# Coefficient Estimates for Certain Families of Analytic Functions Associated with Faber Polynomial

Authors: Adel A. Attiya; Abeer M. Albalahi; Taher S. Hassan
Journal: Journal of Function Spaces
 (2023)

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## Abstract

In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficientsan of bi-univalent functions in the family of analytic functions in the open unit disk. Estimation of the bound value for the initial coefficients of the functions in these classes is also established.


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## Body

## 1. Introduction

LetA denote the class of all functions of the form
(1)fz=z+∑n=2∞anzn,which are analytic in the open unit disc U=z:z∈ℂandz<1. Also, let S be the class of all functions in A which are univalent in U. For f∈A, Airault and Bouali ([1], page 184) used Faber polynomial to show that
(2)zf′zfz=1−∑j=2∞Fj−1a2,a3,⋯,ajzj−1,where Fj−1a2,a3,⋯,aj is the Faber polynomial defined by
(3)Fj−1a2,a3,⋯,aj=∑i1+2i2+⋯+j−1ij−1=j−1Ai1,i2,⋯,ij−1a2i1a3i2⋯,ajij−1,(4)Ai1,i2,⋯,ij−1≔−1j−1+2i1+⋯+jij−1i1+i2+⋯+ij−1−1!j−1i1!i2!⋯ij−1!.The first terms of the Faber polynomialFj−1, j≥2, are given by (e.g., see [2], page 52)
(5)F1=−a2,F2=a22−2a3,F3=−a23+3a2a3−3a4,F4=a24−4a22a3+4a2a4+2a32−4a5,F5=−a25+5a23a3+5a22a4−5a2a32−a5+5a3a4−5a6.The Koebe one-quarter theorem ([3], page 31) ensured that the range of each function of the class S contains the disc w:w<1/4. Therefore, the univalent function f∈S has an inverse f−1, which is defined by
(6)f−1fz=zz∈U,ff−1ω=ωω<14.The inverseg≔f−1 of the function f∈S has Taylor expansion given by (see [1], page 185)
(7)gω=f−1ω=w+∑n=2∞1nKn−1−na2,a3,⋯,anωn=w−a2ω2+2a22−a3ω3−5a22−5a2a3+a4ω4+⋯,where the coefficients Knpa2,a3,⋯,an are given by
(8)K1p=pa2,K2p=pp−12a22+pa3,K3p=pp−1a2a3+pa4+pp−1p−23!a23,K4p=pp−1a2a4+pa5+pp−12a32+pp−1p−22a22a3+p!p−4!4!a24,⋮Knp=p!p−n!n!a2n+p!p−n+1!n−2!a2n−2a3+p!p−n+2!n−3!a2n−3a4+p!p−n+3!n−4!a2n−4a5+p−n+32a32+p!p−n+4!n−5!a2n−4a6+p−n+3a3a4+∑j≥6∞a2n−jVj,and Vj is homogeneous polynomial of degree j in the variables a3,⋯,an (see [4], page 349 and [1], pages 183 and 205).Lemma 1 (Schwarz lemma ([3], page 3).
Ifωz is analytic in the unit disc U, with ω0=0 and ωz<1 in U, then ωz<z and ω′0<1 in U.Letf and g be analytic functions in U; we say that the function f is subordinate to g, written as follows:
(9)f≺gorfz≺gz,if there exists a Schwarz function w, which (by definition) is analytic in U with w0=0 and wz<1z∈U, such that fz=gwz for all z∈U. In particular, if the function g∈S in U, then we have the following equivalence relation (cf., e.g., [5, 6]; see also [3]):
(10)fz≺gz⇔f0≺g0,fU⊂gU.Letϕ be analytic function with positive real part in U, satisfying ϕ0=1,ϕ′0>0. Also, let ϕU be symmetric with respect to the real axis. Such a function has a Taylor series of the form
(11)ϕz=1+B1z+B2z2+B3z3+⋯B1>0.Attiya [7] introduced the operator Hα,βγ,kf, where Hα,βγ,kf:A⟶A is defined by
(12)Hα,βγ,kf=μα,βγ,k∗fzz∈U,with β,γ∈ℂ,Reα>max0,Rek−1 and Rek>0. Also, Reα=0 when Rek=1; β≠0. Here, μα,βγ,k is the generalized Mittag-Leffler function defined by [8] (see also [7]), and the symbol ∗ denotes the Hadamard product or convolution.A detailed investigation of the Mittag-Leffler function has been studied by many authors (see, e.g., [8–12]).Attiya [7] noted that
(13)Hα,βγ,kfz=z+∑n=2∞Γγ+nkΓα+βΓγ+kΓβ+αnn!anzn.From (13), it follows (see [7])
(14)zHα,βγ,kfz′=γ+kkHα,βγ+1,kfz−γkHα,βγ,kfz,(15)αzHα,β+1γ,kfz′=α+βHα,βγ,kfz−βHα,β+1γ,kfz.Also, note the following:(1)
H0,β1,1fz=fz(2)
H0,β2,1fz=1/2fz+zf′z(3)
H0,β0,1fz=∫0z1/tftdt(4)
H1,01,1z/1−z=zez(5)
H1,11,1z/1−z=ez−1(6)
H2,11,1z/1−z=−2+coshzDefinition 2.
A functionfz∈A is said to be in the class Sα,β,γ,k,ϕif it satisfies
(16)γ+kkHα,βγ+1,kfzHα,βγ,kfz−γk≺ϕz,where ϕz is defined by (11).Definition 3.
A functionfz∈A is said to be in the class Tα,β,γ,k,ϕif it satisfies
(17)α+βαHα,βγ,kfzHα,β+1γ,kfz−βα≺ϕz,where ϕz is defined by (11).With the virtue of (14) and (15), we find that S0,β,1,1,1+1−2τz/1−z and T0,β,1,1,1+1−2τz/1−z when τ∈0,1 is the well-known class of starlike function of order τ.A single-valued functionf analytic in a domain D⊂ℂ is said to be univalent, if it never takes the same value twice in D, that is, if fz1≠fz2 for all points z1 and z2 in D with z1≠z2 (see [3], page 26). Also, a function f∈A is said to be bi-univalent in U if f and its inverse map f−1 are univalent in U.We denote byΣ the class of bi-univalent functions in U given by (1). The class of analytic bi-univalent functions was introduced and studied by Lewin [13] and showed that a2<1.51. Recently, many authors found nonsharp estimates on the first two Taylor–Maclaurin’s coefficients a2 and a3.The following are examples of bi-univalent functions inU:
(18)z1−z,−log1−z,12log1−z1+z.For various subclasses of bi-univalent functions, see, for example, [14–26].Definition 4.
A functionf∈Σ given by (1) is said to be in the class SΣα,β,γ,k,ϕ if both f and its inverse map g=f−1 are in Sα,β,γ,k,ϕ. Also, a function f∈Σ given by (1) is said to be in the class TΣα,β,γ,k,ϕ if both f and its inverse map g=f−1 are in Tα,β,γ,k,ϕ.In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficientsan of bi-univalent functions in SΣα,β,γ,k,ϕ and TΣα,β,γ,k,ϕ as well as we estimate the bounds of the initial coefficients of the functions in these classes.Unless otherwise mentioned, we assume throughout this paper thatβ,γ∈ℂ,Reα>max0,Rek−1 and Rek>0. Also, Reα=0 when Rek=1; β≠0.


## 2. Coefficient Estimates ofSΣα,β,γ,k,ϕ and TΣα,β,γ,k,ϕ

Theorem 5.
Let the functionf∈Σ given by (1) be in the class SΣα,β,γ,k,ϕ. Also, let am=0 and ap≠0 for 2≤m,p≤n, where p−1 is a divisor of n−1. Then,
(19)ap≤Γγ+pkΓα+βΓγ+kΓβ+αp1p!B1p−1p−1/n−1,p≥3,where B1 is defined in (11).Proof.
If we setFz=Hα,βγ,kfz, then Fz=z+∑n=2∞δnzn, with δn=Γγ+nkΓα+β/Γγ+kΓβ+αnn!an; using relation (14), we have
(20)f∈SΣα,β,γ,k,ϕifandonlyifzF′zFz≺ϕz.
Sincef and its inverse g=f−1 are in SΣα,β,γ,k,ϕ, gz=z+∑n=2∞bnzn, therefore, there are analytic functions u and vwith u0=v0=0, uz<1 and vz<1, such that
(21)zF′zFz=ϕuzz∈U,wG′wGw=ϕvwz∈U,where Gw=Hα,βγ,kgw; then, Gw=w+∑n=2∞ζnwn, with ζn=Γγ+nkΓα+β/Γγ+kΓβ+αnn!bn; using (7), we have bn=1/nKn−1−na2,a3,⋯,an.
The functionsuz and vz are defined by
(22)uz=∑n=1∞cnzn,vz=∑n=1∞dnznz∈U.
It is well known that (see Duren [3], page 265)
(23)cn≤1,dn≤1,n=2,3,⋯.
Also, we have(24)ϕuz=1−B1∑n=1∞Kn−1c1,−c2,⋯,−1n+1cn,B1,B2,⋯,Bnznz∈U,(25)ϕvω=1−B1∑n=1∞Kn−1d1,−d2,⋯,−1n+1dn,B1,B2,⋯,Bnwnz∈U.
In general (see [21], page 649), the coefficients Knp≔Knpk1,k2,⋯,kn,B1,B2,B3,⋯,Bn are given by
(26)Knp=p!p−n!n!k1nBnB1+p!p−n+1!n−2!k1n−2k2Bn−1B1+p!p−n+2!n−3!k1n−3k3Bn−2B1+p!p−n+3!n−4!k1n−4k4Bn−3B1+p−n+32k22k3Bn−2B1+∑j≥5∞k1n−jXj,where Xj is a homogeneous polynomial of degree j in the variables k2,⋯,kn.
Using the Faber polynomial expansion, (2) yields the following identities:
(27)zF′zFz=1−∑n=2∞Fn−1δ2,δ3,⋯,δnzn−1,(28)wG′wGw=1−∑n=2∞Fn−1ζ2,ζ3,⋯,ζnwn−1.
Comparing the corresponding coefficients of (27) and (24) yields
(29)Fn−1δ2,δ3,⋯,δn=B1Kn−1−1c1,−c2,⋯,−1ncn−1,B1,B2,B3,⋯,Bn−1,and similarly, from (28) and (25), we have
(30)Fn−1ζ2,ζ3,⋯,ζn=B1Kn−1−1d1,−d2,⋯,−1ndn−1,B1,B2,B3,⋯,Bn−1.
Sinceam=0 and ap≠0 for 2≤m,p≤n, and p−1 is a divisor of n−1, then by using (29) and (30), we have
(31)p−1−1n−1/p−1δpn−1/p−1=−B1−1pcp−1,(32)p−1−1n−1/p−1ζpn−1/p−1=−B1−1pdp−1,also, under the condition am=0 and ap≠0 for 2≤m,p≤n and using the relation between δp and ζp, we have ζp=−δp. Then, (31) yields
(33)p−1−1n−1/p−1−δpn−1/p−1=−B1−1pdp−1.
Using either (31) or (33), we have δp≤B1/p−1p−1/n−1, substitute of δp=Γγ+pkΓα+β/Γγ+kΓβ+αpp!ap. This completes the proof of the theorem.Puttingp=n in Theorem 5, we have the following:Corollary 6.
Let the functionf∈Σ given by (1) be in the class SΣα,β,γ,k,ϕ; if am=0 for 2≤m≤n−1, then
(34)an≤Γγ+nkΓα+βΓγ+kΓβ+αn1n!B1n−1,n≥3,where B1 is defined in (11).Using the same technique used in Theorem5, we get the following theorem:Theorem 7.
Let the functionf∈Σ given by (1) be in the class TΣα,β,γ,k,ϕ. Also, let am=0 and ap≠0 for 2≤m,p≤n, where p−1 is a divisor of n−1. Then,
(35)ap≤Γγ+pkΓα+β+1Γγ+kΓβ+1+αp1p!B1p−1p−1/n−1,p≥3,where B1 is defined in (11).Puttingp=n in Theorem 7, we have the following:Corollary 8.
Let the functionf∈Σ given by (1) be in the class TΣα,β,γ,k,ϕ; if am=0 for 2≤m≤n−1, then
(36)an≤Γγ+nkΓα+β+1Γγ+kΓβ+1+αn1n!B1n−1,n≥3,where B1 is defined in (11).To prove our next theorem, we shall need the following lemma.Lemma 9 [3, 21].
Let the functionΦz=∑n=1∞Φnzn be a Schwarz function with Φz<1, z∈U. Then for −∞<η<∞,
(37)Φ2+ηΦ12≤1−1−ηΦ12η>0,1−1+ηΦ12η≤0.Theorem 10.
Let the functionf∈Σ given by (1) be in the class SΣα,β,γ,k,ϕ. Then,
(38)a2≤2B1Γγ+kΓβ+2αΓγ+2kΓα+βB1≥B2,2B2Γγ+kΓβ+2αΓγ+2kΓα+βB1<B2,(39)a3−a22≤B134R3−12R22+14R3−12R22B1≥B2,B234R3−12R22+14R3−12R22B1<B2,where R2=1/2Γγ+2kΓα+β/Γγ+kΓβ+2α and R3=1/6Γγ+3kΓα+β/Γγ+kΓβ+3α.Proof.
Puttingn=2 and n=3 in (29) and (30), respectively, we find that
(40)δ2=c1B1,−ζ2=d1B1,(41)δ22−2δ3=c12B2+c2B1,(42)ζ22−2ζ3=d12B2+d2B1.
Moreover, we haveζ2=−δ2 and ζ3=2δ22−δ2.
Therefore, (41) and (42) imply
(43)a22=−B12R22c2+B2B1c12+d2+B2B1d12.
Apply Lemma9.
Case I: ifB2>0, then both c2+B2/B1c12 and d2+B2/B1d12 have maximum value at 1 when B1>B2. Also, c2+B2/B1c12 and d2+B2/B1d12 have maximum value at B2/B1 when B1≤B2, where c1≤1 and d1≤1
Case II: ifB2≤0, then both c2+B2/B1c12 and d2+B2/B1d12 have maximum value at 1 when B1>−B2 and c2+B2/B1c12 and d2+B2/B1d12 have maximum value at −B2/B1 when B1≤−B2. Then, we have (38)
Moreover, from (41) and (42), we get
(44)a3=−3c2+d24R3B1−3c12+d124R3B2.
It follows from (43) and (44) that
(45)a3−a22=−34R3−12R22c2+B2B1c12B1−14R3−12R22d2+B2B1d12B1.
Applying the cases mentioned above forB2>0 and B2≤0, we have (39). This completes the proof of the theorem.Using the same technique used in Theorem10, we get the following theorem:Theorem 11.
Let the functionf∈Σ given by (1) be in the class TΣα,β,γ,k,ϕ. Then,
(46)a2≤2B1Γγ+kΓβ+1+2αΓγ+2kΓα+β+1B1≥B2,2B2Γγ+kΓβ+1+2αΓγ+2kΓα+β+1B1<B2,(47)a3−a22≤B134R3−12R22+14R3−12R22B1≥B2,B234R3−12R22+14R3−12R22B1<B2,where R2=1/2Γγ+2kΓα+β+1/Γγ+kΓβ+1+2α and R3=1/6Γγ+3kΓα+β+1/Γγ+kΓβ+1+3α.


## 3. Conclusion

By using the Faber polynomial expansion, we obtain bounds for the general coefficientsan of bi-univalent functions for functions in the classes SΣα,β,γ,k,ϕ and TΣα,β,γ,k,ϕ in U=z:z∈ℂandz<1; also, estimation of the bound value for the initial coefficients of the functions in the classes SΣα,β,γ,k,ϕ and TΣα,β,γ,k,ϕ is established.



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