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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2022; 62(2): 257-269

Published online June 30, 2022

### Coefficient Estimates for a Subclass of Bi-univalent Functions Associated with Symmetric q-derivative Operator by Means of the Gegenbauer Polynomials

Ala Amourah, Basem Aref Frasin*, Tariq Al-Hawary

Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid, Jordan
e-mail : alaammour@yahoo.com

Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
e-mail : bafrasin@yahoo.com

Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
e-mail : tariq_amh@bau.edu.jo

Received: July 3, 2021; Revised: November 5, 2021; Accepted: November 16, 2021

In the present paper, a subclass of analytic and bi-univalent functions is defined using a symmetric q-derivative operator by means of Gegenbauer polynomials. Coefficients bounds for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szegö problem for this subclass is solved. A number of known or new results are shown to follow upon specializing the parameters involved in our main results.

Keywords: Gegenbauer polynomials, bi-univalent functions, analytic functions, Fekete-Szegö, problem

Let A denote the class of all analytic functions f defined in the open unit disk U={ξ:ξ<1} and normalized by the conditions f(0)=0 and f(0)=1. Thus each fA has a Taylor-Maclaurin series expansion of the form

f(ξ)=ξ+ n=2anξn,(ξU).

Further, let S denote the class of all functions fA which are univalent in U.

Let the functions f and g be analytic in U. We say that the function f is subordinate to g, written as fg, if there exists a Schwarz function w, which is analytic in U with

w(0)=0 and |w(ξ)|<1(ξU)

such that

f(ξ)=g(w(ξ)).

If the function g is univalent in U, then the following equivalence holds

f(ξ)g(ξ)if and only iff(0)=g(0)

and

f(U)g(U).

It is well known that every function fS has an inverse f1, defined by

f1(f(ξ))=ξ  (ξU)

and

f1(f(w))=w  (w<r0(f); r0(f)14)

where

f1(w)=wa2w2+(2a22a3)w3(5a235a2a3+a4)w4+.

A function is said to be bi-univalent in U if both f(ξ) and f1(ξ) are univalent in U.

Let Σ denote the class of bi-univalent functions in U given by (1.1). Example of functions in the class Σ are

ξ1ξ,log11ξ,log1+ξ1ξ.

However, the familiar Koebe function is not a member of Σ. Other common examples of functions in U such as

2ξξ22 and ξ1ξ2

are also not members of Σ.

Lewin [19] investigated the bi-univalent function class Σ and showed that |a2|<1.51. Subsequently, Brannan and Clunie [9] conjectured that |a2|<2. Netanyahu [22], on the other hand, showed that maxfΣ|a2|=4/3.

The coefficient estimate problem for each of the Taylor-Maclaurin coefficients an(n3;n) is presumably still an open problem.

Similar to the familiar subclasses S(ς) and K(ς) of starlike and convex function of order ς(0ς<1), respectively, Brannan and Taha [10] (see also [29]) introduced certain subclasses of the bi-univalent function class Σ, SΣ(ς) and KΣ(ς) of bi-starlike functions and of bi-convex functions of order ς(0<ς1), respectively. For each of the function classes SΣ(ς) and KΣ(ς), they found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|. For some intriguing examples of functions and characterization of the class Σ, see [1, 13, 21, 27].

Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 [18]. In mathematical treatment of model problems, orthogonal polynomials arise often to find solutions of ordinary differential equations under certain conditions imposed by the model.

The importance of the orthogonal polynomials for the contemporary mathematics, as well as for wide range of their applications in the physics and engineering, is beyond any doubt. It is well-known that these polynomials play an essential role in problems of the approximation theory. They occur in the theory of differential and integral equations as well as in the mathematical statistics. Their applications in the quantum mechanics, scattering theory, automatic control, signal analysis and axially symmetric potential theory are also known [7, 11].

A special case of orthogonal polynomials are Gegenbauer polynomials. They are representatively related with typically real functions TR as discovered in [17], where the integral representation of typically real functions and generating function of Gegenbauer polynomials are using common algebraic expressions. Undoubtedly, this led to several useful inequalities appear from Gegenbauer polynomials realm.

Typically real functions play an important role in the geometric function theory because of the relation TR=co¯SR and its role of estimating coefficient bounds, where SR denotes the class of univalent functions in the unit disk with real coefficients, and co¯SR denotes the closed convex hull of SR.

Very recently, Amourah et al. [4] considered the Gegenbauer polynomials Hα(x,ξ), which are given by

Hα(x,ξ)=112xξ+ξ2α,

where x[1,1] and ξU. For fixed x the function Hα is analytic in U, so it can be expanded in a Taylor series as

Hα(x,ξ)= n=0Cnα(x)ξn,

where Cnα(x) is Gegenbauer polynomial of degree n.

Obviously, Hα generates nothing when α=0. Therefore, the generating function of the Gegenbauer polynomial is set to be

H0(x,ξ)=1log12xξ+ξ2= n=0Cn0(x)ξn

for α=0. Moreover, it is worth to mention that a normalization of α to be greater than -1/2 is desirable [11, 25]. Gegenbauer polynomials can also be defined by the following recurrence relations

Cnα(x)=1n2xn+α1Cn1α(x)n+2α2Cn1α(x),

with the initial values

C0α(x)=1C1α(x)=2αx and C2α(x)=2α1+αx2α.

Special cases of Gegenbauerpolynomials Cnα(x) is Chebyshev Polynomials, when α=1, and if α=12, we get the Legendre Polynomials.

The theory of q-calculus operators are used in describing and solving various problems in applied science such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, as well as geometric function theory of complex analysis. The application of q-calculus was initiated by Jackson [15]. Recently, many researchers studied q-calculus such as Srivastava et al. [28], Muhammad and Darus [20], Kanas and Răducanu [16], Aldweby and Darus [2] (see also, [24, 26, 28]) and also the reference cited therein.

For the convenience, we provide some basic definitions and concept details of q-calculus which are used in this paper. We shall follow the notation and terminology in [14].

Definition 1.1. ([15]) For 0<q<1 the Jackson's q-derivative of a function fA is, by definition, given as follows

Dqf(ξ)=f(ξ)f(qξ)(1q)ξforξ0,f(0)forξ=0,

From (1.8), we have

Dqf(ξ)=1+ n=2[n]qanξn1

where

nq=1qn1q, n=1,2,...,

is sometimes called the basic number n. If q1,[n]qn.

For a function h(ξ)=ξn, we obtain

Dqh(ξ)=Dqξn=1qn1qξn1=[n]qξn1,

and

limq1Dqh(ξ)=limq1[n]qξn1=nξn1=h(ξ),

where h is the ordinary derivative.

Definition 1.2. ([8]) The symmetric q-derivative D˜qf of a function f given by (1.1) is defined as follows:

D˜qf(ξ)=f(qξ)f(q1ξ)qq1ξf(0)ξ0ξ=0.

From (1.11), we deduce that D˜qξn=n~ qξn1, and a power series of D˜qf is

D˜qf(ξ)=1+n=2n~qanξn1,

when f has the form (1.1) and the symbol n~q denotes the number

n~q=qnqnqq1.

Clearly, we have the following relations

D˜qf(ξ)+g(ξ)=D˜qf(ξ)+D˜qg(ξ),
D˜qf(ξ)g(ξ)=gq1ξD˜qf(ξ)+fqξD˜qg(ξ)=gqξD˜qf(ξ)+fq1ξD˜qg(ξ),

and

D˜qf(ξ)=Dqf(q1ξ).

From (1.2) and (1.11), we also deduce that

D˜qg(w)=g(qw)g(q1w)qq1w =1 2~qa2w+ 3~q2a22a3w2 4~q5a235a2a3+a4w3+.

Recently, many researchers have been exploring bi-univalent functions associated with orthogonal polynomials, few to mention (see,[31], [30]). For Gegenbauer polynomial, as far as we know, there is little work associated with bi-univalent functions in the literatures. Inspired by the works of Amourah et al. [4], we introduce the following new subclasses of bi-univalent functions, as follows:

Definition 1.3. Let α is a nonzero real constant. A function fΣ given by (1.1) is said to be in the class B˜Σq(x,α) if the following subordinations are satisfied:

D˜qf(ξ)Hα(x,ξ)

and

D˜qg(w)Hα(x,w),

where x(12,1], the function g(w)=f1(w) is defined by (1.2) and Hα is the generating function of the Gegenbauer polynomial given by (1.3).

We note that limq1B˜Σq(x,α)=BΣ(x,α), where the class BΣ(x,α) defined as follows:

Definition 1.4. Let α is a nonzero real constant. A function fΣ given by (1.1) is said to be in the class BΣ(x,α) if the following subordinations are satisfied:

f(ξ)Hα(x,ξ)

and

g(w)Hα(x,w)

where x(12,1], the function g(w)=f1(w) is defined by (1.2) and Hα is the generating function of the Gegenbauer polynomial given by (1.3).

Remark 1.5. We note that the subclasses B˜Σq(x,1)=H˜Σq(x) and BΣ(x,1)=H(x), were introduced and studied by Altinkaya and Yalçin [3].

The following result will be required for proving our results.

Lemma 1.6. ([23]) Let P be the class of Caratheodory function with positive real part consisting of all analytic functions p:U satisfying p(0)=1 and Re(p(ξ))>0. If the function pP is defined by

p(ξ)=1+p1ξ+p2ξ2+p3ξ3+,

then

pn2, n

In this paper, we use the Gegenbauer polynomial expansions to provide estimates for the initial coefficients of the subclass of bi-univalent functions B˜Σq(x,α) defined by symmetric q-derivative operator. We also solve Fekete-Szegö problem for functions in this class.

Unless otherwise mentioned, we assume in the reminder of this paper that, 0<q<1, x(12,1] and α is a nonzero real constant.

### 2. Coefficient Bounds of the Class B˜Σq(x,α)

This section isdevoted to find initial coefficient bounds of the class B˜Σq(x,α) of bi-univalent functions.

Theorem 2.1. Let fΣ given by (1.1) belongs to the class B˜Σq(x,α) Then

a22αx2αx43~ qα2α 2~ q2(1+α)x2+2α 2~ q2x+α 2~ q2,

and

a34α2x22~q2+2αx3~q.

Proof. Let f B˜Σq(x,α). From Definition 1.3, for some analytic functions ψ,v such that ψ(0)=v(0)=0 and |ψ(ξ)|<1,|v(w)|<1 for all ξ,wU, then we can write

D˜qf(ξ)=Hα(x,w(ξ))

and

D˜qg(w)=Hα(x,v(w)),

Next, define the functions p,qP by

p(ξ)=1+ψ(ξ)1ψ(ξ)=1+c1ξ+c2ξ2+

and

q(w)=1+v(w)1v(w)=1+d1w+d2w2+.

In the following, one can derive

ψ(ξ)=p(ξ)1p(ξ)+1=12c1ξ+12c212c12ξ2+

and

v(w)=q(w)1q(w)+1=12d1w+12d212d12w2+.

From the equalities (2.1), (2.2), (2.3) and (2.4), we obtain that

D˜qf(ξ)=1+12C1α(x)c1ξ+14C2α(x)c12+12C1α(x)c212c12ξ2+

and

D˜qg(w)=1+12C1α(x)d1w+14C2α(x)d12+12C1α(x)d212d12w2+.

Thus, upon comparing the corresponding coefficients in (2.5) and (2.6), we have

2~qa2=12C1α(x)c1,
3~qa3=12C1α(x)c212c12+14C2α(x)c12,
2~qa2=12C1α(x)d1,

and

3~q2a22a3=12C1α(x)d212d12+14C2α(x)d12.

It follows from (2.7) and (2.9) that

c1=d1

and

22~q2a22=14C1α (x)2c12+d12.

If we add (2.8) and (2.10), we get

23~qa22=12C1α(x)c2+d2+14C2α(x)C1α(x)c12+d12.

Substituting the value of c12+d12 from (2.12) in the right hand side of (2.13), we deduce that

23~q2~q2C2α(x)C1α(x) C1α(x)2a22=12C1α(x)c2+d2.

Using (2.6), (1.16) and (2.14), we find that

a22αx2αx43~ qα2α2~ q2(1+α)x2+2α2~ q2x+α2~ q2.

Moreover, if we subtract (2.10) from (2.8), we obtain

43~qa3a22=12C1α(x)c2d2+14C2α(x)C1α(x)c12d12.

Then, in view of (1.7) and (2.12), equation (2.15) becomes

a3=C1α(x)282~q2c12+d12+C1α(x)43~qc2d2.

Thus applying (1.7) and (1.16), we have

a34α2x22~q2+2αx3~q.

### 3. Fekete-Szegö Problem for the Function Class B˜Σq(x,α)

Fekete-Szegö inequality is one of the famous problem related to coefficients of univalent analytic functions. It was first given by [12], who stated that, if fΣ, then

|a3ηa22|1+2e2η/(1μ).

This bound is sharp when η is real.

In this section, we aim to provide Fekete-Szegö inequalities for functions in the class B˜Σq(x,α). These inequalities are given in the following theorem.

Theorem 3.1. Let fΣ given by (1.1) belongs to the class B˜Σq(x,α). Then a3ηa22

2αx3~q,8α3x31η43~qα2α2~q2(1+α)x2+2α2~q2x+α 2~q2,η143~qα2α2~q2(1+α)x2+2α2~q2x+α2~q24α2x23~qη143~qα2α2~q2(1+α)x2+2α2~q2x+α2~q24α2x23~q,

where η.

Proof. From (2.14) and (2.15)

a3ηa22=1ηC1α (x)3c2+d243~qC1α (x)2C2α (x)C1α (x) 2~q2                +C1α(x)43~qc2d2    =C1α(x)h(η)+143~qc2+h(η)143~qd2,

where

h(η)=C1α(x)21η43~qC1α(x)2C2α(x)C1α(x) 2~q2,

Then, in view of (1.7) and (1.16), we conclude that

a3ηa222αx3~q8αxh(η)0h(η)143~q,h(η)143~q.

Which completes the proof of Theorem 3.1.

In this section, we apply our main results in order to deduce each of the following new corollaries and consequences.

Corollary 4.1. Let fΣ given by (1.1) belongs to the class B˜Σq(x,12). Then

a2x2x23~ q322~ q2 x2+2~ q2x+122~ q2,
a3x22~q2+x3~q,

and a3ηa22

x3~q,x31η23~q322~q2x2+2~q2x+122~q2,%η123~q322~q2x2+2~q2x+122~q2x23~qη123~q322~q2x2+2~q2x+122~q2x23~q,

where η.

Corollary 4.2. Let fΣ given by (1.1) belongs to the class BΣ(x,α). Then

a2αx2x12αx2+2x+1,
a3α2x2+2αx3,

and a3ηa222αx3,2α2x31η12αx2+2x+1,%η112αx2+2x+13αx2η112αx2+2x+13αx2,

where η.

By taking α=1, one can deduce the above results for various subclasses of Σ studied by Altinkaya and Yalçin [3].

The authors would like to thank the referees for their helpful comments and suggestions.

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