# Initial Bounds for Certain Classes of Bi-Univalent Functions Defined by Horadam Polynomials

Authors: Chinnaswamy Abirami; Nanjundan Magesh; Jagadeesan Yamini
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

The main purpose of this article is to make use of the Horadam polynomialshnx and the generating function Πx,  z, in order to introduce three new subclasses of the bi-univalent function class σ. For functions belonging to the defined classes, we then derive coefficient inequalities and the Fekete–Szegö inequalities. Some interesting observations of the results presented here are also discussed. We also provide relevant connections of our results with those considered in earlier investigations.


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## Body

## 1. Introduction

LetR be the set of real numbers, C be the set of complex numbers and(1)N≔1,2,3,…be the set of positive integers. LetA denote the class of functions of the form(2)fz=z+∑n=2∞anznwhich are analytic in the open unit diskΔ=z:z∈Candz<1. Further, by S we shall denote the class of all functions in A which are univalent in Δ.It is well known that every functionf∈S has an inverse f−1, defined by(3)f−1fz=zz∈Δand(4)ff−1w=ww<r0f;r0f≧14,where(5)f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+⋯.A functionf∈A is said to be bi-univalent in Δ if both the function f and its inverse f−1 are univalent in Δ. Let σ denote the class of bi-univalent functions in Δ given by (2).In 2010, Srivastava et al. [1] revived the study of bi-univalent functions by their pioneering work on the study of coefficient problems. Various subclasses of the bi-univalent function class σ were introduced and nonsharp estimates on the first two coefficients a2 and a3 in the Taylor–Maclaurin series expansion (2) were found in the recent investigations (see, for example, [2–23]) and including the references therein. The afore-cited all these papers on the subject were actually motivated by the work of Srivastava et al. [1]. However, the problem to find the coefficient bounds on |an|n=3,4,… for functions f∈σ is still an open problem.For analytic functionsf and g in Δ, f is said to be subordinate to g if there exists an analytic function w such that(6)w0=0,wz<1andfz=gwzz∈Δ.This subordination will be denoted here by(7)f≺gz∈Δor, conventionally, by(8)fz≺gzz∈Δ.In particular, wheng is univalent in Δ,(9)f≺gz∈Δ⇔f0=g0andfΔ⊂gΔ.The Horadam polynomialshnx,a,b;p,  q, or briefly hnx are given by the following recurrence relation (see [22, 23]):(10)h1x=a,h2x=bxandhnx=pxhn−1x+qhn−2xn≥3for some real constantsa,b,p, and q.The generating function of the Horadam polynomialshnx (see [23]) is given by(11)Πx,z≔∑n=1∞hnxzn−1=a+b−apxz1−pxz−qz2.Here, and in what follows, the argumentx∈R is independent of the argument z∈C; that is, x≠Rz.Note that for particular values ofa,b,p, and q, the Horadam polynomial hnx leads to various polynomials, among those, we list a few cases here (see, [22, 23] for more details):(1)
Fora=b=p=q=1, we have the Fibonacci polynomials Fnx.(2)
Fora=2 and b=p=q=1, we obtain the Lucas polynomials Lnx.(3)
Fora=q=1 and b=p=2, we get the Pell polynomials Pnx.(4)
Fora=b=p=2 and q=1, we attain the Pell-Lucas polynomials Qnx.(5)
Fora=b=1,p=2 and q=−1, we have the Chebyshev polynomials Tnx of the first kind.(6)
Fora=1,b=p=2 and q=−1, we obtain the Chebyshev polynomials Unx of the second kind.Recently, in literature, the coefficient estimates are found for functions in the class of univalent and bi-univalent functions associated with certain polynomials such as the Faber polynomial [8], the Chebyshev polynomials [6], and the Horadam polynomial [15]. Motivated in these lines, estimates on initial coefficients of the Taylor–Maclaurin series expansion (2) and Fekete–Szegö inequalities for certain classes of bi-univalent functions defined by means of Horadam polynomials are obtained. The classes introduced in this paper are motivated by the corresponding classes investigated in [2, 10, 14, 15].


## 2. Coefficient Estimates and Fekete–Szegö Inequalities

A functionf∈A of the form (2) belongs to the class Sσ∗λ,x for λ≥0 and z,w∈Δ, if the following conditions are satisfied:(12)zf′zfz+λz2f″zfz≺Πx,z+1−aand forgw=f−1w(13)wg′wgw+λw2g″wgw≺Πx,w+1−a,where the real constanta is as in (10).Note thatSσ∗x≡Sσ∗0,x was introduced and studied by Srivastava et al. [15].Remark 1.
Whena=1,b=p=2,q=−1 and x=t, the generating function in (11) reduces to that of the Chebyshev polynomial Unt of the second kind, which is given explicitly by(14)Unt=n+12F1−n,n+2;32;1−t2=sinn+1φsinφ,t=cosφ
in terms of the hypergeometric function2F1.In view of Remark1, the bi-univalent function class Sσ∗x reduces to Sσ∗t and this class was studied earlier in [3, 12]. For functions in the class Sσ∗λ,  x, the following coefficient estimates and Fekete–Szegö inequality are obtained.Theorem 1.
Letfz=z+∑n=2∞anzn be in the class Sσ∗λ,  x. Then(15)a2≤bxbx1+4λb−p1+2λ2bx2−qa1+2λ2,anda3≤bx2+6λ+b2x21+2λ2
and forν∈R(16)a3−νa22≤bx2+6λifν−1≤1+4λb−p1+2λ2bx2−qa1+2λ22b2x21+3λ,bx3ν−11+4λb−p1+2λ2bx2−qa1+2λ2ifν−1≥1+4λb−p1+2λ2bx2−qa1+2λ22b2x21+3λ.Proof.
Letf∈Sσ∗λ,  x be given by the Taylor–Maclaurin expansion (2). Then, there are analytic functions u and v such that(17)u0=0;v0=0,uz<1andvz<1∀z,w∈Δ,
and we can write(18)zf′zfz+λz2f″zfz=Πx,uz+1−a
and(19)wg′wgw+λw2g″wgw=Πx,vw+1−a.
Equivalently,(20)zf′zfz+λz2f″zfz=1+h1x−a+h2xuz+h3xuz2+⋯
and(21)wg′wgw+λw2g″wgw=1+h1x−a+h2xvw+h3xvw2+⋯.
From (20) and (21) and in view of (11), we obtain(22)zf′zfz+λz2f″zfz=1+h2xu1z+h2xu2+h3xu12z2+⋯
and(23)wg′wgw+λw2g″wgw=1+h2xv1w+h2xv2+h3xv12w2+⋯.
If(24)uz=∑n=1∞unznandvz=∑n=1∞vnwn,
then it is well known that(25)un≤1andvn≤1n∈N.
Thus upon comparing the corresponding coefficients in (22) and (23), we have(26)1+2λa2=h2xu1,
(27)
2
1
+
3
λ
a
3
−
1
+
2
λ
a
2
2
=
h
2
x
u
2
+
h
3
x
u
1
2
,
(28)
−
1
+
2
λ
a
2
=
h
2
x
v
1
and(29)3+10λa22−21+3λa3=h2xv2+h3xv12.
From (26) and (28), we can easily see that(30)u1=−v1,providedh2x=bx≠0
and(31)21+2λ2a22=h2x2u12+v12a22=h2x2u12+v1221+2λ2.
If we add (27) to (29), we get(32)21+4λa22=h2xu2+v2+h3xu12+v12.
By substituting (31) in (32), we obtain(33)a22=h2x3u2+v221+4λh2x2−2h3x1+2λ2
and by takingh2x=bx and h3x=bpx2+qa in (33), it further yields(34)a2≤bxbx1+4λb−p1+2λ2bx2−qa1+2λ2.
By subtracting (29) from (27) and in view of (30), we obtain(35)41+3λa3−41+3λa22=h2xu2−v2+h3xu12−v12a3=h2xu2−v241+3λ+a22.
Then in view of (31), (35) becomes(36)a3=h2xu2−v241+3λ+h2x2u12+v1221+2λ2.
Applying (10), we deduce that(37)a3≤bx2+6λ+b2x21+2λ2.
From (35), for ν∈R, we write(38)a3−νa22=h2xu2−v241+3λ+1−νa22.
By substituting (33) in (38), we have(39)a3−νa22=h2xu2−v241+3λ+1−νh2x3u2+v221+4λh2x2−h3x1+2λ2=h2xΩν,x+141+3λu2+Ων,x−141+3λv2,
where(40)Ων,x=1−νh2x221+4λh2x2−2h3x1+2λ2.
Hence, we conclude that(41)a3−νa22≤h2x2+6λ;0≤Ων,x≤141+3λ2h2xΩν,x;Ων,x≥141+3λ
and in view of (10), it evidently completes the proof of Theorem 1.Forλ=0, Theorem 1 readily yields the following coefficient estimates for Sσ∗x.Corollary 1.
Letfz=z+∑n=2∞anzn be in the class Sσ∗x. Then(42)a2≤bxbxb−pbx2−qa,anda3≤bx2+b2x2
and forν∈R(43)a3−νa22≤bx2ifν−1≤b−pbx2−qa2b2x2bx3ν−1b−pbx2−qaifν−1≥b−pbx2−qa2b2x2.
In view of Remark1, Theorem 1 can be shown to yield the following result.Corollary 2.
Letfz=z+∑n=2∞anzn be in the class Sσ∗λ,t. Then(44)a2≤2t2t1+2λ2−16λ2t2,anda3≤t1+3λ+4t21+2λ2
and forν∈R(45)a3−νa22≤t1+3λifν−1≤1+2λ2−16λ2t28t21+3λ,8t3ν−11+2λ2−16λ2t2ifν−1≥1+2λ2−16λ2t28t21+3λ.Remark 2.
Results obtained in Corollary1 coincide with results obtained in [15]. For λ=0. Corollary 2 reduces to the results discussed in [3, 12].
Next, a functionf∈A of the form (2) belongs to the class Mσα,  x for 0≦α≦1 and z,w∈Δ, if the following conditions are satisfied:(46)1−αzf′zfz+α1+zf″zf′z≺Πx,z+1−a
and forgw=f−1w(47)1−αwg′wgw+α1+wg″wg′w≺Πx,w+1−a,
where the real constanta is as in (10).
Note that the classMσα,  x reduces to the classes Sσ∗x and Kσx as Mσ0,x≡Sσ∗x and Mσ1,x≡Kσx. In view of Remark 1, the bi-univalent function classes Mσ∗α,  x would become the class Mσ∗α,  t introduced and studied by Altınkaya and Yalçin [4]. For functions in the class Mσα,  x, the following coefficient estimates and Fekete–Szegö inequality are obtained.Theorem 2.
Letfz=z+∑n=2∞anzn be in the class Mσα,  x. Then(48)a2≤bxbx1+αb−p1+α2bx2−qa1+α2,anda3≤bx2+4α+b2x21+α2
and forν∈R(49)a3−νa22≤bx2+4αifν−1≤1+αb−p1+α2bx2−qa1+α2b2x22+4α,bx3ν−11+αb−p1+α2bx2−qa1+α2ifν−1≥1+αb−p1+α2bx2−qa1+α2b2x22+4α.Proof.
Letf∈Mσα,  x be given by the Taylor–Maclaurin expansion (2). Then, there are analytic functions u and v such that(50)u0=0;v0=0,uz<1andvz<1∀z,w∈Δ
and(51)1−αzf′zfz+α1+zf″zf′z=Πx,uz+1−a
and(52)1−αwg′wgw+α1+wg′wg″w=Πx,vw+1−a.
Equivalently,(53)1−αzf′zfz+α1+zf″zf′z=1+h1x−a+h2xuz+h3xuz2+⋯
and(54)1−αwg′wgw+α1+wg″wg′w=1+h1x−a+h2xvw+h3xvw2+⋯.
From (53), (54) and in view of (11), we obtain(55)1−αzf′zfz+α1+zf″zf′z=1+h2xu1z+h2xu2+h3xu12z2+⋯
and(56)1−αwg′wgw+α1+wg″wg′w=1+h2xv1w+h2xv2+h3xv12w2+⋯.
If(57)uz=∑n=1∞unznandvz=∑n=1∞vnwn,
then it is well known that(58)un≤1andvn≤1n∈N.
Thus upon comparing the corresponding coefficients in (55) and (56), we have(59)1+αa2=h2xu1
(60)
2
1
+
2
α
a
3
−
1
+
3
α
a
2
2
=
h
2
x
u
2
+
h
3
x
u
1
2
(61)
−
1
+
α
a
2
=
h
2
x
v
1
and(62)3+5αa22−21+2αa3=h2xv2+h3xv12.
From (59) and (61), we can easily see that(63)u1=−v1,providedh2x=bx≠0
and(64)21+α2a22=h2x2u12+v12a22=h2x2u12+v1221+α2.
If we add (60) to (62), we get(65)21+αa22=h2xu2+v2+h3xu12+v12.
By substituting (64) in (65), we obtain(66)a22=h2x3u2+v221+αh2x2−2h3x1+α2
and by takingh2x=bx and h3x=bpx2+qa in (66), it further yields(67)a2≤bxbx1+αb−p1+α2bx2−qa1+α2.
By subtracting (62) from (60) and in view of (63), we obtain(68)41+2αa3−41+2αa22=h2xu2−v2+h3xu12−v12a3=h2xu2−v241+2α+a22.
Then in view of (64), (68) becomes(69)a3=h2xu2−v241+2α+h2x2u12+v1221+α2.
Applying (10), we deduce that(70)a3≤bx2+4α+b2x21+α2.
From (68), for ν∈R, we write(71)a3−νa22=h2xu2−v241+2α+1−νa22.
By substituting (66) in (71), we have(72)a3−νa22=h2xu2−v241+2α+1−νh2x3u2+v221+αh2x2−2h3x1+α2=h2xΩν,x+141+2αu2+Ων,x−141+2αv2,
where(73)Ων,x=1−νh2x221+αh2x2−2h3x1+α2.
Hence, we conclude that(74)a3−νa22≤h2x2+4α0≤Ων,x≤141+2α2h2xΩν,xΩν,x≥141+2α,
which in view of (10), evidently completes the proof of Theorem 2.Forα=1, Theorem 2 readily yields the following coefficient estimates for Kσx.Corollary 3.
Letfz=z+∑n=2∞anzn be in the class Kσx. Then(75)a2≤bxbx2b−4pbx2−4qa,anda3≤bx6+b2x24
and forν∈R(76)a3−νa22≤bx6ifν−1≤b−2pbx2−4qa3b2x2bx3ν−12b−4pbx2−4qaifν−1≥b−2pbx2−4qa3b2x2.
In view of Remark1, Theorem 2 yields the following result.Corollary 4.
Letfz=z+∑n=2∞anzn be in the class Mσα,  t. Then(77)a2≤2t2t1+α2−4α1+αt2,anda3≤t1+2α+4t21+α2
and forν∈R(78)a3−νa22≤t1+2αifν−1≤1+α2−4α1+αt28t21+2α8t3ν−11+α2−4α1+αt2ifν−1≥1+α2−4α1+αt28t21+2α.
In view of Remark1, Corollary 3 yields the following result.Corollary 5.
Letfz=z+∑n=2∞anzn be in the class Kσt. Then(79)a2≤t2t1−2t2,anda3≤t3+t2
and forν∈R(80)a3−νa22≤t3ifν−1≤1−2t26t22t3ν−11−2t2ifν−1≥1−2t26t2.Remark 3.
The results obtained in Corollary4 and 5 coincide with results of Altınkaya and Yalçin [4].
Next, a functionf∈σ of the form (2) belongs to the class Lσμ,  x for 0≦μ≦1, and z,w∈Δ, if the following conditions are satisfied:(81)zf′zfzμ1+zf″zf′z1−μ≺Πx,z+1−a
and forgw=f−1w(82)wg′wgwμ1+wg″wg′w1−μ≺Πx,w+1−a,
where the real constantsa is as in (10).This class also reduces toSσ∗x and Kσx. In view of Remark 1, the bi-univalent function class Lσ∗μ,  x would become the class Lσ∗μ,  t. For functions in the class Lσμ,  x, the following coefficient estimates are obtained.Theorem 3.
Letfz=z+∑n=2∞anzn be in the class Lσμ,  x. Then(83)a2≤bx2bxμ2−3μ+4b−2p2−μ2bx2−2qa2−μ2anda3≤bx6−4μ+b2x22−μ2
and forν∈R(84)a3−νa22≤bx6−4μifν−1≤μ2−3μ+4b−2p2−μ2bx2−2qa2−μ24b2x23−2μ2bx3ν−1μ2−3μ+4b−2p2−μ2bx2−2qa2−μ2ifν−1≥μ2−3μ+4b−2p2−μ2bx2−2qa2−μ24b2x23−2μ.Proof.
Letf∈Lσμ,  x be given by the Taylor–Maclaurin expansion (2). Then, there are analytic functions u and v such that(85)u0=0;v0=0,uz<1andvz<1∀z,w∈Δ,
and(86)zf′zfzμ1+zf″zf′z1−μ=Πx,uz+1−a
and(87)wg′wgwμ1+wg″wg′w1−μ=Πx,vw+1−a.
Equivalently,(88)zf′zfzμ1+zf″zf′z1−μ=1+h1x−a+h2xuz+h3xuz2+⋯
and(89)wg′wgwμ1+wg″wg′w1−μ=1+h1x−a+h2xvw+h3xvw2+⋯.
From (88) and (89) and in view of (11), we obtain(90)zf′zfzμ1+zf″zf′z1−μ=1+h2xu1z+h2xu2+h3xu12z2+⋯
and(91)wg′wgwμ1+wg″wg′w1−μ=1+h2xv1w+h2xv2+h3xv12w2+⋯.
If(92)uz=∑n=1∞unznandvz=∑n=1∞vnwn,
then it is well known that(93)un≤1andvn≤1n∈N.
Thus upon comparing the corresponding coefficients in (90) and (91), we have(94)2−μa2=h2xu1
(95)
2
3
−
2
μ
a
3
+
μ
2
+
5
μ
−
8
a
2
2
2
=
h
2
x
u
2
+
h
3
x
u
1
2
(96)
−
2
−
μ
a
2
=
h
2
x
v
1
and(97)μ2−11μ+16a222−23−2μa3=h2xv2+h3xv12.
From (94) and (96), we can easily see that(98)u1=−v1,providedh2x=bx≠0
and(99)22−μ2a22=h2x2u12+v12a22=h2x2u12+v1222−μ2.
If we add (95) to (97), we get(100)μ2−3μ+4a22=h2xu2+v2+h3xu12+v12.
By substituting (99) in (100), we obtain(101)a22=h2x3u2+v2μ2−3μ+4h2x2−2h3x2−μ2
and by takingh2x=bx and h3x=bpx2+qa in (101), it further yields(102)a2≤bx2bxμ2−3μ+4b−2p2−μ2bx2−2qa2−μ2.
By subtracting (97) from (95) and in view of (98), we obtain(103)43−2μa3−43−2μa22=h2xu2−v2+h3xu12−v12a3=h2xu2−v243−2μ+a22.
Then in view of (99), (103) becomes(104)a3=h2xu2−v243−2μ+h2x2u12+v1222−μ2.
Applying (10), we deduce that(105)a3≤bx6−4μ+b2x22−μ2.
From (103), for ν∈R, we write(106)a3−νa22=h2xu2−v243−2μ+1−νa22.
By substituting (101) in (106), we have(107)a3−νa22=h2xu2−v243−2μ+1−νh2x3u2+v2μ2−3μ+4h2x2−2h3x2−μ2=h2xΩν,x+143−2μu2+Ων,x−143−2μv2,
where(108)Ων,x=1−νh2x2μ2−3μ+4h2x2−2h3x2−μ2.
Hence, we conclude that(109)a3−νa22≤h2x6−4μ0≤Ων,x≤143−2μ2h2xΩν,xΩν,x≥143−2μ
which in view of (10) evidently completes the proof of Theorem 2.In view of Remark1, Theorem 3 yields.Corollary 6.
Letfz=z+∑n=2∞anzn be in the class Lσμ,  t. Then(110)a2≤2t2t2−μ2−2μ2−5μ+12t2anda3≤t3−2μ+4t22−μ2
and forν∈R(111)a3−νa22≤t3−2μifν−1≤2−μ2−2μ2−5μ+12t28t23−2μ8t3ν−12−μ2−2μ2−5μ+12t2ifν−1≥2−μ2−2μ2−5μ+12t28t23−2μ.



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*Source: 7391058-2020-01-30.xml*