# Fekete-Szegö Problems for Quasi-Subordination Classes

Authors: Maisarah Haji Mohd; Maslina Darus
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

An analytic functionf is quasi-subordinate to an analytic function g, in the open unit disk if there exist analytic functions φ and w, with |φ(z)|≤1, w(0)=0 and |w(z)|<1 such that f(z)=φ(z)g(w(z)). Certain subclasses of analytic univalent functions associated with quasi-subordination are defined and the bounds for the Fekete-Szegö coefficient functional |a3-μa22| for functions belonging to these subclasses are derived.


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## Body

## 1. Introduction and Motivation

Let𝒜 be the class of analytic function f in the open unit disk 𝔻={z:|z|<1} normalized by f(0)=0 and f′(0)=1 of the form f(z)=z+∑n=2∞‍anzn. For two analytic functions f and g, the function f is subordinate to g, written as follows:
(1.1)f(z)≺g(z),
if there exists an analytic function w, with w(0)=0 and |w(z)|<1 such that f(z)=g(w(z)). In particular, if the function g is univalent in 𝔻, then f(z)≺g(z) is equivalent to f(0)=g(0) and f(𝔻)⊂g(𝔻). For brief survey on the concept of subordination, see [1].Ma and Minda [2] introduced the following class
(1.2)𝒮*(ϕ)={f∈𝒜:zf′(z)f(z)≺ϕ(z)},
where ϕ is an analytic function with positive real part in 𝔻, ϕ(𝔻) is symmetric with respect to the real axis and starlike with respect to ϕ(0)=1 and ϕ′(0)>0. A function f∈𝒮*(ϕ) is called Ma-Minda starlike (with respect to ϕ). The class 𝒞(ϕ) is the class of functions f∈𝒜 for which 1+zf′′(z)/f′(z)≺ϕ(z). The class 𝒮*(ϕ) and 𝒞(ϕ) include several well-known subclasses of starlike and convex functions as special case.In the year 1970, Robertson [3] introduced the concept of quasi-subordination. For two analytic functions f and g, the function f is quasi-subordinate to g, written as follows:
(1.3)f(z)≺qg(z),
if there exist analytic functions φ and w, with |φ(z)|≤1, w(0)=0 and |w(z)|<1 such that f(z)=φ(z)g(w(z)). Observe that when φ(z)=1, then f(z)=g(w(z)), so that f(z)≺g(z) in 𝔻. Also notice that if w(z)=z, then f(z)=φ(z)g(z) and it is said that f is majorized by g and written f(z)≪g(z) in 𝔻. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization. See [4–6] for works related to quasi-subordination.Throughout this paper it is assumed thatϕ is analytic in 𝔻 with ϕ(0)=1. Motivated by [2, 3], we define the following classes.Definition 1.1.
Let the class𝒮q*(ϕ) consists of functions f∈𝒜 satisfying the quasi-subordination
(1.4)zf′(z)f(z)-1≺qϕ(z)-1.Example 1.2.
Since(1.5)zf′(z)f(z)-1=z(ϕ(z)-1)≺qϕ(z)-1,
the function f:𝔻→ℂ defined by  the following:
(1.6)f(z)=zexp(-z+∫0z‍ϕ(ξ)dξ)
belongs to the class 𝒮q*(ϕ).Definition 1.3.
Let the class𝒞q(ϕ) consists of functions f∈𝒜 satisfying the quasi-subordination
(1.7)zf′′(z)f′(z)≺qϕ(z)-1.Example 1.4.
The functionf:𝔻→ℂ defined by  the following:
(1.8)f(z)=∫0z‍exp(-ζ+∫0ζ‍ϕ(ξ)dξ)dζ
belongs to the class 𝒞q(ϕ).The classes𝒮q*(ϕ) and 𝒞q(ϕ) are analogous to the Ma-Minda starlike and convex classes defined in the form of quasi-subordination.Definition 1.5.
Let the classℛq(ϕ) consist of functions f∈𝒜 satisfying the quasi-subordination
(1.9)f′(z)-1≺qϕ(z)-1.Example 1.6.
The functionf:𝔻→ℂ defined by the following:
(1.10)f(z)=z-z22+exp(∫0z‍ϕ(ξ)dξ)
belongs to the class ℛq(ϕ).It is known that a functionf∈𝒜 with Ref'(z)>0 in 𝔻 is univalent. The above class of functions defined in terms of the quasi-subordination is associated with the class of functions with positive real part.Functions in the following classes,ℳq(α,ϕ) and ℒq(α,ϕ) are analogous to the α-convex functions of Miller et al. [7] and α-logarithmically convex functions introduced by Lewandowski et al. [8] (see also [9]), respectively.Definition 1.7.
Let the classℳq(α,ϕ), (α≥0) consist of functions f∈𝒜 satisfying the quasi-subordination
(1.11)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))-1≺qϕ(z)-1.Example 1.8.
The functionf:𝔻→ℂ defined by the following:
(1.12)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))-1=z(ϕ(z)-1)
belongs to the class ℳq(ϕ).Definition 1.9.
Let the classℒq(α,ϕ), (α≥0) consist of functions f∈𝒜 satisfying the quasi-subordination
(1.13)(zf′(z)f(z))α(1+zf′′(z)f′(z))1-α-1≺qϕ(z)-1.Example 1.10.
The functionf:𝔻→ℂ defined by the following:
(1.14)(zf′(z)f(z))α(1+zf′′(z)f′(z))1-α-1=z(ϕ(z)-1)
belongs to the class ℒq(ϕ).It is well known (see [10]) that the n-th coefficient of a univalent function f∈𝒜 is bounded by n. The bounds for coefficient give information about various geometric properties of the function. Many authors have also investigated the bounds for the Fekete-Szegö coefficient for various classes [11–25]. In this paper, we obtain coefficient estimates for the functions in the above defined classes.LetΩ be the class of analytic functions w, normalized by w(0)=0, and satisfying the condition |w(z)|<1. We need the following lemma to prove our results.Lemma 1.11 (see [26]).
Ifw∈Ω, then for any complex number t(1.15)|w2-tw12|≤max{1;|t|}.
The result is sharp for the functions w(z)=z2 or w(z)=z.


## 2. Main Results

Although Theorems2.1 and 2.4 are contained in the corresponding results for the classes ℳq(α,ϕ) and ℒq(α,ϕ), they are stated and proved separately here because of the importance of the classes.Throughout, letf(z)=z+a2z2+a3z3+⋯,ϕ(z)=1+B1z+B2z2+B3z3+⋯,  φ(z)=c0+c1z+c2z2+c3z3+⋯,  B1∈ℝ and B1>0.Theorem 2.1.
Iff∈𝒜 belongs to 𝒮q*(ϕ), then
(2.1)|a2|≤B1,|a3|≤12(B1+max{B1,B12+|B2|}),
and, for any complex number μ,
(2.2)|a3-μa22|≤12(B1+max{B1,|1-2μ|B12+|B2|}).Proof.
Iff∈𝒮q*(ϕ), then there exist analytic functions φ and w, with |φ(z)|≤1,  w(0)=0 and |w(z)|<1 such that
(2.3)zf′(z)f(z)-1=φ(z)(ϕ(w(z))-1).
Since(2.4)zf′(z)f(z)-1=a2z+(-a22+2a3)z2+⋯,ϕ(w(z))-1=B1w1z+(B1w2+B2w12)z2+⋯,(2.5)φ(z)(ϕ(w(z))-1)=B1c0w1z+(B1c1w1+c0(B1w2+B2w12))z2+⋯,
it follows from (2.3) that
(2.6)a2=B1c0w1a3=12(B1c1w1+B1c0w2+c0(B2+B12c0)w12).
Since φ(z) is analytic and bounded in 𝔻, we have [27, page 172]
(2.7)|cn|≤1-|c0|2≤1(n>0).
By using this fact and the well-known inequality, |w1|≤1, we get
(2.8)|a2|≤B1.
Further,
(2.9)a3-μa22=12(B1c1w1+c0(B1w2+(B2+B12c0-2μB12c0)w12)).
Then
(2.10)|a3-μa22|≤12(|B1c1w1|+|B1c0(w2-(2μB1c0-B1c0-B2B1)w12)|).
Again applying |cn|≤1 and |w1|≤1, we have
(2.11)|a3-μa22|≤B12(1+|w2-(-(1-2μ)B1c0-B2B1)w12|).
Applying Lemma 1.11 to
(2.12)|w2-(-(1-2μ)B1c0-B2B1)w12|
yields
(2.13)|a3-μa22|≤B12(1+max{1,|-(1-2μ)B1c0-B2B1|}).
Observe that
(2.14)|-(1-2μ)B1c0-B2B1|≤B1|c0||1-2μ|+|B2B1|,
and hence we can conclude that
(2.15)|a3-μa22|≤B12(1+max{1,B1|1-2μ|+|B2B1|}).
For μ=0, the above will reduce to the estimate of |a3|.Remark 2.2.
Forφ(z)≡1, Theorem 2.1 gives a particular case of the estimates in [13, Theorem 1] for p=1 and [14, Theorem 2.1] for k=1.Theorem 2.3.
Iff∈𝒜 satisfies
(2.16)zf′(z)f(z)-1≪ϕ(z)-1,
then the following inequalities hold:
(2.17)|a2|≤B1,|a3|≤12(B1+B12+|B2|),
and, for any complex number μ,
(2.18)|a3-μa22|≤12(B1+|1-2μ|B12+|B2|).Proof.
The result follows by takingw(z)=z in the proof of Theorem 2.1.Theorem 2.4.
Iff∈𝒜 belongs to 𝒞q(ϕ), then
(2.19)|a2|≤B12,|a3|≤16(B1+max{B1,B12+|B2|}),
and, for any complex number μ,
(2.20)|a3-μa22|≤16(B1+max{B1,|1-32μ|B12+|B2|}).Proof.
Observe that whenzf'∈𝒮q*, equality (2.3) becomes
(2.21)z(zf′(z))′zf′(z)-1=φ(z)(ϕ(w(z))-1),
or equally
(2.22)zf′′(z)f′(z)≺ϕ(w(z))-1,
and the converse can be verified easily. By the Alexander relation, that is f∈𝒞q if and only if zf'∈𝒮q*, we can obtain the required estimates.Theorem 2.5.
Iff∈𝒜 satisfies
(2.23)zf′′(z)f′(z)≪ϕ(z)-1,
then the following inequalities hold:
(2.24)|a2|≤B12,|a3|≤16(B1+B12+|B2|),
and, for any complex number μ,
(2.25)|a3-μa22|≤16(B1+|1-32μ|B12+|B2|).Theorem 2.6.
Iff∈𝒜 belongs to ℛq(ϕ), then
(2.26)|a2|≤B12,|a3|≤13(B1+max{B1,|B2|}),
and, for any complex number μ,
(2.27)|a3-μa22|≤13(B1+max{B1,34|μ|B12+|B2|}).Proof.
Forf∈ℛq(ϕ), we know that by Definition 1.5 there exist analytic functions φ and w, with |φ(z)|≤1, w(0)=0 and |w(z)|<1 such that
(2.28)f′(z)-1=φ(z)(ϕ(w(z))-1).
Since
(2.29)f′(z)-1=2a2z+3a3z2+⋯,
it follows from (2.28) and (2.5) that
(2.30)a2=12B1c0w1,a3=13(B1c1w1+c0(B1w2+B2w12)).
Following the same argument as in Theorem 2.1, where |c0|≤1 and |c1|≤1, we can deduce that
(2.31)|a2|≤B12,|a3-μa22|≤B13(1+|w2-(3B1c04μ-B2B1)w12|).
Applying Lemma 1.11, we get
(2.32)|a3-μa22|≤B13(1+max{1,|3B1c04μ-B2B1|}).
Since
(2.33)|3B1c04μ-B2B1|≤3B14|μ||c0|+|B2B1|,
and |c0|≤1 we can conclude the hypothesis.Theorem 2.7.
Iff∈𝒜 satisfies
(2.34)f′(z)-1≪ϕ(z)-1,
then the following inequalities hold:
(2.35)|a2|≤B12,|a3|≤13(B1+|B2|),
and, for any complex number μ,
(2.36)|a3-μa22|≤13(B1+34|μ|B12+|B2|).Let the classℛqρ(ϕ) consist of functions f∈𝒜 satisfying the quasi-subordination
(2.37)1ρ(f′(z)-1)≺qϕ(z)-1,
where ρ∈ℂ∖{0}. The following corollary gives the results for f∈ℛqρ(ϕ).Corollary 2.8.
Letρ∈ℂ∖{0}. If f∈𝒜 belongs to ℛqρ(ϕ), then
(2.38)|a2|≤|ρ|2B1,|a3|≤|ρ|3(B1+max{B1,|B2|}),
and, for any complex number μ,
(2.39)|a3-μa22|≤|ρ|3(B1+max{B1,34|μρ|B12+|B2|}).Remark 2.9.
(1) Forφ(z)≡1, Corollary 2.8 gives a particular case of the estimates in [13, Theorem 3] for p=1 and [14, Theorem 2.3] for k=1.(2) Forφ(z)≡1 and ϕ(z)=(1+Az)/(1+Bz), (-1≤B<A≤1), Corollary 2.8 reduces to the results in [19, Theorem 4].Theorem 2.10.
Letα≥0. If f∈𝒜 belongs to ℳq(α,ϕ), then
(2.40)|a2|≤B11+α,|a3|≤12(1+2α)(B1+max{B1,1+3α(1+α)2B12+|B2|}),
and, for any complex number μ,
(2.41)|a3-μa22|≤12(1+2α)(B1+max{B1,|2μ(1+2α)-(1+3α)|(1+α)2B12+|B2|}).Proof.
Iff∈ℳq(α,ϕ), for α≥0 then there are analytic functions φ and w, with |φ(z)|≤1, w(0)=0 and |w(z)|<1 such that
(2.42)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))-1=φ(z)(ϕ(w(z))-1).
A computation shows that
(2.43)(1-α)zf′(z)f(z)=(1-α)+(1-α)a2z+(1-α)(-a22+2a3)z2+⋯,α(1+zf′′(z)f′(z))=α+2αa2z+2α(-2a22+3a3)z2+⋯.
Hence from (2.43), we have
(2.44)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))-1=(1+α)a2z+(-(1+3α)a22+2(1+2α)a3)z2+⋯,
It then follows from relation (2.42) and (2.5) that
(2.45)a2=B1c0w11+α,a3=12(1+2α)(B1c1w1+B1c0w2+(B2c0+1+3α(1+α)2B12c02)w12).
We can then conclude the proof by proceeding similarly as previous theorems.Remark 2.11.
(1) Whenα=0, Theorem 2.10 reduces to Theorem 2.1.
(2) Whenα=1, Theorem 2.10 reduces to Theorem 2.4.
(3) Forφ(z)≡1, Theorem 2.10 gives a particular case of the estimates in [14, Theorem 2.9] for k=1.Theorem 2.12.
Letα≥0. If f∈𝒜 satisfies
(2.46)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))-1≪ϕ(z)-1,
then the following inequalities hold:
(2.47)|a2|≤B11+α,|a3|≤12(1+2α)(B1+1+3α(1+α)2B12+|B2|),
and, for any complex number μ,
(2.48)|a3-μa22|≤12(1+2α)(B1+|2μ(1+2α)-(1+3α)|(1+α)2B12+|B2|).Theorem 2.13.
Letα≥0 and β=1-α. If f∈𝒜 belongs to ℒq(α,ϕ), then
(2.49)|a2|≤B1|α+2β|,|a3|≤12|α+3β|(B1+max{B1,|(α+2β)2-3(α+4β)|2(α+2β)2B12+|B2|}),
and, for any complex number μ,
(2.50)|a3-μa22|≤12|α+3β|(B1+max{B1,|(α+2β)2-3(α+4β)-4μ(α+3β)|2(α+2β)2B12+|B2|}).Proof.
Iff∈ℒq(α,ϕ), for α≥0 and β=1-α then there are analytic functions φ and w, with |φ(z)|≤1, w(0)=0 and |w(z)|<1 such that
(2.51)(zf′(z)f(z))α(1+zf′′(z)f′(z))β-1=φ(z)(ϕ(w(z))-1).
A computation shows that
(2.52)(zf′(z)f(z))α=1+αa2z+12((α2-3α)a22+4αa3)z2+⋯,(1+zf′′(z)f′(z))β=1+2βa2z+(2(β2-3β)a22+6βa3)z2+⋯.
Thus (2.52) give
(2.53)(zf′(z)f(z))α(1+zf′′(z)f′(z))β-1=(α+2β)a2z+12(((α+2β)2-3(α+4β))a22+4(α+3β)a3)z2+⋯,
By using the above equation and (2.5) in (2.51) we have
(2.54)a2=B1c0w1α+2βa3=B12(α+3β)(B1c1w1+B1c0w2+(B2c0-(α+2β)2-3(α+4β)2(α+2β)2B12c02)w12).
We can proceed similarly as previous theorems and proof the hypothesis.Remark 2.14.
(1) Whenα=0, Theorem 2.13 reduces to Theorem 2.4.
(2) Whenα=1, Theorem 2.13 reduces to Theorem 2.1.
(3) Forφ(z)≡1, Theorem 2.13 gives a particular case of the estimates in [14, Theorem 2.7] for k=1.Theorem 2.15.
Letα≥0 and β=1-α. If f∈𝒜 satisfies
(2.55)(zf′(z)f(z))α(1+zf′′(z)f′(z))1-α-1≪ϕ(z)-1,
then the following inequalities hold:
(2.56)|a2|≤B1|α+2β|,|a3|≤12|α+3β|(B1+|(α+2β)2-3(α+4β)|2(α+2β)2B12+|B2|),
and, for any complex number μ,
(2.57)|a3-μa22|≤12|α+3β|(B1+|(α+2β)2-3(α+4β)-4μ(α+3β)|2(α+2β)2B12+|B2|).



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