Fractional Differential Operator Based on Quantum Calculus and Bi-Close-to-Convex Functions

Abstract

In this article, we first consider the fractional q-differential operator and the $\left(\lambda ,q\right)$-fractional differintegral operator $D_q^{\lambda }:\mathcal{A}\to \mathcal{A}$. Using the $\left(\lambda ,q\right)$-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ of starlike functions of order $\beta$ and the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions of order $\beta$. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order $\beta$. We also investigate the erratic behavior of the initial coefficients in the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research.

1. Preliminaries and Basic Notations

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 $\mathrm{\Delta }=\{\mu \in \mathbb{C}$:$\left|\mu \right|<1\}$, 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 $\mathrm{\Delta }$ has inversion symmetry about its center (origin), meaning that inverting any complex number $\mu$ within the disk about the origin results in another complex number also within the disk, specifically $\frac{1}{\mu }$. 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 $\mathrm{\Delta }$ 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 $\mathrm{\Delta }=\{\mu \in \mathbb{C}$:$\left|\mu \right|<1\}$, which are normalized by

$$\eta \left(0\right)=0\mathrm{and}{\eta }^{\prime }\left(0\right)=1.$$

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

$$\eta \left(\mu \right)=\mu +\sum _{n=2}^{\infty }{a}_n{\mu }^n.$$

(1)

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

$$\mathcal{P}=\left\{p\in \mathcal{A}:p\left(0\right)=1\mathrm{and}Re\left(p\left(\mu \right)\right)>0\right\}.$$

(2)

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

$${\mathcal{S}}^{*}\left(\alpha \right)=\left\{\eta \in \mathcal{A}:Re\left(\frac{\mu {\eta }^{\prime }\left(\mu \right)}{\eta \left(\mu \right)}\right)>\alpha \right\},0\le \alpha <1,$$

$$\mathcal{Q}\left(\alpha \right)=\left\{\eta \in \mathcal{A}:Re\left(\frac{{\left(\mu {\eta }^{\prime }\left(\mu \right)\right)}^{\prime }}{{\eta }^{\prime }\left(\mu \right)}\right)>\alpha \right\},0\le \alpha <1$$

and

$$\mathcal{C}\left(\alpha \right)=\left\{\eta \in \mathcal{A},\psi \in {\mathcal{S}}^{*}:Re\left(\frac{\mu {\eta }^{\prime }\left(\mu \right)}{\psi \left(\mu \right)}\right)>\alpha \right\},0\le \alpha <1.$$

For $\alpha =0$, then

$${\mathcal{S}}^{*}\left(0\right)={\mathcal{S}}^{*},\mathcal{Q}\left(0\right)=\mathcal{Q}\mathrm{and}\mathcal{C}\left(0\right)=\mathcal{C}.$$

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

$${\eta }_1\left(\mu \right)\prec {\eta }_2\left(\mu \right),\mu \in \mathrm{\Delta }.$$

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

$${\eta }_1\left(\mu \right)={\eta }_2\left(u_0\left(\mu \right)\right),\mu \in \mathrm{\Delta }.$$

The inverse of $\eta \in \mathcal{S}$ is defined as

$${\eta }^{-1}\left(\eta \left(\mu \right)\right)=\mu ,\mu \in \mathrm{\Delta }$$

and

$$\eta \left({\eta }^{-1}\left(w\right)\right)=w,\left|w\right|<r_0\left(\eta \right),\mathrm{and}{r}_0\left(\eta \right)\ge \frac{1}{4}.$$

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

$$\begin{array}{cc}\hfill \mathtt{F}\left(w\right)& =w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\dots \hfill \\ \hfill & =\mu +\sum _{n=2}^{\infty }{A}_n{\mu }^n.\hfill \end{array}$$

(3)

If $\psi \in \mathcal{A}$ and

$$\psi \left(\mu \right)=\mu +\sum _{n=2}^{\infty }{b}_n{\mu }^n$$

(4)

then the series of ${\psi }^{-1}=G$ is given by

$$G\left(\mu \right)=\mu +\sum _{n=2}^{\infty }{B}_n{\mu }^n.$$

If $\eta$ and ${\eta }^{-1}$ are in $\mathcal{S}$, then $\eta$ is considered bi-univalent in $\mathrm{\Delta }$ and such type of functions is denoted by $\mathrm{\Sigma }.$ For $\eta \in \mathrm{\Sigma }$, the author in [1] proved that $\left|a_2\right|<1.51$, and after that the authors in [2] gave the improvement of $\left|a_2\right|$ and proved that $\left|a_2\right|\le \sqrt{2}$. Furthermore, for $\eta \in \mathrm{\Sigma }$, Netanyahu [3] proved that $max\left|a_2\right|=\frac{4}{3}$. Later, researchers like Brannan and Taha [4] and Hayami and Owa [5] explored specific subgroups of bi-univalent functions, denoted as $\mathrm{\Sigma }$, and determined the bounds for their initial coefficients. The study of bi-univalent functions gained significant attention and momentum, particularly after the groundbreaking work of Xi et al. [6]. Only non-sharp estimates on the initial coefficients were achieved in these recent works.

The Faber polynomials expansion method was first presented by Faber [7], who also utilized this approach to study coefficient bound $|a_n|$ for $n\ge 3$. Gong [8] emphasized the significance of Faber polynomials in mathematical sciences, specifically in the context of GFT. Hamidi and Jahangiri [9] employed Faber polynomials to derive coefficient expansions for the class of analytic bi-close-to-convex functions and established estimates for these coefficients. They also revealed the erratic behavior of initial coefficients in subclasses of bi-univalent functions. This class is denoted by $C_{\mathrm{\Sigma }}\left(\alpha \right)$ and was further discussed in references [10,11,12]. Bult [13] investigated a general subclass of analytic bi-univalent functions and established estimates for their Taylor–Maclaurin coefficients using Faber polynomial expansions. Her results refined existing coefficient bounds in certain cases, providing improved estimates. In a subsequent study [14], the authors utilized Faber polynomial expansions to determine coefficient estimates for four prominent classes of bi-univalent functions, defined through subordinations. The authors of [15] utilized the Faber polynomial expansions method to investigate Janowski-type bi-close-to-convex and bi-quasi-convex functions. Meanwhile, the authors of [16] explored the application of the Faber polynomial expansion method to a subclass of analytic and bi-univalent functions related to Janowski functions. Hamidi and Jahangiri [17] utilized Faber polynomial expansions to derive sharp upper bounds for the n-th ($n\ge 3$) coefficients of bi-subordinate functions satisfying a specific gap series condition. Furthermore, they precisely determined the bounds for the first two coefficients of these functions. Serivastava [18] defined novel subclasses of analytic and bi-univalent functions in the open unit disk, leveraging a specific q-integral operator. By seamlessly integrating the Faber polynomial expansion method with q-analysis, they rigorously established bounds for the n-th coefficient in the Taylor–Maclaurin series expansion of functions within these newly defined classes, subject to a well-defined gap series condition.

To express the coefficients of its inverse map $\mathtt{F}$ in terms of the analytic functions $\eta$, use the Faber polynomial method (see [19,20])

$$F\left(w\right)={\eta }^{-1}\left(w\right)=w+\sum _{n=2}^{\infty }\frac{1}{n}{K}_{n-1}^n(a_2,a_3,\dots ,a_n)w^n,$$

where

$$\begin{array}{cc}\hfill K_{n-1}^{-n}& =\frac{(-n)!}{(-2n+1)!(n-1)!}{a}_2^{n-1}+\frac{(-n)!}{\left[2(-n+1)\right]!(n-3)!}{a}_2^{n-3}{a}_3\hfill \\ \hfill & +\frac{(-n)!}{(-2n+3)!(n-4)!}{a}_2^{n-4}{a}_4+\frac{(-n)!}{\left[2(-n+2)\right]!(n-5)!}{a}_2^{n-5}\left[a_5+(-n+2)a_3^2\right]\hfill \\ \hfill & +\frac{(-n)!}{(-2n+5)!(n-6)!}{a}_2^{n-6}\left[a_6+(-2n+5)a_3{a}_4\right]+\sum _{i\ge 7}{a}_2^{n-i}{\mathcal{Q}}_i.\hfill \end{array}$$

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

$$\begin{array}{ccc}& & \frac{1}{2}{K}_1^{-2}=-a_2,\frac{1}{3}{K}_2^{-3}=2a_2^2-a_3,\hfill \\ & & \frac{1}{4}{K}_3^{-4}=-(5a_2^3-5a_2{a}_3+a_4).\hfill \end{array}$$

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

$$K_{n-1}^r=r{a}_n+\frac{r(r-1)}{2}{\mathcal{V}}_{n-1}^2+\frac{r!}{(r-3)!3!}{\mathcal{V}}_{n-1}^3+\dots +\frac{r!}{(r-n+1)!(n-1)!}{\mathcal{V}}_{n-1}^{n-1},$$

where

$${\mathcal{V}}_{n-1}^r={\mathcal{V}}_{n-1}^r(a_2,a_3,\dots )$$

and by [20] we have

$${\mathcal{V}}_{n-1}^v(a_2,\dots ,a_n)=\sum _{n=1}^{\infty }\frac{v!}{{\mu }_{1!},\dots ,{\mu }_n!}{a}_2^{{\mu }_1}\dots a_n^{{\mu }_{n-1}},\mathrm{for}{a}_1=1\mathrm{and}v\le n.$$

The summation for ${\mu }_1,\dots ,{\mu }_n$ satisfies

$$\begin{array}{cc}\hfill & {\mu }_1+{\mu }_2+\dots +{\mu }_{n-1}=v,\hfill \\ \hfill & {\mu }_1+2{\mu }_2+\dots +(n-1){\mu }_{n-1}=n-1.\hfill \end{array}$$

Clearly,

$${\mathcal{V}}_{n-1}^{n-1}(a_1,\dots ,a_n)=a_2^{n-1}$$

and

$${\mathcal{V}}_{n-1}^1(a_1,\dots ,a_n)=a_{n-1}.$$

Fractional calculus is a exciting area of study for physicists and mathematicians. It offers a more powerful and elegant way to solve problems. Fractional differential equations are a new tool for modeling complex phenomena. Different types of derivatives, like Riemann–Liouville, Hadamard, fractional q-derivative operator, and Caputo, have been developed to help with this. The idea of fractional calculus has recently been used in a new way to study analytic functions. Researchers have used old and new definitions of fractional operators for the following:

1:

To understand properties of functions (characterization).

2:

To estimate coefficients (important numbers in functions) [21].

3:

To study distortion inequalities (how functions change shape) [22].

4:

To look at how functions work together (convolutions) for different types of analytic functions.

This work has been explained in detail in research books. In one of these books [23], a mathematician named Srivastava defined new ways to calculate fractional derivatives and integrals in the complex plane (a mathematical space where numbers have both real and imaginary parts).

More research connecting fractional calculus to univalent functions theory has been motivated and encouraged by the recent review study by Srivastava [24], which emphasized the benefits of incorporating fractional calculus into geometric function theory. Geometric characteristics are established for the fractional q-differential operator introduced utilizing the well-known q-derivative operator and the fractional q-derivative operator of order $\lambda$. An in-depth and thorough examination of q-calculus and fractional calculus, involving a meticulous and precise analysis of its principles and applications, was discussed in [25,26]. This study extends the examination of this research area.

Now, the basic definitions and concepts of q-calculus and fractional q-calculus need to be reviewed in order to construct some new subclasses of analytic and bi-univalent functions.

Definition 1.

For $0<q<1$, the q-number n is given by

$$\begin{array}{cc}\hfill {\left[n\right]}_q& =\frac{1-q^n}{1-q},n\in \mathbb{C}\hfill \\ \hfill & =\sum _{k=0}^{n-1}{q}^k=1+q+q^2+\dots +q^{n-1},n\in \mathbb{N}\hfill \end{array}$$

(5)

and

$${\left[0\right]}_q=0.$$

Definition 2 ([24]). 

The q-number shift factorial is given by $\gamma \in \mathbb{C}$, $n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$,

$${(\gamma ,q)}_n=\prod _{k=0}^{n-1}\left(1-\gamma q^k\right),\left(n\in \mathbb{N}\right)$$

and

$${(\gamma ,q)}_0=1.$$

In terms of the q-Gamma function,

$${(q^{\gamma },q)}_n=\frac{{\left(1-q\right)}^n{\mathrm{\Gamma }}_q(\gamma +n)}{{\mathrm{\Gamma }}_q\left(\gamma \right)},\left(n\in {\mathbb{N}}_0\right),$$

where the q-gamma function is defined by

$${\mathrm{\Gamma }}_q\left(\mu \right)=\frac{{\left(1-q\right)}^{1-\mu }{(q,q)}_{\infty }}{{(q^{\mu },q)}_{\infty }},\left|q\right|<1.$$

We note that

$${(\gamma ,q)}_{\infty }=\prod _{k=0}^{\infty }\left(1-\gamma q^k\right),\left|q\right|<1.$$

For the q-gamma function ${\mathrm{\Gamma }}_q\left(\mu \right)$, it is known that (see [27])

$${\mathrm{\Gamma }}_q(\mu +1)={\left[\mu \right]}_q{\mathrm{\Gamma }}_q\left(\mu \right).$$

Jackson [28] introduced the q-difference operator for analytic functions as follows:

Definition 3 ([28]).

For $\eta \in \mathcal{A}$, the q-difference operator is defined as

$${\partial }_q\eta \left(\mu \right)=\frac{\eta \left(q\mu \right)-\eta \left(\mu \right)}{\mu (q-1)}=1+\sum _{n=1}^{\infty }{\left[n\right]}_q{a}_n{\mu }^{n-1},\mu \in \mathrm{\Delta },\mu \ne 0,q\in \left(0,1\right)$$

and

$${\partial }_q\left({\mu }^n\right)={\left[n\right]}_q{\mu }^{n-1},\mathit{and}{\partial }_q\left(\sum _{n=1}^{\infty }{a}_n{\mu }^n\right)=\sum _{n=1}^{\infty }{\left[n\right]}_q{a}_n{\mu }^{n-1},$$

where ${\left[n\right]}_q$ is given by (5) and

$$\underset{q\to 1^{-}}{lim}{\partial }_q\eta \left(\mu \right)={\eta }^{\prime }\left(\mu \right).$$

The q-analog of the class of starlike functions was first introduced by Ismail et al. in [29] by means of the q-difference operator ${\partial }_q\eta \left(\mu \right)$, $(0<q<1)$ and the q-integral is defined by

$$\underset{0}{\overset{\mu }{\int }}\eta \left(t\right)d_{q}t=\mu \left(1-q\right)\sum _{n=0}^{\infty }{q}^n\eta \left(\mu q^n\right).$$

Remark 1.

$$\underset{q\to 1^{-}}{lim}\underset{0}{\overset{\mu }{\int }}\eta \left(t\right)d_{q}t=\underset{0}{\overset{\mu }{\int }}\eta \left(t\right)dt.$$

Definition 4.

Fractional q-integral operator (see [30], page 57, Definition 1): the fractional q-integral operator $I_{q,\mu }^{\lambda }$ of order λ is defined by (see also [31], page 257)

$$I_{q,\mu }^{\lambda }\eta \left(\mu \right)=\frac{1}{{\mathrm{\Gamma }}_q\left(\lambda \right)}\underset{0}{\overset{\mu }{\int }}{\left(\mu -tq\right)}_{\lambda -1}\eta \left(t\right)d_q\left(t\right),\lambda >0,$$

(6)

where, $\eta \left(\mu \right)$ is analytic in a simply connected region of the z-plane containing the origin and the q-binomial function ${\left(\mu -tq\right)}_{\lambda -1}$ is defined by

$${\left(\mu -tq\right)}_{\lambda -1}=\prod _{n=0}^{\infty }\left(\frac{1-\left(\frac{qt}{\mu }\right)q^n}{1-\left(\frac{qt}{\mu }\right)q^{\lambda +n-1}}\right)={\mu }^{\lambda -1}{}_1\mathrm{\Phi }_0\left(q^{-\lambda +1},-,q,\frac{t{q}^{\lambda }}{\mu }\right).$$

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

$${}_1\mathrm{\Phi }_0\left(a,-,q,\mu \right)=1+\sum _{n=1}^{\infty }\frac{{\left(a,q\right)}_n}{{\left(q,q\right)}_n}{\mu }^n,\left(\left|q\right|<1,\left|\mu \right|<1\right).$$

The last equation is known as the q-binomial theorem (see [32] for more information). The series ${}_1\mathrm{\Phi }_0\left(a,-,q,\mu \right)$ is single-valued when $\left|arg\left(\mu \right)\right|<\pi$ and $\left|\mu \right|<1$ (see for detail [27], 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 5 ([30]).

The fractional q-derivative operator $D_q$ of order λ is defined by (see also [31], page 257, Definition 1.2)

$$D_{q,\mu }^{\lambda }\eta \left(\mu \right)=D_{q,\mu }\left(I_{q,\mu }^{1-\lambda }\eta \left(\mu \right)\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(1-\lambda \right)}{\partial }_q\underset{0}{\overset{\mu }{\int }}{\left(\mu -tq\right)}_{-\lambda }\eta \left(t\right)d_q\left(t\right),\left(0\le \lambda <1\right),$$

where $\eta \left(\mu \right)$ is suitably constrained and the multiplicity of ${\left(\mu -tq\right)}_{-\lambda }$ is removed as in Definition 4.

Definition 6.

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

$$D_q^{\lambda }\eta \left(\mu \right)=D_q^m{I}_{q,\mu }^{m-\lambda }\eta \left(\mu \right).$$

(7)

We find from (7) that

$$D_q^{\lambda }{\mu }^n=\frac{{\mathrm{\Gamma }}_q\left(n+1\right)}{{\mathrm{\Gamma }}_q\left(n+1-\lambda \right)}{\mu }^{n-\lambda },(0\le \lambda ,n>-1).$$

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 7 ([33]).

The $\left(\lambda ,q\right)$-fractional differintegral operator $D_q^{\lambda }:\mathcal{A}\to \mathcal{A}$ is defined as follows:

$$\begin{array}{cc}\hfill D_q^{\lambda }\eta \left(\mu \right)& =\frac{{\mathrm{\Gamma }}_q\left(2-\lambda \right)}{{\mathrm{\Gamma }}_q\left(2\right)}{\mu }^{\lambda }{D}_q^{\lambda }\eta \left(\mu \right)=\mu +\sum _{n=2}^{\infty }\frac{{\mathrm{\Gamma }}_q\left(2-\lambda \right){\mathrm{\Gamma }}_q\left(n+1\right)}{{\mathrm{\Gamma }}_q\left(2\right){\mathrm{\Gamma }}_q\left(n+1-\lambda \right)}{a}_n{\mu }^n\hfill \\ \hfill & =\mu +\sum _{n=2}^{\infty }{\mathcal{L}}_n{a}_n{\mu }^n,\mu \in \mathrm{\Delta },\hfill \end{array}$$

(8)

where

$${\mathcal{L}}_n=\frac{{\mathrm{\Gamma }}_q\left(2-\lambda \right){\mathrm{\Gamma }}_q\left(n+1\right)}{{\mathrm{\Gamma }}_q\left(2\right){\mathrm{\Gamma }}_q\left(n+1-\lambda \right)}$$

(9)

and

$$\lambda <2,0<q<1.$$

Note that

i.

 

$$\underset{\lambda \to 1}{lim}{D}_q^{\lambda }\eta \left(\mu \right)=D_q\eta \left(\mu \right)=\mu {\partial }_q\eta \left(\mu \right).$$

ii.

 

$$D_q^{\lambda }\left(D_q^{\delta }\eta \left(\mu \right)\right)=D_q^{\delta }\left(D_q^{\lambda }\eta \left(\mu \right)\right)=\mu +\sum _{n=2}^{\infty }\frac{{\mathrm{\Gamma }}_q\left(2-\lambda \right){\mathrm{\Gamma }}_q\left(2-\delta \right){\left({\mathrm{\Gamma }}_q\left(n+1\right)\right)}^2}{{\mathrm{\Gamma }}_q\left(2\right){\mathrm{\Gamma }}_q\left(n+1-\lambda \right){\mathrm{\Gamma }}_q\left(n+1-\delta \right)}{a}_n{\mu }^n$$

and

$$\begin{array}{cc}\hfill \frac{D_q\left(D_q^{\lambda }\eta \left(\mu \right)\right)}{D_q^{\lambda }\eta \left(\mu \right)}& =\frac{D_q\eta \left(\mu \right)}{\eta \left(\mu \right)}=\frac{\mu {\partial }_q\eta \left(\mu \right)}{\eta \left(\mu \right)},\mathrm{for}\lambda =0\hfill \\ \hfill & =\frac{D_q\left(D_q\eta \left(\mu \right)\right)}{D_q\eta \left(\mu \right)}=\left(1+\frac{\mu {\partial }_q^2\eta \left(\mu \right)}{{\partial }_q\eta \left(\mu \right)}\right),\mathrm{for}\lambda =1.\hfill \end{array}$$

In the subject of GFT, a large number of scholars have developed and investigated various new subclasses of analytic functions using q-calculus and fractional q-calculus. The concept of the q-calculus operator was originally proposed by Jackson [34] in 1909, and the q-difference operator $\left({\partial }_q\right)$ was defined. On the other hand, Ismail et al. was the first to utilize the q-difference operator to construct a class of q-starlike functions in open unit disk $\mathrm{\Delta }$ in [29]. In a research paper [35], mathematicians Kanas and Wińskowa used a mathematical tool called the q-difference operator to define a new operator called the Ruscheweyh q-difference operator. They then investigated some of its properties, specifically those related to a geometric region called a conic domain. Additionally, Aldweby and Darus [36] used the method of differential subordination to investigate several interesting properties of this operator. In [37], Mahmood and Sokol introduced and examined a novel class of analytic functions utilizing the Ruscheweyh q-differential operator. This work involved deriving a structural formula, estimating coefficients, and addressing the Fekete–Szegö problem. In [38], Srivastava applied the principles of q-calculus to define a new subclass of q-starlike functions related to Janowski functions. He also explored new coefficient inequalities associated with this subclass. The authors of the article in [39] used a mathematical tool called the q-derivative operator to create a new group of q-starlike functions connected to a famous mathematical shape called the lemniscate of Bernoulli. They then found the maximum value (upper bound) and third Hankel determinant for this group of functions. Taking the motivations from earlier work [13,17,18], we now introduce a new class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ of q-starlike functions of order $\beta$, which are connected to the $\left(\lambda ,q\right)$-fractional differintegral operator $D_q^{\lambda }$. Using the same operator, we also define a class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions. We explored some fascinating properties and results for functions that belong to these classes.

Definition 8.

Let ψ be of the form (4) and be in the class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$, if

$$Re\left(\frac{D_q\left(D_q^{\lambda }\psi \left(\mu \right)\right)}{\psi \left(\mu \right)}\right)>\beta ,$$

where $0\le \beta <1$, $0<q<1.$

Definition 9.

Let η be of the form (1). Then, $\eta \in C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ if there is a function $\psi \in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ satisfying

$$Re\left(\frac{D_q\left(D_q^{\lambda }\eta \left(\mu \right)\right)}{\psi \left(\mu \right)}\right)>\alpha$$

and

$$Re\left(\frac{D_q\left(D_q^{\lambda }F\left(w\right)\right)}{G\left(w\right)}\right)>\alpha ,$$

where $0\le \alpha <1$, $0<q<1$, $0\le \lambda <2$, $\mu ,w\in \mathrm{\Delta }$.

Remark 2.

For $\lambda =0$, we have a new class ${\mathcal{C}}_{\mathrm{\Sigma }}\left(\alpha ,q\right)$ of bi-close-to-convex functions that can be defined as

$$Re\left(\frac{\mu {\partial }_q\eta \left(\mu \right)}{\psi \left(\mu \right)}\right)>\alpha$$

and

$$Re\left(\frac{\mu {\partial }_{q}F\left(w\right)}{G\left(w\right)}\right)>\alpha ,$$

where $0\le \alpha <1$, $0<q<1$, $\mu ,w\in \mathrm{\Delta }$, ${\eta }^{-1}=F$, and ${\psi }^{-1}=G$.

Remark 3.

For $\lambda =0$, and $q\to 1^{-}$, then we have the known class of bi-close-to-convex functions investigated by Bulut in [40].

The main novelty of the paper is to utilize the Faber polynomial expansion to derive upper bounds for the general Taylor–Maclaurin coefficients $\left|a_n\right|$ of functions in a novel subclass of bi-close-to-convex functions, defined by the $(\lambda ,q)$-fractional differintegral operator $D_q^{\lambda }$ in the open unit disk $\mathrm{\Delta }$. Our primary results, corollaries, and consequences will collectively generalize and refine previously established findings, providing an extension and improvement of existing knowledge in this field.

2. Set of Lemmas

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

Lemma 1 ([41]).

If $p\in \mathcal{P}$ and

$$p\left(\mu \right)=1+\sum _{n=1}^{\infty }{c}_n{\mu }^n,$$

then

$$\left|c_n\right|\le 2.$$

Lemma 2 ([42]).

If $p\in \mathcal{P}$ and $\mu \in \mathbb{C}$, then

$$\left|c_2-\mu c_2^2\right|\le 2max\left\{1;\left|2\mu -1\right|\right\}.$$

We started by explaining the basics of geometric function theory in Section 1 because it is essential to understand our main discovery. We also discussed some advanced mathematical tools like Faber polynomial methods, q-calculus, and $(\lambda ,q)$-fractional differintegral operator. Using the $(\lambda ,q)$-fractional differintegral operator, we defined two new subclasses of analytic and bi-univalent functions in Section 1. We then presented some preliminary results in Section 2. In the next section, we reveal our main findings, including new results about the n-th coefficients, and also explore the erratic behavior of initial coefficient estimates for the function $f\in C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$. Finally, we summarize our conclusions.

3. Main Results

Theorem 1.

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

$$\left|b_n\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)}\prod _{j=1}^{n-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right),n\ge 3$$

(10)

and

$$\left|b_2\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)},$$

(11)

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

Proof. 

Suppose $\psi \in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$, then

$$Re\left(\frac{D_q\left(D_q^{\lambda }\psi \left(\mu \right)\right)}{\psi \left(\mu \right)}\right)>\beta$$

Thus, by setting

$$\frac{\frac{D_q\left(D_q^{\lambda }\psi \left(\mu \right)\right)}{\psi \left(\mu \right)}-\beta }{1-\beta }=p\left(\mu \right)$$

or, equivalently,

$$\begin{array}{cc}\hfill & D_q\left(D_q^{\lambda }\psi \left(\mu \right)\right)=\left(\left(1-\beta \right)p\left(\mu \right)+\beta \right)\psi \left(\mu \right),\hfill \\ \hfill & \left(\mu +\sum _{n=2}^{\infty }{\mathcal{L}}_n{\left[n\right]}_q{b}_n{\mu }^n\right)=\left(1+\left(1-\beta \right)\sum _{n=1}^{\infty }{c}_n{\mu }^n\right)\left(\mu +\sum _{n=2}^{\infty }{b}_n{\mu }^n\right),\hfill \\ \hfill & \sum _{n=2}^{\infty }\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)b_n{\mu }^n=\left(1-\beta \right)\sum _{n=1}^{\infty }{c}_n{\mu }^{n+1}+\left(1-\beta \right)\sum _{n=2}^{\infty }\left(\sum _{j=1}^{n-1}{c}_j{b}_{n-j}\right){\mu }^n.\hfill \end{array}$$

Comparing ${\mu }^n$ on both sides, we have

$$\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)b_n=\left(1-\beta \right)\sum _{j=1}^{n-1}{c}_j{b}_{n-j}$$

and

$$\left|\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)b_n\right|=\left(1-\beta \right)\sum _{j=1}^{n-1}\left|c_j\right|\left|b_{n-j}\right|.$$

Using Lemma 1, we have

$$\left|\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)b_n\right|\le 2\left(1-\beta \right)\sum _{j=1}^{n-1}\left|b_j\right|.$$

(12)

So, for $n=2$ in (12), we have

$$\left|b_2\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}.$$

(13)

This confirms that Equation (10) is true for the base case $n=2$. To establish the general validity of Equation (10), we employ mathematical induction. In the next step, we consider the case $n=3$, and from Equation (12), we obtain

$$\left|b_3\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_3{\left[3\right]}_q-1\right)}\left(1+\left|b_2\right|\right)\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_3{\left[3\right]}_q-1\right)}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\right).$$

This verifies that Equation (10) is true for $n=3$. Moving on to the case $n=4$, we can see from Equation (12) that

$$\left|b_4\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_4{\left[4\right]}_q-1\right)}\left(1+\left|b_2\right|+\left|b_3\right|\right)$$

or this can be written as

$$\left|b_4\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_4{\left[4\right]}_q-1\right)}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\right)\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_3{\left[3\right]}_q-1\right)}\right).$$

This demonstrates that Equation (10) is valid for $n=3$. Next, we assume that Equation (10) is true for all n less than or equal to t; that is,

$$\left|b_t\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_t{\left[t\right]}_q-1\right)}\prod _{j=1}^{t-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right).$$

Consider

$$\begin{array}{cc}\hfill \left|b_{t+1}\right|\le & \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{t+1}{\left[t+1\right]}_q-1\right)}\left(1+\left|b_2\right|+\left|b_3\right|+\dots +\left|b_t\right|\right)\hfill \\ \hfill \le & \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{t+1}{\left[t+1\right]}_q-1\right)}\times \left[1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_3{\left[3\right]}_q-1\right)}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\right)\right.\hfill \\ \hfill & +\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_4{\left[4\right]}_q-1\right)}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\right)\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_3{\left[3\right]}_q-1\right)}\right)\hfill \\ \hfill & \left.+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\prod _{j=1}^{t-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right]\hfill \\ \hfill & =\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{t+1}{\left[t+1\right]}_q-1\right)}\prod _{j=1}^{t-1}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right).\hfill \end{array}$$

Therefore, the result holds for $n=t+1$. Thus, by mathematical induction, we have established that Equation (10) is true for all integers n greater than or equal to 2. This concludes the proof. □

Theorem 2.

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

$$\left|b_3-\mu b_2^2\right|\le \frac{2(1-\beta )}{{\mathcal{L}}_3{\left[3\right]}_q-1}max\left\{1;\left|1+\frac{2(1-\beta )}{{\mathcal{L}}_2{\left[2\right]}_q-1}\left(1-\mu \frac{\left({\mathcal{L}}_3{\left[3\right]}_q-1\right)}{{\mathcal{L}}_2{\left[2\right]}_q-1}\right)\right|\right\},$$

where

$${\mathcal{L}}_3=\left(\frac{{\mathrm{\Gamma }}_q\left(2-\lambda \right){\mathrm{\Gamma }}_q\left(4\right)}{{\mathrm{\Gamma }}_q\left(2\right){\mathrm{\Gamma }}_q\left(4-\lambda \right)}{\left[3\right]}_q-1\right),{\mathcal{L}}_2=\left(\frac{{\mathrm{\Gamma }}_q\left(2-\lambda \right){\mathrm{\Gamma }}_q\left(3\right)}{{\mathrm{\Gamma }}_q\left(2\right){\mathrm{\Gamma }}_q\left(3-\lambda \right)}{\left[2\right]}_q-1\right),$$

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

Proof. 

If $\psi \in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$, then we have

$$Re\left(\frac{D_q\left(D_q^{\lambda }\psi \left(\mu \right)\right)}{\psi \left(\mu \right)}\right)>\beta .$$

Then there exists a positive real part function $p\left(\mu \right)=1+{\sum }_{n=1}^{\infty }{c}_n{\mu }^n\in \mathcal{P}$, such that

$$\frac{D_q\left(D_q^{\lambda }\psi \left(\mu \right)\right)}{\psi \left(\mu \right)}=\beta +(1-\beta )p\left(\mu \right)=1+(1-\beta )\sum _{n=1}^{\infty }{c}_n{\mu }^n.$$

(14)

From (14), we have

$$b_2=\frac{\left(1-\beta \right)c_1}{{\mathcal{L}}_2{\left[2\right]}_q-1}$$

(15)

and

$$b_3=\frac{\left(1-\beta \right)}{{\mathcal{L}}_3{\left[3\right]}_q-1}\left(c_2+\frac{\left(1-\beta \right)c_1^2}{{\mathcal{L}}_2{\left[2\right]}_q-1}\right).$$

(16)

From (15) and (16),

$$\left|b_3-\mu b_2^2\right|=\frac{\left(1-\beta \right)}{{\mathcal{L}}_3{\left[3\right]}_q-1}\left|c_2-v{c}_1^2\right|,$$

where

$$v=-\frac{\left(1-\beta \right)}{{\mathcal{L}}_2{\left[2\right]}_q-1}\left(1-\mu \frac{{\mathcal{L}}_3{\left[3\right]}_q-1}{{\mathcal{L}}_2{\left[2\right]}_q-1}\right).$$

Our result is a direct consequence of Lemma 2, and this completes the proof. □

Theorem 3.

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

$$\begin{array}{cc}\hfill & \left|a_n\right|\le \frac{1}{{\mathcal{L}}_n{\left[n\right]}_q}\left\{\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)}\prod _{j=1}^{n-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right.\hfill \\ \hfill & +2\left(1-\alpha \right)+\sum _{l=1}^{n-2}\left(\frac{2\left(1-\beta \right)}{\left({\left[n-l\right]}_q{\mathcal{L}}_{n-l}-1\right)}\times \left.\prod _{j=1}^{n-l-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right)\mathrm{{\rm Y}}\left(l\right)\right\},\hfill \end{array}$$

where

$$\mathrm{{\rm Y}}\left(l\right)=min\left\{\left|K_l^{-1}(b_2,b_3,\dots ,b_{l+1})\right|;\left|K_l^{-1}(B_2,B_3,\dots ,B_{l+1})\right|\right\}.$$

Proof. 

Let $\eta \in {\mathcal{C}}_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$. Then, there is a function $\psi \left(\mu \right)=\mu +{\displaystyle \sum _{n=2}^{\infty }}{b}_n{\mu }^n$. The Faber polynomial expansion of ${\displaystyle \frac{D_q\left(D_q^{\lambda }\eta \left(\mu \right)\right)}{\psi \left(\mu \right)}}$ is

$$\begin{array}{cc}\hfill \frac{D_q\left(D_q^{\lambda }\eta \left(\mu \right)\right)}{\psi \left(\mu \right)}=1+& \sum _{n=2}^{\infty }[\left({\mathcal{L}}_n{\left[n\right]}_q{a}_n-b_n\right)\hfill \\ \hfill & +\sum _{l=1}^{n-2}{K}_l^{-1}(b_2,b_3,\dots b_{l+1})\left(\left({\left[n\right]}_q-l\right){\mathcal{L}}_n{a}_{n-l}-b_{n-l}\right)]{\mu }^{n-1}.\hfill \end{array}$$

(17)

For $F={\eta }^{-1}$ and $G={\psi }^{-1}$, we obtain

$$\begin{array}{cc}\hfill \frac{D_q\left(D_q^{\lambda }F\left(w\right)\right)}{G\left(w\right)}=1+& \sum _{n=2}^{\infty }[\left({\mathcal{L}}_n{\left[n\right]}_q{A}_n-B_n\right)\hfill \\ \hfill & +\sum _{l=1}^{n-2}{K}_l^{-1}(B_2,B_3,\dots B_{l+1})\left(\left({\left[n\right]}_q-l\right){\mathcal{L}}_n{A}_{n-l}-B_{n-l}\right)]w^{n-1}.\hfill \end{array}$$

(18)

Since $Re{\displaystyle \frac{D_q\left(D_q^{\lambda }\eta \left(\mu \right)\right)}{\psi \left(\mu \right)}}>\alpha$ in $\mathrm{\Delta }$, there is a function with a positive real part

$$p\left(\mu \right)=1+\sum _{n=1}^{\infty }{c}_n{\mu }^n\in \mathcal{P},$$

such that

$$\frac{D_q\left(D_q^{\lambda }\eta \left(\mu \right)\right)}{\psi \left(\mu \right)}=1+\left(1-\alpha \right)p\left(\mu \right)=1+\left(1-\alpha \right)\sum _{n=1}^{\infty }{c}_n{\mu }^n.$$

(19)

Similarly, $Re{\displaystyle \frac{D_q\left(D_q^{\lambda }F\left(w\right)\right)}{G\left(w\right)}}>\alpha$ in $\mathrm{\Delta }$, so there is a function with a positive real part

$$q\left(w\right)=1+\sum _{n=1}^{\infty }{d}_n{w}^n\in \mathcal{P},$$

so that

$$\frac{D_q\left(D_q^{\lambda }F\left(w\right)\right)}{G\left(w\right)}=1+\left(1-\alpha \right)q\left(w\right)=1+\left(1-\alpha \right)\sum _{n=1}^{\infty }{d}_n{w}^n.$$

(20)

Evaluating the coefficients of Equations (17) and (19), for any $n\ge 2$, yields

$$\begin{array}{c}\hfill \left({\left[n\right]}_q{\mathcal{L}}_n{a}_n-b_n\right)+\sum _{l=1}^{n-2}{K}_l^{-1}(b_2,b_3,\dots b_{l+1})\times \left({\mathcal{L}}_{n-l}\left({\left[n\right]}_q-l\right)a_{n-l}-b_{n-l}\right)\\ \hfill =\left(1-\alpha \right)c_{n-1}.\end{array}$$

(21)

Evaluating the coefficients of Equations (18) and (20), for any $n\ge 2$, yields

$$\begin{array}{c}\hfill \left({\left[n\right]}_q{\mathcal{L}}_n{A}_n-B_n\right)+\sum _{l=1}^{n-2}{K}_l^{-1}(B_2,B_3,\dots B_{l+1})\times \left({\mathcal{L}}_{n-l}\left({\left[n\right]}_q-l\right)A_{n-l}-B_{n-l}\right)\\ \hfill =\left(1-\alpha \right)d_{n-1}.\end{array}$$

(22)

But under the assumption $2\le i\le n-1$ and $a_i=0$, respectively, we find from (21) and (22) that

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{n-l}{\left[n\right]}_q{a}_n-b_n\right)-\sum _{l=1}^{n-2}{b}_{n-l}{K}_l^{-1}(b_2,b_3,\dots b_{l+1})=\left(1-\alpha \right)c_{n-1},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & -\left({\mathcal{L}}_{n-l}{\left[n\right]}_q{A}_n-B_n\right)-\sum _{l=1}^{n-2}{B}_{n-l}{K}_l^{-1}(B_2,B_3,\dots B_{l+1})=\left(1-\alpha \right)d_{n-1}.\hfill \end{array}$$

(24)

Also, the equality $a_i=0$ ($2\le i\le n-1)$ implies that

$$A_n=-a_n.$$

Thus, (23) and (24) give

$${\mathcal{L}}_n{\left[n\right]}_q{a}_n=b_n+\left(1-\alpha \right)c_{n-1}+\sum _{l=1}^{n-2}{b}_{n-l}{K}_l^{-1}(b_2,b_3,\dots b_{l+1})$$

and

$$-{\mathcal{L}}_n{\left[n\right]}_q{a}_n=B_n+\left(1-\alpha \right)d_{n-1}+\sum _{l=1}^{n-2}{B}_{n-l}{K}_l^{-1}(B_2,B_3,\dots B_{l+1}),$$

respectively. Taking the absolute value (or modulus) of both sides, we obtain

$$\left|{\mathcal{L}}_n{\left[n\right]}_q\right|\left|a_n\right|=\left|b_n\right|+\left(1-\alpha \right)\left|c_{n-1}\right|+\sum _{l=1}^{n-2}\left|b_{n-l}{K}_l^{-1}(b_2,b_3,\dots b_{l+1})\right|$$

and

$$\left|{\mathcal{L}}_n{\left[n\right]}_q\right|\left|a_n\right|=\left|B_n\right|+\left(1-\alpha \right)\left|d_{n-1}\right|+\sum _{l=1}^{n-2}\left|B_{n-l}{K}_l^{-1}(B_2,B_3,\dots B_{l+1})\right|.$$

Since $\psi ,G\in {\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$, we use Theorem 1 to obtain

$$\begin{array}{cc}\hfill & \left|{\left[n\right]}_q{\mathcal{L}}_n\right|\left|a_n\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)}\times \prod _{j=1}^{n-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)+\left(1-\alpha \right)\left|c_{n-1}\right|\hfill \\ \hfill & +\sum _{l=1}^{n-2}\left(\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{n-l}{\left[n-l\right]}_q-1\right)}\prod _{j=1}^{n-l-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right)\times \left|K_l^{-1}(b_2,b_3,\dots b_{l+1})\right|\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left|{\mathcal{L}}_n{\left[n\right]}_q\right|\left|a_n\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)}\prod _{j=1}^{n-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)+\left(1-\alpha \right)\left|d_{n-1}\right|\hfill \\ \hfill & +\sum _{l=1}^{n-2}\left(\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{n-l}{\left[n-l\right]}_q-1\right)}\prod _{j=1}^{n-l-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right)\times \left|K_l^{-1}(B_2,B_3,\dots B_{l+1})\right|.\hfill \end{array}$$

Using Lemma 1, we obtain

$$\begin{array}{cc}\hfill & \left|{\left[n\right]}_q{\mathcal{L}}_n\right|\left|a_n\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)}\prod _{j=1}^{n-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)+2\left(1-\alpha \right)\hfill \\ \hfill & +\sum _{l=1}^{n-2}\left(\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{n-l}{\left[n-l\right]}_q-1\right)}\prod _{j=1}^{n-l-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right)\times \left|K_l^{-1}(b_2,b_3,\dots b_{l+1})\right|\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill & \left|{\left[n\right]}_q{\mathcal{L}}_n\right|\left|a_n\right|\le \frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_n{\left[n\right]}_q-1\right)}\prod _{j=1}^{n-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)+2\left(1-\alpha \right)\hfill \\ \hfill & +\sum _{l=1}^{n-2}\left(\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{n-l}{\left[n-l\right]}_q-1\right)}\prod _{j=1}^{n-l-2}\left(1+\frac{2\left(1-\beta \right)}{\left({\mathcal{L}}_{j+1}{\left[j+1\right]}_q-1\right)}\right)\right)\times \left|K_l^{-1}(B_2,B_3,\dots B_{l+1})\right|.\hfill \end{array}$$

(26)

By comparing (25) and (26), we obtain $\left|a_n\right|$ as asserted in Theorem 3. □

For $\lambda =0$ and $q\to 1^{-}$, we obtain a recognized corollary in Theorem 3 that was proved in [43].

Corollary 1 ([43]).

Let $\eta \in {\mathcal{C}}_{\mathrm{\Sigma }}\left(\alpha ,\beta \right)$. If $a_i=0$, $2\le i\le n-1$, then for $n\ge 3$

$$\left|a_n\right|\le \frac{1}{n!}\prod _{j=0}^{n-2}\left(j+2\left(1-\beta \right)\right)+\frac{2\left(1-\alpha \right)}{n}+\frac{1}{n}\sum _{l=1}^{n-2}\left(\frac{1}{\left(n-l-1\right)!}\prod _{j=0}^{n-l-2}\left(j+2\left(1-\beta \right)\right)\right)\mathrm{{\rm Y}}\left(l\right),$$

where

$$\mathrm{{\rm Y}}\left(l\right)=min\left\{\left|K_l^{-1}(b_2,b_3,\dots b_{l+1})\right|;\left|K_l^{-1}(B_2,B_3,\dots B_{l+1})\right|\right\}.$$

For $\lambda =0$, $\beta =0$ and $q\to 1^{-}$ we obtain a recognized corollary in Theorem 3 that was proved in [44].

Corollary 2 ([44]).

Let $\eta \in {\mathcal{C}}_{\mathrm{\Sigma }}\left(\alpha \right)$, if $a_i=0$, $2\le i\le n-1$. Then,

$$\left|a_n\right|\le 1+\frac{2\left(1-\alpha \right)}{n}+\frac{1}{n}\sum _{l=1}^{n-2}\left(n-l\right)min\left\{\left|K_l^{-1}(b_2,b_3,\dots b_{l+1})\right|;\left|K_l^{-1}(B_2,B_3,\dots B_{l+1})\right|\right\}.$$

Theorem 4.

Let $\eta \in {\mathcal{C}}_{\mathrm{\Sigma }}\left(m,\alpha ,q\right)$ be given by (1). Then,

$$\begin{array}{cc}\hfill \left|a_2\right|\le min\{& \frac{2\left(1-\alpha \right)}{{\left[2\right]}_q{\mathcal{L}}_2}+\frac{2\left(1-\beta \right)}{{\left[2\right]}_q{\mathcal{L}}_2\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)};\hfill \\ \hfill & \sqrt{\frac{2\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}+\frac{4q\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\left(\frac{\left(1-\alpha \right)}{1}+\frac{\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\right)}\}\hfill \end{array}$$

(27)

and

$$\begin{array}{cc}\hfill \left|a_3\right|\le & min\{\frac{\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}\left[1+max\left\{1,\left|V_1\right|\right\}\right]+\frac{\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\left[3\right]}_q{\mathcal{L}}_3-1\right)}max\left\{1;\left|V_2\right|\right\}\hfill \\ \hfill & +\frac{8\left(1-\alpha \right)\left(1-\beta \right)}{{\left[2\right]}_q^2{\mathcal{L}}_2^2\left({\left[2\right]}_q{\mathcal{L}}_2-1\right)};\frac{2\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}+\frac{4q\left(1-\alpha \right)\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\left[2\right]}_q{\mathcal{L}}_2-1\right)}+\frac{4\left(q-1\right){\left(1-\beta \right)}^2}{{\left[3\right]}_q{\mathcal{L}}_3{\left({\left[2\right]}_q{\mathcal{L}}_2-1\right)}^2}\hfill \\ \hfill & +\frac{2\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\left[3\right]}_q{\mathcal{L}}_3-1\right)}\left(1+\frac{2\left(1-\beta \right)}{{\left[2\right]}_q{\mathcal{L}}_2-1}\right)\}.\hfill \end{array}$$

(28)

Proof. 

Substituting $n=2$ in Equation (21) and $n=3$ in Equation (22), we obtain

$$\begin{array}{ccc}& & a_2=\frac{\left(1-\alpha \right)}{{\left[2\right]}_q{\mathcal{L}}_2}{c}_1+\frac{b_2}{{\left[2\right]}_q{\mathcal{L}}_2},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & a_3=\frac{\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}{c}_2+\frac{q\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2{c}_1+\frac{q-1}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2^2+\frac{b_3}{{\left[3\right]}_q{\mathcal{L}}_3},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & -a_2=\frac{\left(1-\alpha \right)}{{\left[2\right]}_q{\mathcal{L}}_2}{d}_1-\frac{b_2}{{\left[2\right]}_q{\mathcal{L}}_2},\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& & 2a_2^2-a_3=\frac{\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}{d}_2-\frac{q\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2{d}_1+\frac{q+1}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2^2-\frac{b_3}{{\left[3\right]}_q{\mathcal{L}}_3}.\hfill \end{array}$$

(32)

From (29) and (31), we find

$$c_1=-d_1.$$

(33)

On the other hand, from (30) and (32), we obtain

$$a_2^2=\frac{\left(1-\alpha \right)}{2{\left[3\right]}_q{\mathcal{L}}_3}\left(c_2+d_2\right)+\frac{q\left(1-\alpha \right)}{2{\left[3\right]}_q{\mathcal{L}}_3}{b}_2\left(c_1-d_1\right)+\frac{q}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2^2.$$

(34)

Therefore by applying triangle inequality to (29) and (34), using Lemma 1, we obtain

$$\left|a_2\right|\le \frac{2\left(1-\alpha \right)}{{\left[2\right]}_q{\mathcal{L}}_2}+\frac{\left|b_2\right|}{{\left[2\right]}_q{\mathcal{L}}_2}$$

(35)

and

$${\left|a_2\right|}^2\le \frac{2\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}+\frac{2q\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}\left|b_2\right|+\frac{q}{{\left[3\right]}_q{\mathcal{L}}_3}{\left|b_2\right|}^2.$$

(36)

Using inequality (11), we have

$$\left|a_2\right|\le \frac{2\left(1-\alpha \right)}{{\left[2\right]}_q{\mathcal{L}}_2}+\frac{2\left(1-\beta \right)}{{\left[2\right]}_q{\mathcal{L}}_2\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}$$

(37)

and

$$\left|a_2\right|\le \sqrt{\frac{2\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}+\frac{4q\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\left(\frac{\left(1-\alpha \right)}{1}+\frac{\left(1-\beta \right)}{\left({\mathcal{L}}_2{\left[2\right]}_q-1\right)}\right)}.$$

(38)

We obtain the required bound $\left|a_2\right|$ as asserted in (27). Now we subtract (32) from (30), and we thus obtain

$$2a_3-2a_2^2=\frac{\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}\left(c_2-d_2\right)-\frac{2}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2^2+\frac{2}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_3+\frac{q\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}\left(c_1+d_1\right)b_2.$$

By (33), we obtain

$$a_3=a_2^2+\frac{\left(1-\alpha \right)}{2{\left[3\right]}_q{\mathcal{L}}_3}\left(c_2-d_2\right)+\frac{1}{{\left[3\right]}_q{\mathcal{L}}_3}\left(b_3-b_2^2\right).$$

(39)

If we set the value of $a_2^2$ from (29) in (39), then we have

$$\begin{array}{c}\hfill a_3=\frac{\left(1-\alpha \right)}{2{\left[3\right]}_q{\mathcal{L}}_3}\left(c_2-\left(\frac{-2{\left[3\right]}_q{\mathcal{L}}_3\left(1-\alpha \right)}{{\left[2\right]}_q^2{\mathcal{L}}_2^2}\right)c_1^2\right)+\frac{1}{{\left[3\right]}_q{\mathcal{L}}_3}\left(b_3-\left(1-\frac{{\left[3\right]}_q{\mathcal{L}}_3}{{\left[2\right]}_q^2{\mathcal{L}}_2^2}\right)b_2^2\right)\\ \hfill +\frac{2\left(1-\alpha \right)}{{\left[2\right]}_q^2{\mathcal{L}}_2^2}{c}_1{b}_2-\frac{1-\alpha }{2{\left[3\right]}_q{\mathcal{L}}_3}{d}_2.\end{array}$$

So, using Lemma 2, Theorem 2, (1), and (11), we obtain

$$\begin{array}{c}\hfill \left|a_3\right|\le \frac{\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}\left[1+max\left\{1;\left|V_1\right|\right\}\right]+\frac{\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\left[3\right]}_q{\mathcal{L}}_3-1\right)}max\left\{1;\left|V_2\right|\right\}\\ \hfill +\frac{8\left(1-\alpha \right)\left(1-\beta \right)}{{\left[2\right]}_q^2{\mathcal{L}}_2^2\left({\left[2\right]}_q{\mathcal{L}}_2-1\right)},\end{array}$$

(40)

where

$$\begin{array}{cc}\hfill V_1& =1+\frac{4{\left[3\right]}_q{\mathcal{L}}_3\left(1-\alpha \right)}{{\left[2\right]}_q^2{\mathcal{L}}_2^2},\hfill \\ \hfill V_2& =1+\frac{2\left(1-\beta \right)}{{\left[2\right]}_q{\mathcal{L}}_2-1}\left(1-\left(1-\frac{{\left[3\right]}_q{\mathcal{L}}_3}{{\left[2\right]}_q^2{\mathcal{L}}_2^2}\right)\left(\frac{{\left[3\right]}_q{\mathcal{L}}_3-1}{{\left[2\right]}_q{\mathcal{L}}_2-1}\right)\right).\hfill \end{array}$$

If we set the value of $a_2^2$ from (29) in (39), then we have

$$a_3=\frac{\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}{c}_2+\frac{q\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2{c}_1+\frac{q-1}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_2^2+\frac{1}{{\left[3\right]}_q{\mathcal{L}}_3}{b}_3.$$

Using Lemma 2 and Theorem 2, we obtain

$$\begin{array}{cc}\hfill \left|a_3\right|\le & \frac{2\left(1-\alpha \right)}{{\left[3\right]}_q{\mathcal{L}}_3}+\frac{4q\left(1-\alpha \right)\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\left[2\right]}_q{\mathcal{L}}_2-1\right)}+\frac{4\left(q-1\right){\left(1-\beta \right)}^2}{{\left[3\right]}_q{\mathcal{L}}_3{\left({\left[2\right]}_q{\mathcal{L}}_2-1\right)}^2}\hfill \\ \hfill & +\frac{2\left(1-\beta \right)}{{\left[3\right]}_q{\mathcal{L}}_3\left({\left[3\right]}_q{\mathcal{L}}_3-1\right)}\left(1+\frac{2\left(1-\beta \right)}{{\left[2\right]}_q{\mathcal{L}}_2-1}\right).\hfill \end{array}$$

(41)

Hence, (40) and (41) give the required estimate $\left|a_3\right|$, as asserted in (28). □

As a consequence of Theorem 4, we recover the well-known corollary established in [43] when $\lambda =0$ and q approaches $1^{-}$.

Corollary 3 ([43]).

Let $\eta \in {\mathcal{C}}_{\mathrm{\Sigma }}\left(\alpha \right)$ be given by (1). Then,

$$\left|a_2\right|\le min\left\{2-\alpha -\beta ;\sqrt{\frac{1}{3}\left\{4\left(1-\beta \right)\left(2-\alpha -\beta \right)+2\left(1-\alpha \right)\right\}}\right\}$$

and

$$\left|a_3\right|\le \frac{1}{3}\times \left\{\begin{array}{ccc}\left(3-2\beta \right)\left(3-2\alpha -\beta \right),\hfill & \mathrm{if}\hfill & 0\le \alpha \le {\displaystyle \frac{2+\beta }{3}}\hfill \\ \\ \left(1-\alpha \right)\left(5-3\alpha \right)+\left(1-\beta \right)\left(2-\beta \right)+6\left(1-\alpha \right)\left(1-\beta \right),\hfill & \mathrm{if}\hfill & {\displaystyle \frac{2+\beta }{3}}\le \alpha <1.\hfill \end{array}\right.$$

4. Conclusions

Modern research has been significantly influenced by fractional calculus, which has many uses in many areas of science and engineering. It also has implications for many areas of mathematics. For example, it is used in a wide range of complex analysis studies, and it has resulted in some interesting new findings in studies involving analytic functions theory. This article has three parts. As the basics of geometric function theory were necessary to understand our major discovery, we briefly covered them in Section 1. In this section, we also discussed q-calculus, the fractional q-derivative operator, and the $\left(\lambda ,q\right)$-fractional differintegral operator. Using this operator, we defined two new subclasses of analytic and bi-univalent functions. These elements were well recognized, and we appropriately referenced them. The preliminary lemmas were presented in Section 2. In Section 3, utilizing Lemmas 1 and 2, we first investigated Theorems 1 and 2. Then, using these two theorems and the Faber polynomial expansion technique, we determined the upper bound of the n-th coefficient for functions belonging to these newly defined classes. We also explored the erratic behavior of the initial coefficients of bi-close-to-convex functions, which were characterized by the $(\lambda ,q)$-fractional differintegral operator. Furthermore, we investigated Fekete–Szegö problems and provided some notable results, which are well established in the field.

Recent research, such as the study by [45], has revealed that the classes of functions can be further examined and refined when the strong starlikeness of order $\alpha$, a significant property of the $(\lambda ,q)$-fractional differintegral operator, is taken into account. This property enables a more nuanced understanding of the operator’s behavior, allowing for a deeper exploration of the classes and their characteristics.