# Estimates of Initial Coefficients for Bi-Univalent Functions

Authors: Paweł Zaprawa
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider the Fekete-Szegö inequalities for classes which were defined
by Murugusundaramoorthy et al. (2013). These inequalities will result in bounds of the third
coefficient which are better than these obtained by Murugusundaramoorthy et al. (2013). Moreover, we discuss two other classes
of bi-univalent functions. The estimates of initial coefficients in these classes are obtained.


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## Body

## 1. Introduction

LetA denote the class of all functions of the form
(1)f(z)=z+∑n=2∞‍anznanalytic in the unit diskD≡{ζ∈C:|ζ|<1}, and let S denote the class of these functions in A which are univalent. It is known that if f∈S then there exists the inverse function f-1. Because of the normalization f(0)=0, f-1 is defined in some neighbourhood of the origin. In some cases, f-1 can be defined in the whole D. Clearly, f-1 is also univalent. For this reason, the class Σ is defined as follows.A functionf∈A is called bi-univalent in D if both f and f-1 are univalent in D. The set of all bi-univalent functions is usually denoted by Σ (or, following Lewin, by σ).It is easy to check that a bi-univalent functionf given by (1) has the inverse with the Taylor series of the form
(2)f-1(w)=w-a2w2+(2a22-a3)w3+⋯.The research intoΣ was started by Lewin ([1], 1967). It focused on problems connected with coefficients. Many papers concerning bi-univalent functions have been published recently. We owe the revival of these topics to Srivastava et al. ([2], 2010). The investigations in this direction have also been carried out, among others, by Ali et al. [3], Frasin and Aouf [4], and Xu et al. [5]. Hamidi and Jahangiri (e.g., [6]) have revealed the importance of the Faber polynomials in general studies on the coefficients of bi-univalent functions.In fact, little is known about exact bounds of the initial coefficients off∈Σ. For the most general families of functions given by (1) we know that |a2|<1.51 for bi-univalent functions (Lewin, [1]), |a2|≤2 for bi-starlike functions (Kędzierawski, [7]), and |a2|≤1 for bi-convex functions (Brannan and Taha, [8]). Only the last estimate is sharp; equality holds only for f(z)=z/(1-z) and its rotations.In the papers [1–15], the authors present some estimates for a2 and a3, while f is taken from various subclasses of Σ. In 2013 Murugusundaramoorthy et al. (see [16]) obtained some coefficient bounds in two classes: SΣ(α,λ) and MΣ(β,λ). For a function f∈Σ and its inverse function g, let F and G be defined as below:
(3)F(z)≡zf′(z)(1-λ)f(z)+λzf′(z),G(w)≡wg′(w)(1-λ)g(w)+λwg′(w),
where 0≤λ<1, z∈D, and g=f-1.The definitions ofSΣ(α,λ) and MΣ(β,λ) are the following.Definition 1.
A functionf∈Σ is said to be in the class SΣ(α,λ) if the functions F and G, defined by (3), corresponding to f and g=f-1, satisfy
(4)|argF(z)|<απ2,|argG(w)|<απ2,0<α≤1,0≤λ<1,z∈D.Definition 2.
A functionf∈Σ is said to be in the class MΣ(β,λ) if the functions F and G, defined by (3), corresponding to f and g=f-1, satisfy
(5)RF(z)>β,RG(w)>β,0≤β<1,0≤λ<1z∈D.In particular, forλ=0, the classes SΣ(α,λ) and MΣ(β,λ) become the class SΣα of strongly bi-starlike functions of order α and the class SΣ*(β) of bi-starlike functions of order β, respectively. If additionally α=1 or β=0, these two classes reduce to the class SΣ* of bi-starlike functions.Conditions (4) and (5) in the above definitions can be rewritten as follows:
(6)F(z)=[p(z)]α,G(w)=[q(w)]α,(7)F(z)=β+(1-β)p(z),G(w)=β+(1-β)q(w),respectively, wherep and q are functions in P and have the form
(8)p(z)=1+∑k=1∞‍pkzk,q(w)=1+∑k=1∞‍qkwk.Throughout the paper,P stands for the set of all analytic functions h such that h(0)=1 and Rh(z)>0 for z∈D.In [11] the authors proved the following theorems.Theorem 3.
Iff∈SΣ(α,λ), 0<α≤1, and 0≤λ<1, then
(9)|a2|≤2α(1-λ)1+α,|a3|≤4α2(1-λ)2+α1-λ.Theorem 4.
Iff∈MΣ(β,λ), 0≤β<1, and 0≤λ<1, then
(10)|a2|≤2(1-β)1-λ,|a3|≤4(1-β)2(1-λ)2+1-β1-λ.The above results can be improved. In order to do this, we consider the Fekete-Szegö inequalities for the discussed classes. This type of problems has been considered by many authors. The results concerning this problem are given, for example, in [17–20]. Moreover, it seems to be interesting to discuss two other classes defined in a similar way to SΣ(α,λ) and MΣ(β,λ). The results presented in the paper are not sharp, but, unfortunately, no method which gives sharp results with regard to these problems is known.In the proofs of the main theorems we need two lemmas.Lemma 5.
Ifp(z)=1+∑k=1∞‍pkzk∈P then |pk|≤2 for all positive integers k.Lemma 6.
Letk∈R and z1,z2∈C. If |z1|<R and |z2|<R then
(11)|(k+1)z1+(k-1)z2|≤{2|k|Rfor|k|≥1,2Rfor|k|≤1.The proof of Lemma6 is easy. It is enough to observe that
(12)|(k+1)z1+(k-1)z2|≤(|k+1|+|k-1|)R
and to discuss three cases with respect to k.From Lemma6 we immediately obtain the following.Lemma 7.
Letk,l∈R and z1,z2∈C. If |z1|<R and |z2|<R then
(13)|(k+l)z1+(k-l)z2|≤{2|k|Rfor|k|≥|l|,2|l|Rfor|k|≤|l|.


## 2. Results forSΣ(α,λ) and MΣ(β,λ)

Now, we will formulate two theorems concerning the Fekete-Szegö inequalities forSΣ(α,λ) and MΣ(β,λ).Theorem 8.
Iff∈SΣ(α,λ), 0<α≤1, 0≤λ<1, and μ∈R, then
(14)|a3-μa22|≤{4α2(1-λ)2(1+α)|1-μ|for4α|1-μ|≥(1+α)(1-λ),α1-λfor4α|1-μ|≤(1+α)(1-λ).Theorem 9.
Iff∈MΣ(β,λ), 0≤β<1, 0≤λ<1, and μ∈R, then
(15)|a3-μa22|≤{2(1-β)(1-λ)2|1-μ|for2|1-μ|≥1-λ,1-β1-λfor2|1-μ|≤1-λ.Proof of Theorem8.
Letf given by (1) be in SΣ(α,λ) and let 0<α≤1, 0≤λ<1, and μ∈R. From Definition 1 and from (6) we know that
(16)zf′(z)(1-λ)f(z)+λzf′(z)=[p(z)]α,wg′(w)(1-λ)g(w)+λwg′(w)=[q(w)]α,
where p and q are functions in P which have the form (8).
Comparing the coefficients in each equality in (16), it follows that
(17)(1-λ)a2=αp1,(18)2(1-λ)a3-(1-λ2)a22=12α(α-1)p12+αp2,(19)-(1-λ)a2=αq1,(20)-2(1-λ)a3+(λ2-4λ+3)a22=12α(α-1)q12+αq2.
From (17) and (19), there is p1=-q1. Summing and subtracting (18) and (20), we have two equalities
(21)2(1-λ)2a22=12α(α-1)(p12+q12)+α(p2+q2),(22)a3=a22+α4(1-λ)(p2-q2).
Applying (17) and (19) we dispose of p1 and q1 in (21). Hence
(23)a3-μa22=p2[h(α)(1-μ)+α4(1-λ)]+q2[h(α)(1-μ)-α4(1-λ)],
where h(α)=α2/(1-λ)2(1+α) is nonnegative. From Lemmas 5 and 7 we conclude
(24)|a3-μa22|≤{4h(α)|1-μ|forh(α)|1-μ|≥α4(1-λ),α1-λforh(α)|1-μ|≤α4(1-λ).The proof of Theorem9 is similar to that of Theorem 8 and can be omitted. From Theorems 8 and 9 we get the following corollaries.Corollary 10.
Iff∈SΣ(α,λ), 0<α≤1, and 0≤λ<1, then
(25)|a3|≤{4α2(1-λ)2(1+α)for4α≥(1+α)(1-λ)α1-λfor4α≤(1+α)(1-λ).Corollary 11.
Iff∈MΣ(β,λ), 0≤β<1, and 0≤λ<1, then
(26)|a3|≤2(1-β)(1-λ)2.The result in Corollary10 improves the corresponding result in Theorem 3. Similarly, for 0≤β≤1/2 the bound in Corollary 11 is better that the one obtained in Theorem 4.Ifλ=0 we get the bounds for SΣα and SΣ*(β) which are better than these obtained in [3, 11]. It is worth mentioning that recently Hamidi and Jahangiri ([6]) and Srivastava et al. ([13]) have provided an improvement of the result from Corollaries 10 and 11.If additionallyα=1, we obtain that |a3|≤2 for the class SΣ* of bi-starlike functions (see [3, 6, 13]).


## 3. Results forKΣ(α,λ) and NΣ(β,λ)

To begin with, we can observe that the operators which were used by Murugusundaramoorthy et al. in the definitions ofSΣ(α,λ) and MΣ(β,λ) can be written as the weighted harmonic mean of two expressions: zf′(z)/f(z) and 1; that is,
(27)F(z)=[1-λzf′(z)/f(z)+λ1]-1,G(w)=[1-λwg′(w)/g(w)+λ1]-1,
where g=f-1.Let us define two new classes. In definitions ofF and G we consider the weighted harmonic mean of 1+zf′′(z)/f′(z) and zf′(z)/f(z); namely,(28)F(z)=[1-λzf′(z)/f(z)+λ1+zf′′(z)/f′(z)]-1,G(w)=[1-λwg′(w)/g(w)+λ1+wg′′(w)/g′(w)]-1,
where 0≤λ≤1, z∈D, and g=f-1. In fact, in the above functions the range of λ can be extended to the set [0,∞).Now, we can define the classesKΣ(α,λ) and NΣ(β,λ).Definition 12.
A functionf∈Σ is said to be in the class KΣ(α,λ) if the functions F and G, defined by (28), corresponding to f and g=f-1, satisfy
(29)|argF(z)|<απ2,|argG(w)|<απ2,0<α≤1,λ≥0,z∈D.Definition 13.
A functionf∈Σ is said to be in the class NΣ(β,λ) if the functions F and G, defined by (28), corresponding to f and g=f-1, satisfy
(30)RF(z)>β,RG(w)>β,0≤β<1,λ≥0,z∈D.The idea of considering the weighted mean of1+zf′′(z)/f′(z) and zf′(z)/f(z) first appeared in the paper by Miller et al. (see [16]). They did their research into the class of so-called α-convex functions defined as the arithmetic weighted mean of the expressions mentioned above.Now we are ready to establish the main theorems of this section.Theorem 14.
Iff∈KΣ(α,λ), 0<α≤1, λ≥0, and μ∈R, then (1)
(31)|a2|≤2αα(1-λ)2+(1+λ)2,(2)
(32)|a3-μa22|≤{4α2α(1-λ)2+(1+λ)2|1-μ|for4α(1+2λ)|1-μ|≥α(1-λ)2+(1+λ)2α1+2λfor4α(1+2λ)|1-μ|≤α(1-λ)2+(1+λ)2.Theorem 15.
Iff∈NΣ(β,λ), 0≤β<1, λ≥0, and μ∈R, then (1)
(33)|a2|≤{2(1-β)1+λforβ≥(1-λ)22(1+λ2),2(1-β)1+λ2forβ≤(1-λ)22(1+λ2),(2)
(34)|a3-μa22|≤{2(1-β)1+λ2|1-μ|for2(1+2λ)|1-μ|≥1+λ2,1-β1+2λfor2(1+2λ)|1-μ|≤1+λ2.Proof of Theorem14.
Assume that0<α≤1, λ≥0, and μ∈R. From Definition 12 it follows that if f∈KΣ(α,λ) then
(35)zf′(z)[f′(z)+zf′′(z)]λz(f′(z))2+(1-λ)f(z)[f′(z)+zf′′(z)]=[p(z)]α,wg′(z)[g′(w)+wg′′(z)]λw(g′(z))2+(1-λ)g(w)[g′(w)+wg′′(w)]=[q(w)]α,
where p and q are functions in P and have the form (8).
Hence, comparing the coefficients in each equality in (35), we can write
(36)(1+λ)a2=αp1,(37)2(1+2λ)a3+4a22-(5-λ)a2αp1=12α(α-1)p12+αp2,(38)-(1+λ)a2=αq1,(39)-2(1+2λ)a3+8(1+λ)a22+(5-λ)a2αq1=12α(α-1)q12+αq2.
From (36) and (38), it yields that p1=-q1. Putting (36) into (37) and (38) into (39), we obtain
(40)2(1+2λ)a3+(λ2-4λ-1)a22=12α(α-1)p12+αp2-2(1+2λ)a3+(λ2+4λ+3)a22=12α(α-1)q12+αq2.
Now, summing and subtracting (40) we have two equalities
(41)2(1+λ2)a22=12α(α-1)(p12+q12)+α(p2+q2),(42)a3=a22+α4(1+2λ)(p2-q2).
Substituting in (41) p1 and q1 taken from (36) and (38), we get
(43)a22=α2α(1-λ)2+(1+λ)2(p2+q2).
By Lemma5, the first part of our assertion follows.
From (41) and (42),
(44)a3-μa22=p2[α2α(1-λ)2+(1+λ)2(1-μ)+α4(1+2λ)]+q2[α2α(1-λ)2+(1+λ)2(1-μ)-α4(1+2λ)].
Applying Lemmas5 and 7 completes the second part of the assertion.Theorem14 gives the following corollaries.Corollary 16.
Iff∈KΣ(α,λ), 0<α≤1, and λ≥0, then
(45)|a3|≤{4α2α(1-λ)2+(1+λ)2for4α(1+2λ)≥α(1-λ)2+(1+λ)2,α1+2λfor4α(1+2λ)≤α(1-λ)2+(1+λ)2.Forλ=0 we obtain the bounds for SΣα. For λ=1, the class KΣ(α,λ) reduces to the class KΣα of strongly bi-convex functions of order α. Hence we have the following.Corollary 17.
Iff∈KΣα and 0<α≤1, then(1)
(46)|a2|≤α,(2)
(47)|a3|≤{α3forα∈(0,13],α2forα∈[13,1].This result improves the result given in [3].Proof of Theorem15.
Letf∈NΣ(α,λ) with 0≤β<1 and λ≥0. From Definition 13 we obtain
(48)zf′(z)[f′(z)+zf′′(z)]λz(f′(z))2+(1-λ)f(z)[f′(z)+zf′′(z)]=β+(1-β)p(z),wg′(z)[g′(w)+wg′′(z)]λw(g′(z))2+(1-λ)g(w)[g′(w)+wg′′(w)]=β+(1-β)q(w),
where p,q∈P.
Hence,(49)(1+λ)a2=(1-β)p1,(50)2(1+2λ)a3+4a22-(5-λ)(1-β)a2p1=(1-β)p2,(51)-(1+λ)a2=(1-β)q1,(52)2(1+2λ)(2a22-a3)+4a22+(5-λ)(1-β)a2q1=(1-β)q2.
Applying the same method as in the proof of Theorem15 we get
(53)a22=1-β2(1+λ2)(p2+q2),(54)a3=a22+1-β4(1+2λ)(p2-q2).
Observe that the estimate for|a2| obtained from (53), that is, |a2|≤2(1-β)/(1+λ2), is not always better than the estimate which follows directly from (49), that is, |a2|≤2(1-β)/(1+λ). The comparison of these two bounds completes the first part of the proof.From (53) and (54),
(55)a3-μa22=p2[(1-β)(1-μ)2(1+λ2)+1-β4(1+2λ)]mmm+q2[(1-β)(1-μ)2(1+λ2)-1-β4(1+2λ)],which results in the second part of the assertion.From Theorem15 we get the following corollaries.Corollary 18.
Iff∈NΣ(β,λ), 0≤β<1, and λ≥0, then |a3|≤2(1-β)/(1+λ2).Forλ=0 we obtain the bounds for SΣ*(β). For λ=1, the set NΣ(β,λ) is the class KΣ(β) of bi-convex functions of order β. Hence we have the following.Corollary 19.
Iff∈KΣ(β) and 0≤β<1, then (1)
(56)|a2|≤1-β,(2)
(57)|a3|≤1-β.This bound is better than the one proved in [3].



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