In mathematics, symmetry is defined as the property of two shapes being identical when one is moved, rotated, or flipped. The open unit disk, denoted by $\Delta =\{\mu :\left|\mu \right|<1\}$, exhibits a rich set of symmetries, consisting of inversion, rotational and reflection symmetry. Specifically, inversion symmetry means that the disk remains unchanged when inverted about a particular point, maintaining its overall appearance and structure. The disk $\Delta$ has inversion symmetry about its center (origion), meaning that inverting any complex number $\mu$ within the disk about the origin results in another complex number also within the disk, specifically ${\displaystyle \frac{1}{\mu }}$. This disk exhibits a rich set of symmetries, making it valuable in various mathematical and geometric contexts. Our goal is to explore additional geometric properties within this symmetric domain.

A function is considered starlike (or convex) if it transforms $\Delta$ into a star-shaped (or convex) region, centered at a fixed point, through scaling and rotation. This means the function’s image is contained within a star-shaped (or convex) domain, formed by connecting the fixed point to all other points with straight lines. Starlike and univalent functions are crucial subclasses of analytic functions with numerous applications and properties. Univalent functions are used in geometric function theory (GFT) for conformal mappings, while starlike functions model phenomena like electrostatics and fluid flow in GFT. Another important subclass of analytic functions is the class of close-to-convex functions. In this article we will focus on the study of bi-close-to-convex functions.

Let $\mathcal{A}$ indicate a collection of all analytic functions $\eta \left(\mu \right)$ in the region $\Delta =\{\mu :\left|\mu \right|<1\}$, which are normalized by

Thus, every $\eta \in \mathcal{A}$ can be expressed as:

Moreover, $\mathcal{S}$ is the subclass of $\mathcal{A}$ whose members in $\Delta$ are univalent. Let the class $\mathcal{P}$ be defined by

The following are some notable subclasses of the univalent functions in class $\mathcal{S}$: and

These classes defined in the following articles [1,2,3,4,5]. For $\alpha =0,$ then

For ${\eta }_1$, ${\eta }_2\in \mathcal{A},$ and ${\eta }_1$ subordinate to ${\eta }_2$ in $\eta$, denoted by

If we have a function $u_0\in \mathcal{A},$ satisfy the condition $\left|u_0\left(\mu \right)\right|<1,$$u_0\left(0\right)=0,$ and

The inverse of $\eta \in \mathcal{S},$ defined as: and

The series of ${\eta }^{-1}=\mathtt{F}$ is given by

If $\psi \in \mathcal{A}$ and then the series of ${\psi }^{-1}=G$ is given by

If $\eta$ and ${\eta }^{-1}$are in $\mathcal{S},$ then $\eta$ is considered the bi-univalent in $\Delta$ and such type of functions are denoted by $\Sigma .$ For $\eta \in \Sigma ,$ author in [6] proved that $\left|a_2\right|<1.51$ and after that authors in [7] gave the improvement of $\left|a_2\right|$ and proved that $\left|a_2\right|\le \sqrt{2}$. Furthermore, for $\eta \in \Sigma ,$ Netanyahu [8] proved that $max\left|a_2\right|={\displaystyle \frac{4}{3}}.$ (see for details [9,10,11]). Only non-sharp estimates on the initial coefficients were achieved in these recent works.

The Faber polynomials expansion method was first presented by Faber [12], who also utilized this approach to study coefficient bound $|a_n|$ for $n\ge 3$. Gong [13] emphasized the significance of Faber polynomials in mathematical sciences, specifically in the context of GFT. In the cited work [14,15,16,17], authors introduced new subcategories of bi-univalent functions and using the Faber polynomials expansion approach to establish coefficient bounds. Additionally, a number of researchers have used the Faber polynomials approach [18,19,20,21,22] to derive some intriguing findings for bi-univalent functions (see for detail [23,24]). In 2014, a group of researchers [25] in the field of geometric function theory developed a new class of functions called bi-close-to-convex functions of order $\alpha$, where $0\le \alpha <1$. This class is denoted by $C_{\Sigma }\left(\alpha \right)$ and was previously discussed in references [26,27,28].

To express the coefficients of its inverse map $\mathtt{F}$ in terms of the analytic functions $\eta$, use the Faber polynomial method (see [19,29]). where

For $7\le i\le n$, ${\mathcal{Q}}_i$ is a homogeneous polynomial in the variables $a_2,a_3,...,a_n$. Particularly, the first three terms of $K_{n-1}^{-n}$ are

For integers r, that is, $r=0,\pm 1,\pm 2,...,$ and integers $n\ge 2$, the quantity $K_{n-1}^r$ (see [19]) admits an expansion of the form: where, and by [29], we have

Taking the summation for ${\mu }_1,...,{\mu }_n$ which satisfying

Clearly, and

In the realm of (GFT), many scholars have built upon the foundational work of Jackson [30], who introduced the $q-$calculus operator in 1909. Ismail et al. [31] subsequently employed this operator to define $q-$starlike functions in $\Delta$. (Recent contributions to this field can be found in references [32,33,34,35,36,37]). To further expand this research, we must revisit the core definitions and principles of $q-$calculus and fractional $q-$calculus, paving the way for the construction of new subclasses of analytic and bi-univalent functions.

Definition 1.1. For $(0<q<1)$, the q-number n is given by and

Definition 1.2. [38]. The q-number shift factorial is given by for $\gamma \in \mathbb{C},$$n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},$ and

In terms of the q-Gamma function where the q-gamma function is defined by

We note that

For the q-gamma function ${\Gamma }_q\left(\mu \right),$ it is known that (see [39])

Jackson [40] introduced the q-difference operator for analytic functions as follows:

Definition 1.3. [40]. For $\eta \in \mathcal{A},$ the q-difference operator is defined as: and where ${\left[n\right]}_q$ given by (5) and

The q-analogous of the class of starlike functions was first introduced by Ismail et al. in [31] by means of the q-difference operator ${\partial }_q\eta \left(\mu \right),$$(0<q<1)$ and the q-integral is defined by

Remark 1.4.

Definition 1.5. Fractional q-integral operator, (see [41], page 57, Definition 1) the Fractional q-integral operator $I_{q,\mu }^{\lambda }$ of order $\lambda$ is defined by (see also [42], page 257) where, $\eta \left(\mu \right)$ is analytic in a simply connected region of the $\mu$-plane containing the origin and the q-binomial function ${\left(\mu -tq\right)}_{\lambda -1}$ is defined by

The definition of series ${ }_1{\Phi }_0$ is

The last equation is known as the q-binomial theorem (see [43] for more information). The series ${ }_1{\Phi }_0\left(a,-,q,\mu \right)$ is single valued, when $\left|arg\left(\mu \right)\right|<\pi$ and $\left|\mu \right|<1,$ (see for detail [39], pages 104–106) and ${\left(\mu -tq\right)}_{\lambda -1}$in (6) is single valued, when $\left|arg(-t{q}^{\lambda }/\mu )\right|<\pi ,\left|(t{q}^{\lambda }/\mu )\right|<\pi$ and $\left|arg\left(\mu \right)\right|<\pi$.

Definition 1.6. [41]. The fractional q-derivative operator $D_q$ of order $\lambda$ is defined by (see also [42], page 257, Definition 1.2) where $\eta \left(\mu \right)$ is suitably constrained and the multiplicity of ${\left(\mu -tq\right)}_{-\lambda }$ is removed as in Definition Section 1.

Definition 1.7. For m be the smallest integer. The extended fractional q-derivative $\left(D_q^{\lambda }\right)$ of order $\lambda$ defined by

We find from (7) that

Note that: When $-\infty <\lambda <0,$ then $D_q^{\lambda }$ represents a fractional q-integral of $\eta$ of order $\lambda$. For $0\le \lambda <2,$ then $D_q^{\lambda }$ represents a fractional q-derivative of $\eta$ of order $\lambda .$

Definition 1.8. ([44]). The $\left(\lambda ,q\right)$-fractional-differintegral operator $D_q^{\lambda }:\mathcal{A}\to \mathcal{A}$ defined as follows: where, and

Note that:

i. ii. and

First, we define a class of q-starlike functions of order $\beta$ associated with $\left(\lambda ,q\right)$-fractional-differintegral operator $D_q^{\lambda }$ and then we define the class of close-to-convex function by using the same operator. We start by creating a class of q-starlike functions of order $\beta$ that are linked to the $\left(\lambda ,q\right)$-fractional-differintegral operator $D_q^{\lambda }$. Then, we use the same operator to create a class $C_{\Sigma }^{\lambda ,q}\left(\alpha \right)$ of close-to-convex functions.

Definition 1.9. Let $\psi$ be of the form (4) be in the class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right),$ if where $0\le \beta <1,$$0<q<1.$

Definition 1.10. Let $\eta$ be of the form (1). Then $\eta \in C_{\Sigma }^{\lambda ,q}\left(\alpha \right)$ if there is a function $\psi \in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ satisfying and where, $0\le \alpha <1,$$0<q<1,$$0\le \lambda <2,$$\mu ,w\in \Delta$.

Remark 1.11. For $\lambda =0,$ we have a new class ${\mathcal{C}}_{\Sigma }\left(\alpha ,q\right)$ of bi-close-to-convex functions and be defined as: and where, $0\le \alpha <1,$$0<q<1,$$\mu ,w\in \eta$, ${\eta }^{-1}=F$ and ${\psi }^{-1}=G.$

Remark 1.12. For $\lambda =0,$ and $q\to 1-,$ then we have the known class of bi-close-to-convex functions investigated by Bulut in [45].

To show our primary points, we need the following Lemmas:

Lemma 2.1. [46]. If $p\in \mathcal{P}$ and

then

Lemma 2.2. [1]. If $p\in \mathcal{P}$and$\mu \in \mathbb{C}$, then

The aim of this work is to investigate a new subclass of the bi-univalent functions defined by $\left(\lambda ,q\right)$-fractional operator and by using the Faber Polynomials technique to obtain coefficients $\left|a_n\right|$, and initial coefficients bounds $\left|a_2\right|$ and $\left|a_3\right|$.

Theorem 3.1.Let an analytic function$\psi$be of the form (4). If$\psi \in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right),$ then

and

where${\mathcal{L}}_n$is given by (9) and$0\le \beta <1,$$0<q<1,$ $0\le \lambda <1.$

Theorem 3.2.If$\psi$is of the form (4) and$\psi \in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right),$ then

where

and$0\le \beta <1,$$0<q<1,$$0\le \lambda <2,$$\mu \in \mathbb{C}.$

Theorem 3.3.Let$\eta \in {\mathcal{C}}_{\Sigma }^{\lambda ,q}\left(\alpha \right)$be given by (1), if $a_i=0,$$2\le i\le n-1.$ Then for $n\ge 3$

where