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

Further, let S denote the class of all functions f∈A 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 f≺g, if there exists a Schwarz function w, which is analytic in U with

such that

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

and

It is well known that every function f∈S has an inverse f−1, defined by

and

where

A function is said to be bi-univalent in U if both f(ξ) and f−1(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

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

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 maxf∈Σ|a2|=4/3.

The coefficient estimate problem for each of the Taylor-Maclaurin coefficients an(n≥3;n∈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(0≤ς<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<ς≤1), 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 T_R 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 S_R denotes the class of univalent functions in the unit disk with real coefficients, and co¯SR denotes the closed convex hull of S_R.

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

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

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

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

with the initial values

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 f∈A is, by definition, given as follows

From (1.8), we have

where

is sometimes called the basic number n. If q→1−,[n]q→n.

For a function h(ξ)=ξn, we obtain

and

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:

From (1.11), we deduce that D˜qξn=n~ qξn−1, and a power series of D˜qf is

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

Clearly, we have the following relations

and

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

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 f∈Σ given by (1.1) is said to be in the class B˜Σq(x,α) if the following subordinations are satisfied:

and

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

We note that limq→1−B˜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 f∈Σ given by (1.1) is said to be in the class BΣ(x,α) if the following subordinations are satisfied:

and

where x∈(12,1], the function g(w)=f−1(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→ℂ satisfying p(0)=1 and Re(p(ξ))>0. If the function p∈P is defined by

then

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.

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

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

and

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,w∈U, then we can write

and

Next, define the functions p,q∈P by

and

In the following, one can derive

and

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

and

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

and

It follows from (2.7) and (2.9) that

and

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

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

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

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

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

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

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 f∈Σ, then

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 f∈Σ given by (1.1) belongs to the class B˜Σq(x,α). Then a3−ηa22≤

where η∈ℝ.

Proof. From (2.14) and (2.15)

where

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

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 f∈Σ given by (1.1) belongs to the class B˜Σq(x,12). Then

and  a3−ηa22≤

where η∈ℝ.

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

and  a3−ηa22≤2αx3,2α2x31−η1−2αx2+2x+1,%η−1≤1−2αx2+2x+13αx2η−1≥1−2αx2+2x+13αx2,

where η∈ℝ.

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.