# Initial Coefficients of Biunivalent Functions

Authors: See Keong Lee; V. Ravichandran; Shamani Supramaniam
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

An analytic functionf defined on the open unit disk is biunivalent if the function f and its inverse f-1 are univalent in 𝔻. Estimates for the initial coefficients of biunivalent functions f are investigated when f and f-1, respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.


---

## Body

## 1. Introduction

Let𝒮 be the class of all univalent analytic functions f in the open unit disk 𝔻:={z∈ℂ:|z|<1} and normalized by the conditions f(0)=0 and f′(0)=1. For f∈𝒮, it is well known that the nth coefficient is bounded by n. The bounds for the coefficients give information about the geometric properties of these functions. Indeed, the bound for the second coefficient of functions in the class 𝒮 gives rise to the growth, distortion and covering theorems for univalent functions. In view of the influence of the second coefficient in the geometric properties of univalent functions, it is important to know the bounds for the (initial) coefficients of functions belonging to various subclasses of univalent functions. In this paper, we investigate this coefficient problem for certain subclasses of biunivalent functions.Recall that the Koebe one-quarter theorem [1] ensures that the image of 𝔻 under every univalent function f∈𝒮 contains a disk of radius 1/4. Thus, every univalent function f has an inverse f-1 satisfying f-1(f(z))=z, (z∈𝔻), and
(1)f(f-1(w))=w,(|w|<r0(f),r0(f)≥14).
A function f∈𝒮 isbiunivalent in 𝔻 if both f and f-1 are univalent in 𝔻. Let σ denote the class of biunivalent functions defined in the unit disk 𝔻. Lewin [2] investigated this class σ and obtained the bound for the second coefficient of the biunivalent functions. Several authors subsequently studied similar problems in this direction (see [3, 4]). A function f∈σ is bistarlike or strongly bistarlike or biconvex of order α if f and f-1 are both starlike, strongly starlike, or convex of order α, respectively. Brannan and Taha [5] obtained estimates for the initial coefficients of bistarlike, strongly bistarlike, and biconvex functions. Bounds for the initial coefficients of several classes of functions were also investigated in [6–24].An analytic functionf issubordinate to an analytic function g, written f(z)≺g(z), if there is an analytic function w:𝔻→𝔻 with w(0)=0 satisfying f(z)=g(w(z)). Ma and Minda [25] unified various subclasses of starlike (𝒮*) and convex functions (𝒞) by requiring that either the quantity zf′(z)/f(z) or 1+zf′′(z)/f′(z) is subordinate to a more general superordinate function φ with positive real part in the unit disk 𝔻, φ(0)=1, φ′(0)>0, φ maps 𝔻 onto a region starlike with respect to 1 and symmetric with respect to the real axis. The class 𝒮*(φ) of Ma-Minda starlike functions with respect to φ consists of functions f∈𝒮 satisfying the subordination zf′(z)/f(z)≺φ(z). Similarly, the class 𝒞(φ) of Ma-Minda convex functions consists of functions f∈𝒮 satisfying the subordination 1+zf′′(z)/f′(z)≺φ(z). Ma and Minda investigated growth and distortion properties of functions in 𝒮*(φ) and 𝒞(φ) as well as Fekete-Szegö inequalities for 𝒮*(φ) and 𝒞(φ). Their proof of Fekete-Szegö inequalities requires the univalence of φ. Ali et al. [7] investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of φ. In particular, their results are valid even if one just assumes the function φ to have a series expansion of the form φ(z)=1+B1z+B2z2+⋯,  B1>0. So, in this paper, we assume that φ has series expansion φ(z)=1+B1z+B2z2+⋯, B1, B2 are real, and B1>0. A function f is Ma-Minda bistarlike or Ma-Minda biconvex if both f and f-1 are, respectively, Ma-Minda starlike or convex. Motivated by the Fekete-Szegö problem for the classes of Ma-Minda starlike and Ma-Minda convex functions [25], Ali et al. [26] recently obtained estimates of the initial coefficients for biunivalent Ma-Minda starlike and Ma-Minda convex functions.The present work is motivated by the results of Kędzierawski [27] who considered functions f belonging to certain subclasses of univalent functions while their inverses f-1 belong to some other subclasses of univalent functions. Among other results, he obtained the following coefficient estimates.Theorem 1 (see [27]).
Letf∈σ with Taylor series f(z)=z+a2z2+⋯ and g=f-1. Then,
(2)|a2|≤{1.5894iff∈𝒮,g∈𝒮,2iff∈𝒮*,g∈𝒮*,1.507iff∈𝒮*,g∈𝒮,1.224iff∈𝒞,g∈𝒮.We need the following classes investigated in [6, 7, 26].Definition 2.
Letφ:𝔻→ℂ be analytic and φ(z)=1+B1z+B2z2+⋯ with B1>0 and B2∈ℝ. For α≥0, let
(3)ℳ(α,φ):={f∈𝒮:(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))≺φ(z)},ℒ(α,φ):={f∈𝒮:(zf′(z)f(z))α(1+zf′′(z)f′(z))1-α≺φ(z)},𝒫(α,φ):={f∈𝒮:zf′(z)f(z)+αz2f′′(z)f(z)≺φ(z)}.In this paper, we obtain the estimates for the second and third coefficients of functionsf when(i)
f∈𝒫(α,φ) and g:=f-1∈𝒫(β,ψ), or g∈ℳ(β,ψ), or g∈ℒ(β,ψ),(ii)
f∈ℳ(α,φ) and g∈ℳ(β,ψ), or g∈ℒ(β,ψ),(iii)
f∈ℒ(α,φ) and g∈ℒ(β,ψ).


## 2. Coefficient Estimates

In the sequel, it is assumed thatφ and ψ are analytic functions of the form
(4)φ(z)=1+B1z+B2z2+B3z3+⋯(B1>0),ψ(z)=1+D1z+D2z2+D3z3+⋯(D1>0).Theorem 3.
Letf∈σ and g=f-1. If f∈𝒫(α,φ), g∈𝒫(β,ψ) and f is of the form
(5)f(z)=z+∑n=2∞‍anzn,
then
(6)|a2|≤(B1D1B1(1+3β)+D1(1+3α))×(|σB12D12-(1+2α)2(1+3β)(B2-B1)D12-(1+2β)2(1+3α)(D2-D1)B12|)-1/2,(7)2σ|a3|≤B1(3+10β)+D1(1+2α)+(3+10β)|B2-B1|+(1+2β)2B12|D2-D1|D12(1+2α),
where σ:=2+7α+7β+24αβ.Proof.
Sincef∈𝒫(α,φ) and g∈𝒫(β,ψ), g=f-1, then there exist analytic functions u,v:𝔻→𝔻, with u(0)=v(0)=0, satisfying
(8)zf′(z)f(z)+αz2f′′(z)f(z)=φ(u(z)),wg′(w)g(w)+βw2g′′(w)g(w)=ψ(v(w)).
Define the functions p1 and p2 by
(9)p1(z):=1+u(z)1-u(z)=1+c1z+c2z2+⋯,p2(z):=1+v(z)1-v(z)=1+b1z+b2z2+⋯,
or, equivalently,
(10)u(z)=p1(z)-1p1(z)+1=12(c1z+(c2-c122)z2+⋯),v(z)=p2(z)-1p2(z)+1=12(b1z+(b2-b122)z2+⋯).
Then, p1 and p2 are analytic in 𝔻 with p1(0)=1=p2(0). Since u,v:𝔻→𝔻, the functions p1 and p2 have positive real part in 𝔻, and |bi|≤2 and |ci|≤2. In view of (8) and (10), it is clear that
(11)zf′(z)f(z)+αz2f′′(z)f(z)=φ(p1(z)-1p1(z)+1),wg′(w)g(w)+βw2g′′(w)g(w)=ψ(p2(w)-1p2(w)+1).
Using (10) together with (4), it is evident that
(12)φ(p1(z)-1p1(z)+1)=1+12B1c1z+(12B1(c2-c122)+14B2c12)z2+⋯,ψ(p2(w)-1p2(w)+1)=1+12D1b1w+(12D1(b2-b122)+14D2b12)w2+⋯.
Since f has the Maclaurin series given by (5), a computation shows that its inverse g=f-1 has the expansion
(13)g(w)=f-1(w)=w-a2w2+(2a22-a3)w3+⋯.
Since
(14)zf′(z)f(z)+αz2f′′(z)f(z)=1+a2(1+2α)z+(2(1+3α)a3-(1+2α)a22)z2+⋯,wg′(w)g(w)+βw2g′′(w)g(w)=1-(1+2β)a2w+((3+10β)a22-2(1+3β)a3)w2+⋯,
it follows from (11) and (12) that
(15)a2(1+2α)=12B1c1,(16)2(1+3α)a3-(1+2α)a22=12B1(c2-c122)+14B2c12,(17)-(1+2β)a2=12D1b1,(18)(3+10β)a22-2(1+3β)a3=12D1(b2-b122)+14D2b12.
It follows from (15) and (17) that
(19)b1=-B1(1+2β)D1(1+2α)c1.
Equations (15), (16), (18), and (19) lead to
(20)a22=B12D12[B1(1+3β)c2+D1(1+3α)b2]×(2[σB12D12-(1+2α)2(1+3β)(B2-B1)D12-(1+2β)2(1+3α)(D2-D1)B12])-1,
where σ:=2+7α+7β+24αβ, which, in view of |b2|≤2 and |c2|≤2, gives us the desired estimate on |a2| as asserted in (6).
By using (16), (18), and (19), we get
(21)2σa3=12[B1(3+10β)c2+D1(1+2α)b2]+c124[(3+10β)(B2-B1)+(1+2β)2B12(D2-D1)D12(1+2α)],
and this yields the estimate given in (7).Remark 4.
Whenα=β=0 and B1=B2=2, D1=D2=2, then (6) reduces to Theorem 1. When β=α and ψ=φ, Theorem 3 reduces to [26, Theorem 2.2].Theorem 5.
Letf∈σ and g=f-1. If f∈𝒫(α,φ) and g∈ℳ(β,ψ), then
(22)|a2|≤(B1D1B1(1+2β)+D1(1+3α))×(|(1+β)2(1+3α)(D2-D1)B12σB12D12-(1+2α)2(1+2β)(B2-B1)D12-(1+β)2(1+3α)(D2-D1)B12|)-1/2,(23)2σ|a3|≤B1(3+5β)+D1(1+2α)+(3+5β)|B2-B1|+(1+β)2B12|D2-D1|D12(1+2α),
where σ:=2+7α+3β+11αβ.Proof.
Letf∈𝒫(α,φ) and g∈ℳ(β,ψ), g=f-1. Then, there exist analytic functions u,v:𝔻→𝔻, with u(0)=v(0)=0, such that
(24)zf′(z)f(z)+αz2f′′(z)f(z)=φ(u(z)),(1-β)wg′(w)g(w)+β(1+wg′′(w)g′(w))=ψ(v(w)).
Since
(25)zf′(z)f(z)+αz2f′′(z)f(z)=1+a2(1+2α)z+(2(1+3α)a3-(1+2α)a22)z2+⋯,(1-β)wg′(w)g(w)+β(1+wg′′(w)g′(w))=1-(1+β)a2w+((3+5β)a22-2(1+2β)a3)w2+⋯,
(12) and (24) yield
(26)a2(1+2α)=12B1c1,(27)2(1+3α)a3-(1+2α)a22=12B1(c2-c122)+14B2c12,(28)-(1+β)a2=12D1b1,(29)(3+5β)a22-2(1+2β)a3=12D1(b2-b122)+14D2b12.
It follows from (26) and (28) that
(30)b1=-B1(1+β)D1(1+2α)c1.
Hence, (26), (27), (29), and (30) lead to
(31)a22=(B12D12[B1(1+2β)c2+D1(1+3α)b2])×(-(1+2β)2(1+3α)(D2-D1)B12]2[σB12D12-(1+2α)2(1+2β)(B2-B1)D12-(1+2β)2(1+3α)(D2-D1)B12])-1,
which gives us the desired estimate on |a2| as asserted in (22) when |b2|≤2 and |c2|≤2.
Further, (27), (29), and (30) give
(32)2σa3=12[B1(3+5β)c2+D1(1+2α)b2]+c124[(3+5β)(B2-B1)+(1+β)2B12(D2-D1)D12(1+2α)],
and this yields the estimate given in (23).Theorem 6.
Letf∈σ and g=f-1. If f∈𝒫(α,φ) and g∈ℒ(β,ψ), then
(33)|a2|≤(B1D12[B1(3-2β)+D1(1+3α)])×(|σB12D12-2(1+2α)2(3-2β)(B2-B1)D12-2(2-β)2(1+3α)(D2-D1)B12|)-1/2,|σa3|≤12B1(β2-11β+16)+D1(1+2α)+12(β2-11β+16)|B2-B1|+(2-β)2B12|D2-D1|D12(1+2α),
where σ:=10+36α-7β-25αβ+β2+3αβ2.Proof.
Letf∈𝒫(α,φ) and g∈ℒ(β,ψ), g=f-1. Then, there are analytic functions u,v:𝔻→𝔻, with u(0)=v(0)=0, satisfying
(34)zf′(z)f(z)+αz2f′′(z)f(z)=φ(u(z)),(wg′(w)g(w))β(1+wg′′(w)g′(w))1-β=ψ(v(w)).
Using
(35)zf′(z)f(z)+αz2f′′(z)f(z)=1+a2(1+2α)z+(2(1+3α)a3-(1+2α)a22)z2+⋯,(wg′(w)g(w))β(1+wg′′(w)g′(w))1-β=1-(2-β)a2w+((8(1-β)+12β(β+5))a22-2(3-2β)a3)w2+⋯, and (12) and (34) will yield
(36)a2(1+2α)=12B1c1,2(1+3α)a3-(1+2α)a22=12B1(c2-c122)+14B2c12,-(2-β)a2=12D1b1,[8(1-β)+β2(β+5)]a22-2(3-2β)a3=12D1(b2-b122)+14D2b12.
Further implication of (36) and applying the fact that |b2|≤2 and |c2|≤2 give the estimates in (33).Theorem 7.
Letf∈σ and g=f-1. If f∈ℳ(α,φ), g∈ℳ(β,ψ), then
(37)|a2|≤(B1D1B1(1+2β)+D1(1+2α))×(|σB12D12-(1+α)2(1+2β)(B2-B1)D12-(1+β)2(1+2α)(D2-D1)B12|)-1/2,2σ|a3|≤B1(3+5β)+D1(1+3α)+(3+5β)|B2-B1|+(1+β)2(1+3α)B12|D2-D1|D12(1+α)2,
where σ:=2+3α+3β+4αβ.Proof.
Forf∈ℳ(α,φ) and g∈ℳ(β,ψ), g=f-1, there exist analytic functions u,v:𝔻→𝔻, with u(0)=v(0)=0, satisfying
(38)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))=φ(u(z)),(1-β)wg′(w)g(w)+β(1+wg′′(w)g′(w))=ψ(v(w)).
Since
(39)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))=1+(1+α)a2z+(2(1+2α)a3-(1+3α)a22)z2+⋯,(1-β)wg′(w)g(w)+β(1+wg′′(w)g′(w))=1-(1+β)a2w+((3+5β)a22-2(1+2β)a3)w2+⋯,
then (12) and (38) yield
(40)a2(1+α)=12B1c1,2(1+2α)a3-(1+3α)a22=12B1(c2-c122)+14B2c12,-(1+β)a2=12D1b1,(3+5β)a22-2(1+2β)a3=12D1(b2-b122)+14D2b12.
Further implication of (40) and applying the fact that |b2|≤2 and |c2|≤2 give the estimates in (37).Remark 8.
Whenβ=α and ψ=φ, Theorem 7 reduces to [26, Theorem 2.3].The following theorems give the estimates for the second and third coefficients of functionsf when (i) f∈ℳ(α,φ) and g∈ℒ(β,ψ) and (ii) f∈ℒ(α,φ) and g∈ℒ(β,ψ). The proofs are similar as for the theorems above; hence, they are omitted here.Theorem 9.
Letf∈σ and g=f-1. If f∈ℳ(α,φ) and g∈ℒ(β,ψ), then
(41)|a2|≤(B1D12[B1(3-2β)+D1(1+2α)])×(|σB12D12-2(1+α)2(3-2β)(B2-B1)D12-2(2-β)2(1+2α)(D2-D1)B12|)-1/2,|σa3|≤B12(β2-11β+16)+D1(1+3α)+12(β2-11β+16)|B2-B1|+(2-β)2(1+3α)B12|D2-D1|D12(1+α)2,
where σ:=10+14α-7β+β2+2αβ2-10αβ.Theorem 10.
Letf∈σ and g=f-1. If f∈ℒ(α,φ) and g∈ℒ(β,ψ), then
(42)|a2|≤(B1D12[B1(3-2β)+D1(3-2α)])×(|σB12D12-2(2-α)2(3-2β)(B2-B1)D12-2(2-β)2(3-2α)(D2-D1)B12|)-1/2,(43)2|σa3|≤B1(β2-11β+16)+D1(8-5α-α2)+(β2-11β+16)|B2-B1|+(2-β)2(α2+5α-8)B12|D2-D1|D12(2-α)2,
where σ:=24+3α2+3β2-17α-17β-2βα2-2αβ2+12αβ.Remark 11.
Whenβ=α and ψ=φ, Theorem 10 reduces to [26, Theorem 2.4].



---

*Source: 640856-2014-04-08.xml*