfied. We have
(24)|z(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′-1|=(1-η)|∑n=2∞(n-1)Φn(α,β,m)anzn-11+∑n=2∞(1-η+ηn)Φn(α,β,m)anzn-1|<(1-η)∑n=2∞(n-1)Φn(α,β,m)|an|1-∑n=2∞(1-η+ηn)Φn(α,β,m)|an|.
The last expression is bounded previously by(1-λ)cosγ, if
(25)∑n=2∞‍(1-η)(n-1)Φn(α,β,m)|an|≤(1-λ)cosγ(1-∑n=2∞‍(1-η+ηn)Φn(α,β,m)|an|),
which is equivalent to
(26)∑n=2∞‍[(1-η)(n-1)secγ+(1-λ)(1+η(n-1))]∑n=2∞‍×Φn(α,β,m)|an|≤1-λ.For special values ofm, η, γ, and λ, from Theorem 4, we can derive the following sufficient conditions for a function f∈𝒜 to be in the classes 𝒮α,β0(η,γ,λ),𝒮α,β0(0,γ,λ)=𝒮*(γ,λ), and 𝒮α,β0(0,γ,0)=𝒮*(γ), respectively.Corollary 5.
Letf∈𝒜. If
(27)∑n=2∞‍[(1-η)(n-1)secγ+(1-λ)(1+η(n-1))]|an|≤1-λ,
where 0≤η<1, 0≤λ<1, and |γ|<π/2, then f∈𝒮α,β0(η,γ,λ).Corollary 6 (see [17]).
Letf∈𝒜. If
(28)∑n=2∞‍[(n-1)secγ+1-λ]|an|≤1-λ,
where 0≤λ<1, |γ|<π/2, then f∈𝒮*(γ,λ).Corollary 7 (see [18]).
Letf∈𝒜. If
(29)∑n=2∞‍[1+(n-1)secγ]|an|≤1,
where |γ|<π/2, then f∈𝒮*(γ).A necessary and sufficient condition for a function to be in the class𝒮α,βm(η,γ,λ) can be given in terms of integral representation.Theorem 8.
A functionf∈𝒜 is in the class 𝒮α,βm(η,γ,λ) if and only if there exists w∈ℬ such that
(30)f(z)=gα,β(-1)(z)*zexp(∫0z‍[pλ,γ(w(ζ))-11-ηpλ,γ(w(ζ))]dζζ),z∈𝒰,
where pλ,γ(z) and gα,β(-1)(z) are defined by (3) and (13), respectively.Proof.
In virtue of (15), f∈𝒮α,βm(η,γ,λ) if and only if there exists w∈ℬ such that
(31)z(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′=pλ,γ(w(z)).
From the last equality, we obtain
(32)Dα,βmf(z)=zexp(∫0z‍[pλ,γ(w(ζ))-11-ηpλ,γ(w(ζ))]dζζ).
Making use of (14) and (32), we have
(33)f(z)=gα,β(-1)(z)*zexp(∫0z‍[pλ,γ(w(ζ))-11-ηpλ,γ(w(ζ))]dζζ),z∈𝒰,
and thus, the proof is completed.For0≤θ≤2π, 0≤τ≤1, define the function
(34)Ψ(z,θ,τ)=gα,β(-1)(z)=*zexp(∫0z‍[pλ,γ(eiθζ(ζ+τ)/(1+τζ))-11-ηpλ,γ(eiθζ(ζ+τ)/(1+τζ))]dζζ),
where pλ,γ(z) and gα,β(-1)(z) are defined by (3) and (13), respectively.In virtue of Theorem8, the function Ψ(z,θ,τ) belongs to the class 𝒮α,βm(η,γ,λ). Note that Ψ(z,0,0) is an odd function.


## 3. The Fekete-Szegö Problem

In order to obtain sharp upper bounds for the Fekete-Szegö functional for the class𝒮α,βm(η,γ,λ), the following lemma is required (see, e.g., [19, page 108]).Lemma 9.
Let the functionw∈ℬ be given by
(35)w(z)=∑n=1∞‍wnzn,z∈𝒰.
Then
(36)|w1|≤1,|w2|≤1-|w1|2,(37)|w2-sw12|≤max{1,|s|},foranycomplexnumbers.
The functionsw(z)=z and w(z)=z2, or one of their rotations, show that both inequalities (36) and (37) are sharp.First we obtain sharp upper bounds for the Fekete-Szegö functional|a3-μa22| with μ real parameter.Theorem 10.
Letf∈𝒮α,βm(η,γ,λ) be given by (1), and let μ be a real number. Then
(38)|a3-μa22|≤{(1-λ)cosγ(1-η)2Φ3(α,β,m)×[4(1-λ)Φ3(α,β,m)Φ22(α,β,m)η+3-2λ(1+η)-μ4(1-λ)Φ3(α,β,m)Φ22(α,β,m)],ifμ≤σ1,(1-λ)cosγ(1-η)Φ3(α,β,m),ifσ1≤μ≤σ2,(1-λ)cosγ(1-η)2Φ3(α,β,m)×[μ4(1-λ)Φ3(α,β,m)Φ22(α,β,m)+2λ(1+η)-η-34(1-λ)Φ3(α,β,m)Φ22(α,β,m)],ifμ≥σ2,
where
(39)σ1=(1+η)Φ22(α,β,m)2Φ3(α,β,m),(40)σ2=2-λ(1+η)1-λΦ22(α,β,m)2Φ3(α,β,m),
and Φ2(α,β,m), Φ3(α,β,m) are defined by (10) with n=2 and n=3, respectively.
All estimates are sharp.Proof.
Suppose thatf∈𝒮α,βm(η,γ,λ) is given by (1). Then, from the definition of the class 𝒮α,βm(η,γ,λ), there exist w∈ℬ, w(z)=w1z+w2z2+w3z3+⋯ such that
(41)z(Dα,βmf(z))′(1-η)Dα,βmf(z)+ηz(Dα,βmf(z))′=pλ,γ(w(z)),z∈𝒰.
Set pλ,γ(z)=1+p1z+p2z2+p3z3+⋯. Equating the coefficients of z and z2 on both sides of (41), we obtain
(42)a2=p1w1(1-η)Φ2(α,β,m),a3=12(1-η)Φ3(α,β,m)[(1+η1-ηp12+p2)w12+p1w2].
From (5), we have p1=p2=2e-iγ(1-λ)cosγ, and thus we obtain
(43)a2=2e-iγ(1-λ)cosγ(1-η)Φ2(α,β,m)w1,a3=e-iγ(1-λ)cosγ(1-η)Φ3(α,β,m)a3=×[(2e-iγ(1-λ)cosγ1+η1-η+1)w12+w2].
It follows that(44)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m)≤×{|2e-iγ(1-λ)cosγ1-η(1+η-μ2Φ3(α,β,m)Φ22(α,β,m))+1|×|w1|2+|w2|2e-iγ(1-λ)cosγ1-η}.
Making use of Lemma9 (36), we have
(45)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m)≤×{1+[|2e-iγ(1-λ)cosγ1-η×(1+η-μ2Φ3(α,β,m)Φ22(α,β,m))+1|-1]×|w1|2[|2e-iγ(1-λ)cosγ1-η}
or
(46)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m)|a3-μa22|≤×[1+(1+M(2+M)cos2γ-1)|w1|2],
where
(47)M=2(1-λ)1-η(1+η-μ2Φ3(α,β,m)Φ22(α,β,m)).
Denote
(48)F(x,y)=1+(1+M(2+M)x2-1)y2,
where x=cosγ, y=|w1|, and (x,y)∈[0,1]×[0,1].
Simple calculation shows that the functionF(x,y) does not have a local maximum at any interior point of the open rectangle (0,1)×(0,1). Thus, the maximum must be attained at a boundary point. Since F(x,0)=1, F(0,y)=1, and F(1,1)=|1+M|, it follows that the maximal value of F(x,y) may be F(0,0)=1 or F(1,1)=|1+M|.
Therefore, from (46), we obtain
(49)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m)max{1,|1+M|},
where M is given by (47).
Consider first the case|1+M|≥1. If μ≤σ1, where σ1 is given by (39), then M≥0, and from (49), we obtain
(50)|a3-μa22|≤(1-λ)cosγ(1-η)2Φ3(α,β,m)≤×[η+3-2λ(1+η)-μ4(1-λ)Φ3(α,β,m)Φ22(α,β,m)],
which is the first part of the inequality (38). If μ≥σ2, where σ2 is given by (40), then M≤-2, and it follows from (49) that
(51)|a3-μa22|≤(1-λ)cosγ(1-η)2Φ3(α,β,m)≤×[μ4(1-λ)Φ3(α,β,m)Φ22(α,β,m)+2λ(1+η)-η-3],
and this is the third part of (38).
Next, suppose thatσ1≤μ≤σ2. Then, |1+M|≤1, and thus, from (49), we obtain
(52)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m),
which is the second part of the inequality (38).
In view of Lemma9, the results are sharp for w(z)=z and w(z)=z2 or one of their rotations. From (41), we obtain that the extremal functions are Ψ(z,θ,1) and Ψ(z,θ,0) defined by (34) with τ=1 and τ=0.Next, we consider the Fekete-Szegö problem for the class𝒮α,βm(η,γ,λ) with μ complex parameter.Theorem 11.
Letf∈𝒮α,βm(η,γ,λ) be given by (1), and let μ be a complex number. Then,
(53)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m)≤×max{1,|2(1-λ)cosγ1-η≤×max===×(μ2Φ3(α,β,m)Φ22(α,β,m)-1-η)-eiγ2(1-λ)cosγ1-η|}.
The result is sharp.Proof.
Assume thatf∈𝒮α,βm(η,γ,λ). Making use of (43), we obtain
(54)|a3-μa22|≤(1-λ)cosγ(1-η)Φ3(α,β,m)≤×|w2-[2e-iγ(1-λ)cosγ1-η|a3-μa22|≤×(μ2Φ3(α,β,m)Φ22(α,β,m)-1-η)-1(μ2Φ3(α,β,m)Φ22(α,β,m)-1-η)]w12|.
The inequality (53) follows as an application of Lemma 9 (37) with
(55)s=2e-iγ(1-λ)cosγ1-η(μ2Φ3(α,β,m)Φ22(α,β,m)-1-η)-1.
The functions Ψ(z,θ,1) and Ψ(z,θ,0) defined by (34) with τ=1 and τ=0 show that the inequality (53) is sharp.Our Theorems10 and 11 include several various results for special values of m, η, γ, and λ. For example, taking m=η=γ=λ=0, in Theorem 10, we obtain the Fekete-Szegö inequalities for the class 𝒮* (see [2, 11]). The special case m=η=λ=0 leads to the Fekete-Szegö inequalities for the class 𝒮*(γ) (see [2]). The Fekete-Szegö inequalities for the class 𝒮*(γ,λ) (see [2]) are also included in Theorems 10 and 11.



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