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Research article

Emergent behavior of Cucker–Smale model with time-varying topological structures and reaction-type delays

  • This paper studies the continuous Cucker–Smale model with time-varying topological structures and reaction-type delay. The goal of this paper is to establish a sufficient framework for flocking behaviors. Our method combines strict Lyapunov design with the derivation of an appropriate persistence condition for multi-agent systems. First, to prove that position fluctuations are uniformly bounded, a strict and trajectory-dependent Lyapunov functional is constructed via reparametrization of the time variable. Then, by constructing a global Lyapunov functional and using a novel backward-forward estimate, it is deduced that velocity fluctuations converge to zero. Finally, flocking behaviors are analyzed separately in terms of time delays and communication failures.

    Citation: Qin Xu, Xiao Wang, Yicheng Liu. Emergent behavior of Cucker–Smale model with time-varying topological structures and reaction-type delays[J]. Mathematical Modelling and Control, 2022, 2(4): 200-218. doi: 10.3934/mmc.2022020

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  • This paper studies the continuous Cucker–Smale model with time-varying topological structures and reaction-type delay. The goal of this paper is to establish a sufficient framework for flocking behaviors. Our method combines strict Lyapunov design with the derivation of an appropriate persistence condition for multi-agent systems. First, to prove that position fluctuations are uniformly bounded, a strict and trajectory-dependent Lyapunov functional is constructed via reparametrization of the time variable. Then, by constructing a global Lyapunov functional and using a novel backward-forward estimate, it is deduced that velocity fluctuations converge to zero. Finally, flocking behaviors are analyzed separately in terms of time delays and communication failures.



    In his survey-cum-expository review article, Srivastava [35] included also a brief overview of the classical q-analysis versus the so-called (p,q)-analysis with an obviously redundant additional parameter p (see, for details, [35,p. 340]). The present sequel to Srivastava's widely-cited review article [35], we apply the concept of q-convolution in order to introduce and study the general Taylor-Maclaurin coefficient estimates for functions belonging to a new class of normalized analytic and bi-close-to-convex functions in the open unit disk, which we have defined here.

    Let A denote the class of analytic functions of the form:

    f(z)=z+n=2anzn(zΔ), (1.1)

    where Δ denotes the open unit disk in the complex z-plane given by

    Δ:={z:zCand|z|<1}.

    Also let SA consist of functions which are also univalent in Δ.

    If the function f is given by (1.1) and the function ΥA is given by

    Υ(z)=z+n=2ψnzn(zΔ), (1.2)

    then the Hadamard product (or convolution) of the functions f and Υ is defined by defined by

    (fΥ)(z):=z+n=2anψnzn=:(Υf)(z)(zΔ).

    For 0α<1, we let S(α) denote the class of functions gS which are starlike of order α in Δ such that

    (zg(z)g(z))>α(zΔ).

    We denote by C(α) the class of functions fS which are close-to-convex of order α in Δ such that (see [10,24])

    (zf(z)g(z))>α(zΔ),

    where

    gS(0)=:S.

    We note that

    S(α)C(α)Sand|an|<n(fS;nN{1})

    by the Bieberbach conjecture or the De Branges Theorem (see [3,10]), N being the set of natural numbers (or the positive integers).

    In the above-cited review article, Srivastava [35] made use of various operators of q-calculus and fractional q-calculus. We begin by recalling the definitions and notations as follows (see also [33] and [45,pp. 350–351]).

    The q-shifted factorial is defined, for λ,qC and nN0=N{0}, by

    (λ;q)n={1(n=0)(1λ)(1λq)(1λqn1)(nN).

    By using the q-gamma function Γq(z), we get

    (qλ;q)n=(1q)n Γq(λ+n)Γq(λ)(nN0),

    where (see [19,33])

    Γq(z)=(1q)1z(q;q)(qz;q)(|q|<1).

    We note also that

    (λ;q)=n=0(1λqn)(|q|<1),

    and that the q-gamma function Γq(z) satisfies the following recurrence relation:

    Γq(z+1)=[z]qΓq(z),

    where [λ]q denotes the basic (or q-) number defined as follows:

    [λ]q:={1qλ1q(λC)1+1j=1qj(λ=N). (1.3)

    Using the definition in (1.3), we have the following consequences:

    (ⅰ) For any non-negative integer nN0, the q-shifted factorial is given by

    [n]q!:={1(n=0)nk=1[k]q(nN).

    (ⅱ) For any positive number r, the generalized q-Pochhammer symbol is defined by

    [r]q,n:={1(n=0)r+n1k=r[k]q(nN).

    In terms of the classical (Euler's) gamma function Γ(z), it is easily seen that

    limq1{Γq(z)}=Γ(z).

    We also observe that

    limq1{(qλ;q)n(1q)n}=(λ)n,

    where (λ)n is the familiar Pochhammer symbol defined by

    (λ)n={1(n=0)λ(λ+1)(λ+n1)(nN).

    For 0<q<1, the q-derivative operator (or, equivalently, the q-difference operator) Dq is defined by (see [22]; see also [14,16,21])

    Dq(fΥ)(z)=Dq(z+n=2anψnzn):=(fΥ)(z)(fΥ)(qz)z(1q)=1+n=2[n]qanψnzn1(zΔ),

    where, as in the definition (1.3),

    [n]q={1qn1q=1+n1j=1qj(nN)0(n=0). (1.4)

    Remark 1. Whereas a q-extension of the class of starlike functions was introduced in 1990 in [20] by means of the q-derivative operator Dq, a firm footing of the usage of the q-calculus in the context of Geometric Function Theory was actually provided and the generalized basic (or q-) hypergeometric functions were first used in Geometric Function Theory in an earlier book chapter published in 1989 by Srivastava (see, for details, [34]; see also the recent works [25,27,32,36,37,39,40,46,51,52,53,55,56,57]).

    For λ>1 and 0<q<1, El-Deeb et al. [14] defined the linear operator Hλ,qΥ:AA by

    Hλ,qΥf(z)Mq,λ+1(z)=zDq(fΥ)(z)(zΔ),

    where the function Mq,λ(z) is given by

    Mq,λ(z)=z+n=2[λ]q,n1[n1]q!zn(zΔ).

    A simple computation shows that

    Hλ,qΥf(z)=z+n=2[n]q![λ+1]q,n1anψnzn(λ>1;0<q<1;zΔ). (1.5)

    From the defining relation (1.5), we can easily verify that the following relations hold true for all fA:

    (i)[λ+1]qHλ,qΥf(z)=[λ]qHλ+1,qΥf(z)+qλz Dq(Hλ+1,qΥf(z))(zΔ);(ii)IλΥf(z):=limq1Hλ,qΥf(z)=z+n=2n!(λ+1)n1anψnzm(zΔ). (1.6)

    Remark 2. If we take different particular cases for the coefficients ψn, we obtain the following special cases for the operator Hλ,qh:

    (ⅰ) For ψn=1, we obtain the operator Jλq defined by Arif et al. [2] as follows (see also Srivastava [47]):

    Jλqf(z):=z+n=2[n]q![λ+1]q,n1anzn(zΔ); (1.7)

    (ⅱ) For

    ψn=(1)n1Γ(υ+1)4n1(n1)!Γ(n+υ)andυ>0,

    we obtain the operator Nλυ,q defined by El-Deeb and Bulboacǎ [12] and El-Deeb [11] as follows (see also [16]):

    Nλυ,qf(z):=z+n=2(1)n1Γ(υ+1)4n1(n1)!Γ(n+υ)[n]q![λ+1]q,n1anzn=z+n=2[n]q![λ+1]q,n1ϕnanzn (1.8)
    (υ>0;λ>1;0<q<1; zΔ),

    where

    ϕn:=(1)n1Γ(υ+1)4n1(n1)!Γ(n+υ)(nN{1}); (1.9)

    (ⅲ) For

    ψn=(n+1m+n)α,α>0andnN0,

    we obtain the operator Mλ,αm,q defined by El-Deeb and Bulboacǎ (see [13,43]) as follows:

    Mλ,αm,qf(z):=z+n=2(m+1m+n)α[n]q![λ+1]q,n1anzn(zΔ); (1.10)

    (ⅳ) For

    ψn=ρn1(n1)!eρandρ>0,

    we obtain a q-analogue of the Poisson operator defined in [30] by

    Iλ,ρqf(z):=z+n=2ρn1(n1)!eρ[n]q![λ+1]q,n1anzn(zΔ); (1.11)

    (ⅴ) For

    \psi_{n} = \binom{m+n-2}{n-1}\; \theta^{n-1}\left(1-\theta \right)^{m}\qquad \left(m\in \mathbb{N};\; 0\leqq \theta \leqq 1\right),

    we get a q -analogue \Psi_{q, \theta}^{\lambda, m} of the Pascal distribution operator as follows (see [15]):

    \begin{equation} \Psi_{q,\theta}^{\lambda,m}f(z): = z+\sum\limits_{n = 2}^{\infty} \binom{m+n-2}{n-1}\theta^{n-1}\left(1-\theta \right)^{m}\cdot \frac{[n]_{q}!}{[\lambda+1]_{q,n-1}}\;a_{n}\; z^{n} \end{equation} (1.12)
    (z\in \Delta).

    If f and F are analytic functions in \Delta , we say that the function f is subordinate to the function F , written as f(z)\prec F(z) , if there exists a Schwarz function s , which is analytic in \Delta with s(0) = 0 and \left\vert s(z)\right\vert < 1 for all z\in \Delta , such that

    f(z) = F\big(s(z)\big) \qquad (z\in \Delta).

    Furthermore, if the function F is univalent in \Delta , then we have the following equivalence (see, for example, [7,28])

    \begin{equation*} f(z)\prec F(z)\; \Longleftrightarrow \; f(0) = F(0)\qquad \text{and} \qquad f(\Delta)\subset F(\Delta). \end{equation*}

    The Koebe one-quarter theorem (see [10]) asserts that the image of \Delta under every univalent function f\in \mathcal{S} contains the disk of radius \dfrac{1}{4} . Therefore, every function f\in \mathcal{S} has an inverse f^{-1} which satisfies the following inequality:

    \begin{equation*} f\big(f^{-1}(w)\big) = w\qquad \left(\left\vert w\right\vert \lt r_{0}\left(f\right); \;r_{0}\left(f\right) \geqq \frac{1}{4}\right) , \end{equation*}

    where

    \begin{align*} g(w) & = f^{-1}(w) = w-a_{2}w^{2}+\left(2a_{2}^{2}-a_{3}\right) w^{3}-\left( 5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right) w^{4}+\cdots \\ & = w+\sum\limits_{n = 2}^{\infty}A_{n}\; w^{n}. \end{align*}

    A function f\in \mathcal{A} is said to be bi-univalent in \Delta if both f and f^{-1} are univalent in \Delta . Let \Sigma denote the class of normalized analytic and bi-univalent functions in \Delta given by (1.1). The class \Sigma of analytic and bi-univalent functions was introduced by Lewin [26], where it was shown that

    f\in \Sigma \;\Longrightarrow\;\left\vert a_{2}\right\vert \lt 1.51.

    Brannan and Clunie [4] improved Lewin's result to the following form:

    f\in \Sigma \;\Longrightarrow\; \left\vert a_{2}\right\vert \lt \sqrt{2}

    and, subsequently, Netanyahu [29] proved that

    f\in \Sigma \;\Longrightarrow\;\left\vert a_{2}\right\vert \lt \frac{4}{3}.

    It should be noted that the following functions:

    f_{1}(z) = \dfrac{z}{1-z},\quad f_{2}(z) = \dfrac{1}{2}\log\left(\dfrac{1+z}{1-z}\right) \qquad \text{and} \qquad f_{3}(z) = -\log (1-z),

    together with their corresponding inverses given by

    f_{1}^{-1}(w) = \dfrac{w}{1+w}, \quad f_{2}^{-1}(w) = \dfrac{e^{2w}-1}{e^{2w}+1} \qquad \text{and} \qquad f_{3}^{-1}(w) = \dfrac{e^{w}-1}{e^{w}},

    are elements of the analytic and bi-univalent function class \Sigma (see [14,48]). A brief history and interesting examples of the analytic and bi-univalent function class \Sigma can be found in (for example) [5,48].

    Brannan and Taha [6] (see also [48]) introduced certain subclasses of the bi-univalent function class \Sigma similar to the familiar subclasses S^{\ast }\left(\alpha \right) and K\left(\alpha \right) of starlike and convex functions of order \alpha \; \left(0\leqq \alpha < 1\right) , respectively (see [5]). Indeed, following Brannan and Taha [6], a function f\in \mathcal{A} is said to be in the class S_{\Sigma}^{\ast }\left(\alpha \right) of bi-starlike functions of order \alpha \; \left(0 < \alpha \leqq 1\right) , if each of the following conditions is satisfied:

    \begin{equation*} f\in \Sigma \qquad \text{and}\qquad \left\vert \arg\left(\frac{zf^{\prime}(z)}{ f(z)}\right)\right\vert \lt \frac{\alpha \pi}{2}\qquad (z\in \Delta) \end{equation*}

    and

    \begin{equation*} \left\vert \arg\left(\frac{z\mathcal{F}^{\prime}(w)}{\mathcal{F}(w)}\right\vert\right) \lt \frac{\alpha \pi }{2} \qquad (w\in \Delta), \end{equation*}

    where the function \mathcal{F} is the analytic extension of f^{-1} to \Delta , given by

    \begin{equation} \mathcal{F}(w) = w-a_{2}w^{2}+\left( 2a_{2}^{2}-a_{3}\right) w^{3}-\left( 5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right) w^{4}+\cdots \qquad (w\in \Delta). \end{equation} (1.13)

    A function f\in A is said to be in the class K_{\Sigma}^{\ast }\left(\alpha \right) of bi-convex functions of order \alpha \; \left(0 < \alpha \leqq 1\right) , if each of the following conditions is satisfied:

    \begin{equation*} f\in \Sigma ,\quad \text{with}\quad \left\vert \arg\left(1+\frac{ zf^{\prime \prime }(z)}{f^{\prime}(z)}\right) \right\vert \lt \frac{\alpha \pi }{2} \qquad (z\in \Delta) \end{equation*}

    and

    \begin{equation*} \left\vert \arg\left(1+\frac{zg^{\prime \prime}(w)}{g^{\prime}(w)} \right) \right\vert \lt \frac{\alpha \pi }{2}\qquad (w\in \Delta). \end{equation*}

    The classes S_{\Sigma}^{\ast}\left(\alpha \right) and K_{\Sigma}\left(\alpha \right) of bi-starlike functions of order \alpha in \Delta and bi-convex functions of order \alpha \; \left(0 < \alpha \leqq 1\right) in \Delta , corresponding to the function classes S^{\ast}\left(\alpha \right) and K\left(\alpha \right) , were also introduced analogously. For each of the function classes S_{\Sigma}^{\ast}\left(\alpha \right) and K_{\Sigma}\left(\alpha \right) , non-sharp estimates on the first two Taylor-Maclaurin coefficients \left\vert a_{2}\right\vert and \left\vert a_{3}\right\vert are known (see [6,35,48]). In fact, this pioneering work by Srivastava et al. [48] happens to be one of the most important studies of the bi-univalent function class \Sigma . It not only revived the study of the bi-univalent function class \Sigma in recent years, but it has also inspired remarkably many investigations in this area including the present paper. Some of these many recent papers dealing with problems involving the analytic and bi-univalent functions such as those considered in this article include [1,9,17,23,48], and indeed also many other works (see, for example, [38,44,54]).

    Sakar and Güney [31] introduced and studied the following class:

    \mathcal{T}_{\Sigma}\left( \lambda,\beta \right) \;\; \left(0\leqq \lambda \leqq 1;\; 0\leqq \beta \lt 1\right).

    In the same way, we define the following subclass of bi-close-to-convex functions \mathcal{H}_{\Sigma}^{q, \lambda }\left(\eta, \beta, \Upsilon \right) as follows.

    Definition 1. For 0\leqq \eta < 1 and 0\leqq \beta \leqq 1, \; a function f\in \Sigma has the form (1.1) and the function \Upsilon given by (1.2), the function f is said to be in the class \mathcal{H}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon\right) if there exists a function g \in \mathcal{S}^{\ast} such that

    \begin{equation} \Re\left(\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}f(z)\right) ^{{\prime }}+\beta z^{2}\left( \mathcal{H}_{\Upsilon }^{\lambda ,q}f(z)\right)^{\prime\prime}}{\left(1-\beta \right) \mathcal{H}_{\Upsilon }^{\lambda,q}g(z)+\beta z\left(\mathcal{H}_{\Upsilon }^{\lambda,q}g(z)\right)^{\prime}}\right) \gt \eta \qquad (z\in \Delta) \end{equation} (1.14)

    and

    \begin{equation} \Re\left(\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda,q}\mathcal{F }(w)\right)^{{\prime}}+\beta z^{2}\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}\mathcal{F}(w)\right) ^{^{{\prime \prime }}}}{\left(1-\beta \right) \mathcal{H}_{\Upsilon }^{\lambda,q}\mathcal{G}(w)+\beta z\left(\mathcal{H} _{\Upsilon }^{\lambda ,q}\mathcal{G}(w)\right)^{{\prime}}}\right) \gt \eta \qquad (w\in \Delta), \end{equation} (1.15)

    where the function \mathcal{F} is the analytic extension of f^{-1} to \Delta , and is given by (1.13), and \mathcal{G} is the analytic extension of g^{-1} to \Delta as follows:

    \begin{equation} \mathcal{G}(w) = w-b_{2}w^{2}+\left(2b_{2}^{2}-b_{3}\right)w^{3}-\left( 5b_{2}^{3}-5b_{2}b_{3}+b_{4}\right) w^{4}+\cdots \qquad (w\in \Delta). \end{equation} (1.16)

    We note that, if b_{n} = a_{n}\; \; (n\in \mathbb{N}\setminus\{1\}) , \mathcal{S}_{\Sigma }^{q, \lambda}\left(\eta, \beta, \Upsilon\right) becomes the class of bi-starlike functions satisfying the following inequalities:

    \begin{equation} \Re\left(\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}f(z)\right)^{{\prime}}+\beta z^{2}\left(\mathcal{H}_{\Upsilon }^{\lambda ,q}f(z)\right) ^{\prime \prime}}{\left( 1-\beta \right) \mathcal{H}_{\Upsilon }^{\lambda ,q}f(z)+\beta z\left( \mathcal{H}_{\Upsilon }^{\lambda ,q}f(z)\right)^{\prime}}\right) \gt \eta \qquad (z\in \Delta). \end{equation} (1.17)

    and

    \begin{equation} \Re\left(\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}\mathcal{F }(w)\right) ^{{\prime}}+\beta z^{2}\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}\mathcal{F}(w)\right)^{\prime \prime}}{\left(1-\beta \right) \mathcal{H}_{\Upsilon}^{\lambda,q}\mathcal{F}(w)+\beta z\left( \mathcal{H} _{\Upsilon }^{\lambda ,q}\mathcal{F}(w)\right)^{\prime}}\right) \gt \eta \qquad (w\in \Delta). \end{equation} (1.18)

    Remark 3. Each of the following limit cases when q\rightarrow 1{-} is worthy of note.

    (ⅰ) Putting q\rightarrow 1{-} , we obtain

    \lim\limits_{q\rightarrow 1{-}}\mathcal{H}_{\Sigma }^{q,\lambda}\left(\eta,\beta,h\right) = : \mathcal{P}_{\Sigma}^{\lambda}\left(\eta,\beta,h\right),

    where \mathcal{P}_{\Sigma}^{\lambda }\left(\eta, \beta, \Upsilon \right) represents the functions f\in \Sigma that satisfy (1.14) and (1.15) with \mathcal{H}_{\Upsilon}^{\lambda, q} replaced by \mathcal{I}_{\Upsilon}^{\lambda} as in (1.6).

    (ⅱ) Putting

    \psi_{n} = \dfrac{(-1)^{n-1}\Gamma(\upsilon+1)}{ 4^{n-1}\; (n-1)! \;\Gamma(m+\upsilon)}\qquad (\upsilon \gt 0),

    we obtain the class \mathcal{B}_{\Sigma}^{q, \lambda} \left(\eta, \beta, \upsilon\right) representing the functions f\in \Sigma that satisfy (1.14) and (1.15) with \mathcal{H}_{\Upsilon}^{\lambda, q} replaced by \mathcal{N}_{\upsilon, q}^{\lambda} as in (1.8).

    (ⅲ) Putting

    \psi_{n} = \left(\dfrac{n+1}{m+n}\right)^{\alpha}\qquad (\alpha \gt 0;\; m\geqq \mathbb{N}_0),

    we obtain the class \mathcal{L}_{\Sigma}^{\lambda, q} \left(\eta, \beta, m, \alpha \right) consisting of the functions f\in \Sigma that satisfy (1.14) and (1.15) with \mathcal{H} _{\Upsilon}^{\lambda, q} replaced by \mathcal{M}_{m, q}^{\lambda, \alpha } as in (1.10).

    (ⅳ) Putting

    \psi_{n} = \dfrac{\rho^{n-1}}{(n-1)!}\;e^{-\rho }\qquad (\rho \gt 0),

    we obtain the class \mathcal{M}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \rho \right) representing the functions f\in \Sigma which satisfy the inequalities in (1.14) and (1.15) with \mathcal{H}_{\Upsilon}^{\lambda, q} replaced by \mathcal{I}_{q}^{\lambda, \rho} as in (1.11).

    (ⅴ) Putting

    \psi_{n} = \binom{m+n-2}{n-1}\; \theta^{n-1}\left(1-\theta \right)^{m}\qquad \left(m\in \mathbb{N};\; 0\leqq \theta \leqq 1\right),

    we get the class \mathcal{W}_{\Sigma}^{q, \lambda}\left(\eta, \beta, m, \theta \right) of the functions f\in \Sigma which satisfy the inequalities in (1.14) and (1.15) with \mathcal{H}_{\Upsilon}^{\lambda, q} replaced by \Psi_{q, \theta}^{\lambda, m} occurring in (1.12).

    Using the Faber polynomial expansion of functions f\in \mathcal{A} which have the normalized form (1.1), the coefficients of its inverse map may be expressed as follows (see [18]):

    \begin{equation} \mathcal{F}(w) = f^{-1}(w) = w+\sum\limits_{n = 2}^{\infty}\frac{1}{n}\; K_{n-1}^{-n}(a_{2},a_{3},\cdots)\;w^{n} = w+\sum\limits_{n = 2}^{\infty}A_{n}\;w^{n}, \end{equation} (1.19)

    where

    \begin{align} \mathcal{K}_{n-1}^{-n}(a_{2},a_{3},\cdots) & = \frac{(-n)!}{(-2n+1)!\; (n-1)!} a_{2}^{n-1} \\ &\qquad +\frac{(-n)!}{(2(-n+1))!\; (n-3)!}\;a_{2}^{n-3}\;a_{3} \\ &\qquad +\frac{(-n)!}{(-2n+3)!\; (n-4)!}\;a_{2}^{n-4}\;a_{4} \\ &\qquad +\frac{(-n)!}{ (2(-n+2))!\; (n-5)!}\;a_{2}^{n-5}\;\left[a_{5}+\left(-n+2\right)a_{3}^{2}\right] \\ &\qquad +\frac{(-n)!}{(-2n+5)!\; (n-6)!}\;a_{2}^{n-6} \left[a_{6}+\left(-2n+5\right) a_{3}\;a_{4}\right] \\ &\qquad +\sum\limits_{j\geqq 7}a_{2}^{n-j}U_{j} \end{align} (1.20)

    such that U_{j} with 7\leqq j\leqq n is a homogeneous polynomial in the variables a_{2}, a_{3}, \cdots, a_{n} . Here such expressions as (for example) (-n)! are to be interpreted symbolically by

    \begin{equation*} (-n)!\equiv \Gamma(1-n): = (-n)(-n-1)(-n-2)\cdots \qquad \big(n\in \mathbb{N}_0\big). \end{equation*}

    In particular, the first three terms of \mathcal{K}_{n-1}^{-n} are given by

    \begin{align*} \mathcal{K}_{1}^{-2} = -2a_{2}, \end{align*}
    \begin{align*} \mathcal{K}_{2}^{-3} = 3\left(2a_{2}^{2}-a_{3}\right) \end{align*}

    and

    \begin{align*} \mathcal{K}_{3}^{-4} = -4\left(5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right). \end{align*}

    In general, an expansion of \mathcal{K}_{m}^{-n} \; (n\in \mathbb{N}) is given by (see [1,8,41,42,47,49,50])

    \begin{equation*} \mathcal{K}_{m}^{-n} = na_{m}+\frac{n\left(n-1\right)}{2}\;\mathcal{D}_{m}^{2}+ \frac{n!}{3!\;\left(n-3\right)!}\;\mathcal{D}_{m}^{3}+\cdots+\frac{n!}{m!\left( n-m\right)!}\;\mathcal{D}_{m}^{m}, \end{equation*}

    where

    \mathcal{D}_{m}^{n} = \mathcal{D}_{m}^{n}(a_{2},a_{3},a_{4},\cdots)

    and, alternatively,

    \begin{equation*} \mathcal{D}_{m}^{n}(a_{2},a_{3},\cdots,a_{m+1}) = \sum\limits_{i_1,\cdots,i_m}\left(\frac{n!}{ i_{1}!\;\cdots\; i_{m}!}\right)\;a_{2}^{i_{1}}\;\cdots\; a_{m+1}^{i_{m}}, \end{equation*}

    where a_{1} = 1 and the sum is taken over all non-negative integers i_{1}, \cdots, i_{m} satisfying the following constraints:

    i_{1}+i_{2}+\cdots+i_{m} = n

    and

    i_{1}+2i_{2}+\cdots+mi_{m} = m.

    Evidently, we have

    \begin{equation*} \mathcal{D}_{m}^{m}(a_{2},a_{3},\cdots,a_{m+1}) = a_{2}^{m}. \end{equation*}

    The following Lemma will be needed to prove our results.

    The Carathéodory Lemma. (see [10]) If \phi \in \mathfrak{P} and

    \phi(z) = 1+\sum\limits_{n = 1}^{\infty}c_{n}\;z^{n},

    then

    |c_{n}|\leqq 2 \qquad (n \in \mathbb{N}).

    This inequality is sharp for all positive integers n . Here \mathfrak{P} is the family of all functions \phi, which analytic and have positive real part in \Delta, with \phi(0) = 1 .

    In this section, we apply the above-described Faber polynomial expansion method, we derive bounds for the general Taylor-Maclaurin coefficients of functions in \mathcal{H}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon \right) .

    Theorem 1. Let the function f given by (1.1) belong to the class \mathcal{H}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon \right) . Suppose also that

    0\leqq \eta \lt 1,\quad 0\leqq \beta \leqq 1,\quad \lambda \gt -1 \qquad \mathit{\text{and}} \qquad 0 \lt q \lt 1.

    If a_{k} = 0 for 2\leqq k\leqq n-1, then

    \begin{equation*} \left\vert a_{n}\right\vert \leqq \frac{2\left(1-\eta \right) [\lambda +1]_{q,n-1}}{n\left[1+\left(n-1\right)\beta\right] \ [n]_{q}!\,\psi _{n}} +1. \end{equation*}

    Proof. If f\in \mathcal{H}_{\Sigma }^{q, \lambda }\left(\eta, \beta, \Upsilon \right) , then there exists a function g(z) , given by

    g(z): = z+\sum\limits_{n = 2}^{\infty}b_{n}\;z^{n}\in S^{\ast},

    such that

    \begin{equation*} \Re\left(\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}f(z)\right)^{\prime}+\beta z^{2}\left( \mathcal{H}_{\Upsilon }^{\lambda ,q}f(z)\right)^{\prime \prime}}{\left(1-\beta \right) \mathcal{H}_{\Upsilon}^{\lambda,q}g(z)+\beta z\left(\mathcal{H}_{\Upsilon }^{\lambda,q}g(z)\right)^{{\prime}}}\right) \gt \eta \qquad (z\in \Delta). \end{equation*}

    Moreover, by using the Faber polynomial expansion, we have

    \begin{align} &\frac{z\left(\mathcal{H}_{\Upsilon }^{\lambda,q}f(z)\right)^{\prime} +\beta z^{2}\left(\mathcal{H}_{\Upsilon}^{\lambda ,q}f(z)\right) ^{\prime \prime}}{\left(1-\beta \right) \mathcal{H}_{\Upsilon }^{\lambda ,q}g(z)+\beta z\left(\mathcal{H}_{\Upsilon }^{\lambda ,q}g(z)\right)^{\prime}} \\ &\qquad = 1+\sum\limits_{n = 2}^{\infty}\Bigg(\left[1+\beta \left(n-1\right) \right]\; \frac{[n]_{q}!}{[\lambda+1]_{q,n-1}}\,\psi _{n}\left(na_{n}-b_{n}\right) \\ &\qquad \qquad \qquad +\sum\limits_{t = 1}^{n-2}\dfrac{[n,q]!} {[\lambda +1,q]_{n-1}}\,\psi _{n}\left[1+\left(n-t-1\right)\beta \right] \\ &\qquad \qquad \qquad \qquad \cdot K_{t}^{-1}\left[\left( 1+\beta \right) b_{2},\left(1+2\beta \right) b_{3},\cdots, \left(1+t\beta \right) b_{t+1}\right] \\ &\qquad \qquad \qquad \qquad \cdot \left[\left(n-t\right)\ a_{n-t}-b_{n-t}\right] \Bigg) z^{n-1}\qquad (z\in \Delta). \end{align} (2.1)

    Also, for the inverse map \mathcal{F} = f^{-1}, there exists a function \mathcal{G}(w) , given by

    \mathcal{G}(w) = w+\sum\limits_{n = 2}^{\infty}B_{n}\;w^{n}\in S^{\ast},

    such that

    \begin{equation*} \Re\left(\frac{z\left( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F }(w)\right) ^{{\prime }}+\beta z^{2}\left( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(w)\right) ^{{{\prime \prime }}}}{\left( 1-\beta \right) \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{G}(w)+\beta z\left( \mathcal{H} _{\Upsilon }^{\lambda ,q}\mathcal{G}(w)\right) ^{{\prime }}}\right) \gt \eta \qquad (w\in \Delta), \end{equation*}

    the Faber polynomial expansion of the inverse map \mathcal{F} = f^{-1} is given by

    \mathcal{F}(w) = w+\sum\limits_{n = 2}^{\infty}A_{n}\;w^{n},

    so we have

    \begin{align} &\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda,q}\mathcal{F}(w)\right) ^{\prime}+\beta z^{2}\left(\mathcal{H}_{\Upsilon}^{\lambda,q}\mathcal{ F}(w)\right)^{\prime \prime}}{\left(1-\beta \right) \mathcal{H} _{\Upsilon }^{\lambda ,q}\mathcal{G}(w)+\beta z\left(\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{G}(w)\right)^{\prime}} \\ &\qquad = 1+\sum\limits_{n = 2}^{\infty}\Bigg(\left[1+\beta \left(n-1\right)\right]\;\frac{[n]_{q}!}{[\lambda +1]_{q,n-1}}\,\psi _{n}\left(nA_{n}-B_{n}\right) \\ &\qquad \qquad +\sum\limits_{t = 1}^{n-2}\frac{[n]_{q}!}{[\lambda +1]_{q,n-1}}\psi _{n}\left[1+\left(n-t-1\right) \beta \right] \\ &\qquad \qquad \qquad \cdot K_{t}^{-1}\left[\left( 1+\beta \right) B_{2},\left( 1+2\beta \right) B_{3},\cdots,\left(1+t\beta \right) B_{t+1}\right] \\ &\qquad \qquad \qquad \cdot \left[\left(n-t\right) A_{n-t}-B_{n-t}\right] \Bigg)\; w^{n-1}\qquad (w\in \Delta). \end{align} (2.2)

    Now, since

    \begin{equation*} f\in \mathcal{H}_{\Sigma}^{q,\lambda}\left(\eta,\beta,\Upsilon \right) \qquad \text{and}\qquad \mathcal{F} = f^{-1}\in \mathcal{H}_{\Sigma}^{q,\lambda }\left(\eta,\beta,\Upsilon \right) , \end{equation*}

    there are the following two positive real part functions:

    \begin{equation*} U(z) = 1+\sum\limits_{n = 1}^{\infty}c_{n}\;z^{n} \end{equation*}

    and

    \begin{equation*} V(w) = 1+\sum\limits_{n = 1}^{\infty}d_{n}\;w^{n}, \end{equation*}

    for which

    \begin{equation*} \Re\big(U(z)\big) \gt 0\qquad \text{and} \qquad \Re\big( V(w)\big) \gt 0 \qquad (z,w\in \Delta), \end{equation*}

    so that

    \begin{align} &\frac{z\left(\mathcal{H}_{\Upsilon}^{\lambda,q}\mathcal{F}(w)\right) ^{\prime}+\beta z^{2}\left(\mathcal{H}_{\Upsilon }^{\lambda,q}\mathcal{ F}(w)\right)^{{{\prime \prime }}}}{\left( 1-\beta \right) \mathcal{H} _{\Upsilon}^{\lambda,q}\mathcal{G}(w)+\beta z\left( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{G}(w)\right)^{\prime}} = \eta +\left(1-\eta \right) \; U(z) \\ &\qquad = 1+\left(1-\eta \right) \sum\limits_{n = 1}^{\infty}c_{n}\;z^{n} \end{align} (2.3)

    and

    \begin{align} &\frac{z\left(\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(w)\right) ^{{\prime }}+\beta z^{2}\left(\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{ F}(w)\right)^{\prime \prime}}{\left(1-\beta \right) \mathcal{H} _{\Upsilon }^{\lambda ,q}\mathcal{G}(w)+\beta z\left(\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{G}(w)\right)^{\prime}} = \eta +\left(1-\eta \right)\; V(w) \\ &\qquad = 1+\left(1-\eta \right)\sum\limits_{n = 1}^{\infty}d_{n}\;w^{n}. \end{align} (2.4)

    Now, under the assumption that a_{k} = 0 for 0\leqq k\leqq n-1, we obtain A_{n} = -a_{n}. Then, by using (2.1) and comparing the corresponding coefficients in (2.3), we obtain

    \begin{equation} \left[1+\beta \left(n-1\right)\right] \;\frac{[n]_{q}!}{[\lambda+1]_{q,n-1} }\,\psi_{n}\left(na_{n}-b_{n}\right) = \left(1-\eta \right)\; c_{n-1}. \end{equation} (2.5)

    Similarly, by using (2.2) in the Eq (2.4), we find that

    \begin{equation} \left[1+\beta \left(n-1\right) \right]\; \frac{[n]_{q}!}{[\lambda +1]_{q,n-1} }\,\psi_{n}\left(nA_{n}-B_{n}\right) = \left(1-\eta \right)\; d_{n-1}, \end{equation} (2.6)
    \begin{equation} \left[1+\beta \left(n-1\right) \right]\; \frac{[n]_{q}!}{[\lambda +1]_{q,n-1} }\,\psi_{n}\left(na_{n}-b_{n}\right) = \left(1-\eta \right) \;c_{n-1} \end{equation} (2.7)

    and

    \begin{equation} -\left[ 1+\beta \left(n-1\right) \right] \frac{[n]_{q}!}{[\lambda +1]_{q,n-1}}\,\psi_{n}\left(-na_{n}-B_{n}\right) = \left(1-\eta \right)\; d_{n-1}. \end{equation} (2.8)

    Taking the moduli of both members of (2.7) and (2.8) for

    \left\vert b_{n}\right\vert \leqq n\qquad \text{and} \qquad \left\vert B_{n}\right\vert \leqq n,

    and applying the Carathéodory Lemma, we conclude that

    \begin{equation*} \left\vert a_{n}\right\vert \leqq \frac{2\left(1-\eta \right) [\lambda +1]_{q,n-1}}{n\left[1+\left(n-1\right) \beta \right] \; [n]_{q}!\,\psi_{n}} +1, \end{equation*}

    which completes the proof of Theorem 1.

    If we set

    \psi_{n} = \dfrac{(-1)^{n-1}\Gamma (\upsilon+1)}{4^{n-1}\;(n-1)!\;\Gamma (n+\upsilon)}\qquad (\upsilon \gt 0)

    in Theorem 1, we obtain the following special case.

    Corollary 1. Let the function f given byt (1.1) belong to the class \mathcal{B}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \upsilon \right) . Suppose also tha

    0\leqq \eta \lt 1,\quad 0\leqq \beta \leqq 1,\quad \lambda \gt -1,\quad \upsilon \gt 0\qquad \mathit{\text{and}} \qquad 0 \lt q \lt 1.

    If a_{k} = 0 for 2\leqq k\leqq n-1, then

    \begin{equation*} \left\vert a_{n}\right\vert \leqq \frac{2\left(1-\eta \right)[\lambda +1]_{q,n-1}}{n\left[1+\left( n-1\right) \beta \right] \; [n]_{q}!\;\phi_{n}} +1, \end{equation*}

    where \phi_{n} is given by (1.9).

    Upon putting

    \psi_{n} = \left(\dfrac{n+1}{m+n}\right)^{\alpha} \qquad (\alpha \gt 0;\; m\in \mathbb{N}_0)

    in Theorem 1, we obtain the following result.

    Corollary 2. Let the function f given by (1.1) belong to the class \mathcal{L}_{\Sigma}^{q, \lambda}\left(\eta, \beta, m, \alpha \right) . Suppose also that

    0\leqq \eta \lt 1,\quad 0\leqq \beta \leqq 1,\quad \lambda \gt -1,\quad \alpha \gt 0,\quad m\in \mathbb{N}_0 \qquad \mathit{\text{and}} \qquad 0 \lt q \lt 1.

    If a_{k} = 0 for 2\leqq k\leqq n-1, then

    \begin{equation*} \left\vert a_{n}\right\vert \leqq \frac{2\left(1-\eta \right) \left( m+n\right)^{\alpha}\;[\lambda +1]_{q,n-1}}{n\left[1+\left(n-1\right) \beta \right] \; [n]_{q}!\,\left(n+1\right)^{\alpha}}+1. \end{equation*}

    If we take

    \psi_{n} = \dfrac{\rho^{n-1}}{(n-1)!}\;e^{-\rho}\qquad (\rho \gt 0)

    in Theorem 1, we obtain the following special case.

    Corollary 3. Let the function f given by (1.1) belong to the class \mathcal{M}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \rho \right) . Suppose also that

    0\leqq \eta \lt 1,\quad 0\leqq \beta \leqq 1,\quad\lambda \gt -1,\quad \rho \gt 0 \qquad \mathit{\text{and}} \qquad 0 \lt q \lt 1.

    If a_{k} = 0 for 2\leqq k\leqq n-1, then

    \begin{equation*} \left\vert a_{n}\right\vert \leqq \frac{2\left(1-\eta \right)\;(n-1)!\;[\lambda +1]_{q,n-1}}{n\left[1+\left(n-1\right) \beta \right] \; [n]_{q}!\;\rho ^{n-1}\;e^{-\rho }}+1. \end{equation*}

    Upon setting

    \psi_{n} = \binom{m+n-2}{n-1}\ \theta^{n-1}\;\left(1-\theta \right)^{m}\qquad \left(m\in\mathbb{N};\; 0\leqq \theta \leqq 1\right)

    in Theorem 1, we are led to the following result for the above-defined class \mathcal{W}_{\Sigma}^{q, \lambda}\left(\eta, \beta, m, \theta\right) .

    Corollary 4. Let the function f given by (1.1) belong to the following class :

    \mathcal{W}_{\Sigma}^{q,\lambda}\left(\eta,\beta,m,\theta \right)
    (0\leqq \eta \lt 1;\; 0\leqq \beta \leqq 1;\;\lambda \gt -1;\; 0 \lt q \lt 1;\; m\in \mathbb{N};\; 0\leqq\theta \leqq 1).

    If a_{k} = 0 for 2\leqq k\leqq n-1, then

    \left\vert a_{n}\right\vert \leqq \frac{2\left(1-\eta\right) [\lambda +1]_{q,n-1}}{n\left[1+\left(n-1\right)\beta \right] \; [n]_{q}!\; \binom{m+n-2}{n-1}\; \theta^{n-1}\;\left(1-\theta \right)^{m}}+1.

    In particular, if we let g(z) = f(z) , we obtain the class \mathcal{S}_{\Sigma }^{q, \lambda}\left(\eta, \beta, \Upsilon \right) , which is a subclass of \mathcal{H}_{\Sigma }^{q, \lambda}\left(\eta, \beta, \Upsilon\right) . We then give the next theorem, which involves the coefficients of this subclass of the analytic and bi-starlike functions in \Delta .

    Theorem 2. Let the function f given by (1.1) belong to the class \mathcal{S}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon \right) . Suppose also that

    \gamma \geqq 1,\quad \eta \geqq 0,\quad \lambda \gt -1, \quad 0\leqq\beta \lt 1 \qquad \mathit{\text{and}} \qquad 0 \lt q \lt 1.

    Then

    \begin{equation} \left\vert a_{2}\right\vert \leqq \left\{ \begin{array}{ll} \frac{2\left( 1-\eta \right) [\lambda +1]_{q}}{\left( 1+\beta \right) \ [2]_{q}!\,\psi _{2}}& \qquad \left(0\leqq \eta \lt 1-\frac{\left( 1+\beta \right) ^{2}\ \left( [2]_{q}!\right) ^{2}\,[\lambda +2]_{q}\ \psi _{2}^{2}}{ 2\left( 1+2\beta -\beta ^{2}\right) \ [3]_{q}!\,[\lambda +1]_{q}\ \psi _{3}} \right) \\ \\ \sqrt{\frac{2\left( 1-\eta \right) [\lambda +1]_{q,2}}{\left( 1+2\beta -\beta ^{2}\right) \ [3]_{q}!\,\psi _{3}}} &\qquad \left(1-\frac{\left(1+\beta \right)^{2}\ \left([2]_{q}!\right)^{2}\,[\lambda +2]_{q}\ \psi_{2}^{2}}{ 2\left(1+2\beta -\beta ^{2}\right) \ [3]_{q}!\,[\lambda +1]_{q} \psi_{3}} \leqq \eta \lt 1\right) \end{array} \right. \end{equation} (2.9)

    and

    \begin{equation} \left\vert a_{3}\right\vert \leqq \left\{ \begin{array}{ll} \frac{2\left(1-\eta \right) [\lambda +1]_{q,2}}{\left( 1+2\beta -\beta ^{2}\right) \ [3]_{q}!\,\psi _{3}} &\quad \left(0\leqq \eta \lt 1-\frac{\left(1+\beta \right)^{2}\ \left( [2]_{q}!\right)^{2}\,[\lambda +2]_{q}\ \psi _{2}^{2}}{2\left(1+2\beta -\beta ^{2}\right) \ [3]_{q}!\,[\lambda +1]_{q}\ \psi _{3}}\right) \\ \\ \frac{\left(1-\eta \right)}{\left(1+2\beta \right)}\left(\tfrac{ [\lambda +1]_{q,2}}{[3]_{q}!\,\psi_{3}}+\tfrac{2\left(1-\eta\right) [\lambda +1]_{q}^{2}}{\left([2]_{q}!\right)^{2}\ \psi_{2}^{2}}\right) &\quad \left(1-\tfrac{\left(1+\beta \right)^{2}\ \left([2]_{q}!\right) ^{2}\,[\lambda +2]_{q}\ \psi_{2}^{2}}{2\left(1+2\beta -\beta ^{2}\right) \ [3]_{q}![\lambda +1]_{q}\ \psi _{3}}\leqq \eta \lt 1\right). \end{array} \right. \end{equation} (2.10)

    Proof. Putting n = 2 and n = 3 in (2.5) and (2.6), we have

    \begin{equation} \left( 1+\beta \right) \ \frac{[2]_{q}!}{[\lambda +1]_{q}}\,\psi _{2}a_{2} = \left( 1-\eta \right) c_{1}, \end{equation} (2.11)
    \begin{equation} \left[ 2\left( 1+2\beta \right) \ a_{3}-\left(1+\beta \right)^{2}a_{2}^{2} \right] \frac{[3]_{q}!}{[\lambda +1]_{q,2}}\,\psi_{3} = \left( 1-\eta \right) c_{2}, \end{equation} (2.12)
    \begin{equation} -\left( 1+\beta \right) \ \frac{[2]_{q}!}{[\lambda +1]_{q}}\,\psi _{2}a_{2} = \left( 1-\eta \right) d_{1} \end{equation} (2.13)

    and

    \begin{equation} \left[ -2\left( 1+2\beta \right) \ a_{3}+\left(3+6\beta -\beta ^{2}\right) a_{2}^{2}\right] \frac{[3]_{q}!}{[\lambda +1]_{q,2}}\,\psi _{3} = \left( 1-\eta \right) d_{2}. \end{equation} (2.14)

    From (2.11) and (2.13), by using the Carathéodory Lemma, we obtain

    \begin{align} \left\vert a_{2}\right\vert & = \frac{\left( 1-\eta \right) [\lambda +1]_{q}\left\vert c_{1}\right\vert }{\left( 1+\beta \right) [2]_{q}!\psi _{2} } = \frac{\left( 1-\beta \right) [\lambda +1]_{q}\left\vert d_{1}\right\vert }{ \left( 1+\gamma +2\eta \right) [2]_{q}!\psi _{2}} \\ &\leq \frac{2\left( 1-\eta \right) [\lambda +1]_{q}}{\left( 1+\beta \right) [2]_{q}!\psi _{2}}. \end{align} (2.15)

    Also, from (2.12) and (2.14), we have

    \begin{equation*} 2\left(1+2\beta -\beta ^{2}\right) \ \frac{[3]_{q}!}{[\lambda +1]_{q,2}} \,\psi _{3}a_{2}^{2} = \left( 1-\beta \right) \left( c_{2}+d_{2}\right). \end{equation*}

    Thus, by using the Carathéodory Lemma, we obtain

    \begin{equation} \left\vert a_{2}\right\vert \leqq \sqrt{\frac{2\left( 1-\beta \right) [\lambda +1]_{q,2}}{\left( 1+2\beta -\beta ^{2}\right) \ [3]_{q}!\,\psi _{3}} }. \end{equation} (2.16)

    From (2.15) and (2.16), we obtain the desired estimate on the coefficient \left\vert a_{2}\right\vert as asserted in (2.9).

    In order to find the bound on the coefficient \left\vert a_{3}\right\vert, we subtract (2.14) from (2.12), so that

    \begin{equation*} 4\left(1+2\beta \right) \ \frac{[3]_{q}!}{[\lambda +1]_{q,2}}\,\psi _{3}\left( a_{3}-a_{2}^{2}\right) = \left( 1-\eta \right) \left( c_{2}-d_{2}\right), \end{equation*}

    that is,

    \begin{equation} a_{3} = a_{2}^{2}+\frac{\left(1-\eta \right) \left( c_{2}-d_{2}\right) [\lambda +1]_{q,2}}{4\left(1+2\beta \right)\; [3]_{q}!\; \psi_{3}}. \end{equation} (2.17)

    Now, upon substituting the value of a_{2}^{2} from (2.16) into (2.17) and using the Carathéodory Lemma, we find that

    \begin{equation} \left\vert a_{3}\right\vert \leqq \frac{2\left( 1-\beta \right) [\lambda +1]_{q,2}}{\left( 1+2\beta -\beta ^{2}\right) \ [3]_{q}!\;\psi _{3}}. \end{equation} (2.18)

    Moreover, upon substituting the value of a_{2}^{2} from (2.11) into (2.12), we have

    \begin{equation*} a_{3} = \frac{\left(1-\eta \right)}{2\left(1+2\beta \right)}\left(\frac{ [\lambda+1]_{q,2}\ c_{2}}{[3]_{q}!\psi_{3}}+\frac{\left(1-\eta \right) [\lambda+1]_{q}^{2}c_{1}^{2}}{\left([2]_{q}!\right)^{2}\psi_{2}^{2}} \right). \end{equation*}

    Applying the Carathéodory Lemma, we obtain

    \begin{equation} \left\vert a_{3}\right\vert \leqq \frac{\left( 1-\eta \right) }{\left( 1+2\beta \right) }\left(\frac{[\lambda +1]_{q,2}\ }{[3]_{q}!\psi _{3}}+ \frac{2\left(1-\eta \right) [\lambda +1]_{q}^{2}}{\left([2]_{q}!\right) ^{2}\psi_{2}^{2}}\right). \end{equation} (2.19)

    Finally, by combining (2.18) and (2.19), we have the desired estimate on the coefficient \left\vert a_{3}\right\vert as asserted in (2.10). The proof of Theorem 2 is thus completed.

    In our present investigation, we have made use of the concept of q -convolution with a view to introducing a new class of analytic and bi-close-to-convex functions in the open unit disk. For functions belonging to this analytic and bi-univalent function class, we have derived estimates for the general coefficients in their Taylor-Maclaurin series expansions in the open unit disk. Our methodology is based essentially upon the Faber polynomial expansion method. We have also presented a number of corollaries and consequences of our main results.

    In his recently-published review-cum-expository review article, in addition to applying the q -analysis to Geometric Function Theory of Complex Analysis, Srivastava [35] pointed out the fact that the results for the q -analogues can easily (and possibly trivially) be translated into the corresponding results for the (p, q) -analogues (with 0 < q < p \leqq 1 ) by applying some obvious parametric and argument variations, the additional parameter p being redundant. Of course, this exposition and observation of Srivastava [35,p. 340] would apply also to the results which we have considered in our present investigation for 0 < q < 1 .

    The authors received no funding for the investigation leading to the completion of this article.

    The authors declare that there is no conflict of interest in respect of this article.



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