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Research article

Soliton solutions and stability analysis of the stochastic nonlinear reaction-diffusion equation with multiplicative white noise in soliton dynamics and optical physics

  • In this article, we explored the stochastic nonlinear reaction-diffusion (RD) equation under the influence of multiplicative white noise. To obtain novel soliton solutions, we employed two powerful analytical techniques: the unified Riccati equation expansion method and the modified Kudryashov method. These methods yield a diverse set of soliton solutions, including combo-dark solitons, dark solitons, singular solitons, combo-bright-singular solitons, and periodic wave solutions. We also performed a comprehensive stability analysis of the stochastic nonlinear RD equation with multiplicative white noise. The findings provide valuable insights into the behavior of solitons in stochastic nonlinear systems, with significant implications for fields such as mathematical physics, nonlinear science, and applied mathematics. These results hold particular relevance for soliton dynamics in optical physics, where they can be applied to improve understanding of wave propagation in noisy environments, signal transmission, and the design of robust optical communication systems.

    Citation: Nafissa T. Trouba, Huiying Xu, Mohamed E. M. Alngar, Reham M. A. Shohib, Haitham A. Mahmoud, Xinzhong Zhu. Soliton solutions and stability analysis of the stochastic nonlinear reaction-diffusion equation with multiplicative white noise in soliton dynamics and optical physics[J]. AIMS Mathematics, 2025, 10(1): 1859-1881. doi: 10.3934/math.2025086

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  • In this article, we explored the stochastic nonlinear reaction-diffusion (RD) equation under the influence of multiplicative white noise. To obtain novel soliton solutions, we employed two powerful analytical techniques: the unified Riccati equation expansion method and the modified Kudryashov method. These methods yield a diverse set of soliton solutions, including combo-dark solitons, dark solitons, singular solitons, combo-bright-singular solitons, and periodic wave solutions. We also performed a comprehensive stability analysis of the stochastic nonlinear RD equation with multiplicative white noise. The findings provide valuable insights into the behavior of solitons in stochastic nonlinear systems, with significant implications for fields such as mathematical physics, nonlinear science, and applied mathematics. These results hold particular relevance for soliton dynamics in optical physics, where they can be applied to improve understanding of wave propagation in noisy environments, signal transmission, and the design of robust optical communication systems.



    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

    ψn=(m+n2n1)θn1(1θ)m(mN;0θ1),

    we get a q-analogue Ψλ,mq,θ of the Pascal distribution operator as follows (see [15]):

    Ψλ,mq,θf(z):=z+n=2(m+n2n1)θn1(1θ)m[n]q![λ+1]q,n1anzn (1.12)
    (zΔ).

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

    f(z)=F(s(z))(zΔ).

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

    f(z)F(z)f(0)=F(0)andf(Δ)F(Δ).

    The Koebe one-quarter theorem (see [10]) asserts that the image of Δ under every univalent function fS contains the disk of radius 14. Therefore, every function fS has an inverse f1 which satisfies the following inequality:

    f(f1(w))=w(|w|<r0(f);r0(f)14),

    where

    g(w)=f1(w)=wa2w2+(2a22a3)w3(5a325a2a3+a4)w4+=w+n=2Anwn.

    A function fA is said to be bi-univalent in Δ if both f and f1 are univalent in Δ. Let Σ denote the class of normalized analytic and bi-univalent functions in Δ given by (1.1). The class Σ of analytic and bi-univalent functions was introduced by Lewin [26], where it was shown that

    fΣ|a2|<1.51.

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

    fΣ|a2|<2

    and, subsequently, Netanyahu [29] proved that

    fΣ|a2|<43.

    It should be noted that the following functions:

    f1(z)=z1z,f2(z)=12log(1+z1z)andf3(z)=log(1z),

    together with their corresponding inverses given by

    f11(w)=w1+w,f12(w)=e2w1e2w+1andf13(w)=ew1ew,

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

    Brannan and Taha [6] (see also [48]) introduced certain subclasses of the bi-univalent function class Σ similar to the familiar subclasses S(α) and K(α) of starlike and convex functions of order α(0α<1), respectively (see [5]). Indeed, following Brannan and Taha [6], a function fA is said to be in the class SΣ(α) of bi-starlike functions of order α(0<α1), if each of the following conditions is satisfied:

    fΣand|arg(zf(z)f(z))|<απ2(zΔ)

    and

    |arg(zF(w)F(w)|)<απ2(wΔ),

    where the function F is the analytic extension of f1 to Δ, given by

    F(w)=wa2w2+(2a22a3)w3(5a325a2a3+a4)w4+(wΔ). (1.13)

    A function fA is said to be in the class KΣ(α) of bi-convex functions of order α(0<α1), if each of the following conditions is satisfied:

    fΣ,with|arg(1+zf(z)f(z))|<απ2(zΔ)

    and

    |arg(1+zg(w)g(w))|<απ2(wΔ).

    The classes SΣ(α) and KΣ(α) of bi-starlike functions of order α in Δ and bi-convex functions of order α(0<α1) in Δ, corresponding to the function classes S(α) and K(α), were also introduced analogously. For each of the function classes SΣ(α) and KΣ(α), non-sharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3| 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 Σ. It not only revived the study of the bi-univalent function class Σ 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:

    TΣ(λ,β)(0λ1;0β<1).

    In the same way, we define the following subclass of bi-close-to-convex functions Hq,λΣ(η,β,Υ) as follows.

    Definition 1. For 0η<1 and 0β1, a function fΣ has the form (1.1) and the function Υ given by (1.2), the function f is said to be in the class Hq,λΣ(η,β,Υ) if there exists a function gS such that

    (z(Hλ,qΥf(z))+βz2(Hλ,qΥf(z))(1β)Hλ,qΥg(z)+βz(Hλ,qΥg(z)))>η(zΔ) (1.14)

    and

    (z(Hλ,qΥF(w))+βz2(Hλ,qΥF(w))(1β)Hλ,qΥG(w)+βz(Hλ,qΥG(w)))>η(wΔ), (1.15)

    where the function F is the analytic extension of f1 to Δ, and is given by (1.13), and G is the analytic extension of g1 to Δ as follows:

    G(w)=wb2w2+(2b22b3)w3(5b325b2b3+b4)w4+(wΔ). (1.16)

    We note that, if bn=an(nN{1}), Sq,λΣ(η,β,Υ) becomes the class of bi-starlike functions satisfying the following inequalities:

    (z(Hλ,qΥf(z))+βz2(Hλ,qΥf(z))(1β)Hλ,qΥf(z)+βz(Hλ,qΥf(z)))>η(zΔ). (1.17)

    and

    (z(Hλ,qΥF(w))+βz2(Hλ,qΥF(w))(1β)Hλ,qΥF(w)+βz(Hλ,qΥF(w)))>η(wΔ). (1.18)

    Remark 3. Each of the following limit cases when q1 is worthy of note.

    (ⅰ) Putting q1, we obtain

    limq1Hq,λΣ(η,β,h)=:PλΣ(η,β,h),

    where PλΣ(η,β,Υ) represents the functions fΣ that satisfy (1.14) and (1.15) with Hλ,qΥ replaced by IλΥ as in (1.6).

    (ⅱ) Putting

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

    we obtain the class Bq,λΣ(η,β,υ) representing the functions fΣ that satisfy (1.14) and (1.15) with Hλ,qΥ replaced by Nλυ,q as in (1.8).

    (ⅲ) Putting

    ψn=(n+1m+n)α(α>0;mN0),

    we obtain the class Lλ,qΣ(η,β,m,α) consisting of the functions fΣ that satisfy (1.14) and (1.15) with Hλ,qΥ replaced by Mλ,αm,q as in (1.10).

    (ⅳ) Putting

    ψn=ρn1(n1)!eρ(ρ>0),

    we obtain the class Mq,λΣ(η,β,ρ) representing the functions fΣ which satisfy the inequalities in (1.14) and (1.15) with Hλ,qΥ replaced by Iλ,ρq as in (1.11).

    (ⅴ) Putting

    ψn=(m+n2n1)θn1(1θ)m(mN;0θ1),

    we get the class Wq,λΣ(η,β,m,θ) of the functions fΣ which satisfy the inequalities in (1.14) and (1.15) with Hλ,qΥ replaced by Ψλ,mq,θ occurring in (1.12).

    Using the Faber polynomial expansion of functions fA which have the normalized form (1.1), the coefficients of its inverse map may be expressed as follows (see [18]):

    F(w)=f1(w)=w+n=21nKnn1(a2,a3,)wn=w+n=2Anwn, (1.19)

    where

    Knn1(a2,a3,)=(n)!(2n+1)!(n1)!an12+(n)!(2(n+1))!(n3)!an32a3+(n)!(2n+3)!(n4)!an42a4+(n)!(2(n+2))!(n5)!an52[a5+(n+2)a23]+(n)!(2n+5)!(n6)!an62[a6+(2n+5)a3a4]+j7anj2Uj (1.20)

    such that Uj with 7jn is a homogeneous polynomial in the variables a2,a3,,an. Here such expressions as (for example) (n)! are to be interpreted symbolically by

    (n)!Γ(1n):=(n)(n1)(n2)(nN0).

    In particular, the first three terms of Knn1 are given by

    K21=2a2,
    K32=3(2a22a3)

    and

    K43=4(5a325a2a3+a4).

    In general, an expansion of Knm(nN) is given by (see [1,8,41,42,47,49,50])

    Knm=nam+n(n1)2D2m+n!3!(n3)!D3m++n!m!(nm)!Dmm,

    where

    Dnm=Dnm(a2,a3,a4,)

    and, alternatively,

    Dnm(a2,a3,,am+1)=i1,,im(n!i1!im!)ai12aimm+1,

    where a1=1 and the sum is taken over all non-negative integers i1,,im satisfying the following constraints:

    i1+i2++im=n

    and

    i1+2i2++mim=m.

    Evidently, we have

    Dmm(a2,a3,,am+1)=am2.

    The following Lemma will be needed to prove our results.

    The Carathéodory Lemma. (see [10]) If ϕP and

    ϕ(z)=1+n=1cnzn,

    then

    |cn|2(nN).

    This inequality is sharp for all positive integers n. Here P is the family of all functions ϕ, which analytic and have positive real part in Δ, with ϕ(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 Hq,λΣ(η,β,Υ).

    Theorem 1. Let the function f given by (1.1) belong to the class Hq,λΣ(η,β,Υ). Suppose also that

    0η<1,0β1,λ>1and0<q<1.

    If ak=0 for 2kn1, then

    |an|2(1η)[λ+1]q,n1n[1+(n1)β] [n]q!ψn+1.

    Proof. If fHq,λΣ(η,β,Υ), then there exists a function g(z), given by

    g(z):=z+n=2bnznS,

    such that

    (z(Hλ,qΥf(z))+βz2(Hλ,qΥf(z))(1β)Hλ,qΥg(z)+βz(Hλ,qΥg(z)))>η(zΔ).

    Moreover, by using the Faber polynomial expansion, we have

    z(Hλ,qΥf(z))+βz2(Hλ,qΥf(z))(1β)Hλ,qΥg(z)+βz(Hλ,qΥg(z))=1+n=2([1+β(n1)][n]q![λ+1]q,n1ψn(nanbn)+n2t=1[n,q]![λ+1,q]n1ψn[1+(nt1)β]K1t[(1+β)b2,(1+2β)b3,,(1+tβ)bt+1][(nt) antbnt])zn1(zΔ). (2.1)

    Also, for the inverse map F=f1, there exists a function G(w), given by

    G(w)=w+n=2BnwnS,

    such that

    (z(Hλ,qΥF(w))+βz2(Hλ,qΥF(w))(1β)Hλ,qΥG(w)+βz(Hλ,qΥG(w)))>η(wΔ),

    the Faber polynomial expansion of the inverse map F=f1 is given by

    F(w)=w+n=2Anwn,

    so we have

    z(Hλ,qΥF(w))+βz2(Hλ,qΥF(w))(1β)Hλ,qΥG(w)+βz(Hλ,qΥG(w))=1+n=2([1+β(n1)][n]q![λ+1]q,n1ψn(nAnBn)+n2t=1[n]q![λ+1]q,n1ψn[1+(nt1)β]K1t[(1+β)B2,(1+2β)B3,,(1+tβ)Bt+1][(nt)AntBnt])wn1(wΔ). (2.2)

    Now, since

    fHq,λΣ(η,β,Υ)andF=f1Hq,λΣ(η,β,Υ),

    there are the following two positive real part functions:

    U(z)=1+n=1cnzn

    and

    V(w)=1+n=1dnwn,

    for which

    (U(z))>0and(V(w))>0(z,wΔ),

    so that

    z(Hλ,qΥF(w))+βz2(Hλ,qΥF(w))(1β)Hλ,qΥG(w)+βz(Hλ,qΥG(w))=η+(1η)U(z)=1+(1η)n=1cnzn (2.3)

    and

    z(Hλ,qΥF(w))+βz2(Hλ,qΥF(w))(1β)Hλ,qΥG(w)+βz(Hλ,qΥG(w))=η+(1η)V(w)=1+(1η)n=1dnwn. (2.4)

    Now, under the assumption that ak=0 for 0kn1, we obtain An=an. Then, by using (2.1) and comparing the corresponding coefficients in (2.3), we obtain

    [1+β(n1)][n]q![λ+1]q,n1ψn(nanbn)=(1η)cn1. (2.5)

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

    [1+β(n1)][n]q![λ+1]q,n1ψn(nAnBn)=(1η)dn1, (2.6)
    [1+β(n1)][n]q![λ+1]q,n1ψn(nanbn)=(1η)cn1 (2.7)

    and

    [1+β(n1)][n]q![λ+1]q,n1ψn(nanBn)=(1η)dn1. (2.8)

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

    |bn|nand|Bn|n,

    and applying the Carathéodory Lemma, we conclude that

    |an|2(1η)[λ+1]q,n1n[1+(n1)β][n]q!ψn+1,

    which completes the proof of Theorem 1.

    If we set

    ψn=(1)n1Γ(υ+1)4n1(n1)!Γ(n+υ)(υ>0)

    in Theorem 1, we obtain the following special case.

    Corollary 1. Let the function f given byt (1.1) belong to the class Bq,λΣ(η,β,υ). Suppose also tha

    0η<1,0β1,λ>1,υ>0and0<q<1.

    If ak=0 for 2kn1, then

    |an|2(1η)[λ+1]q,n1n[1+(n1)β][n]q!ϕn+1,

    where ϕn is given by (1.9).

    Upon putting

    ψn=(n+1m+n)α(α>0;mN0)

    in Theorem 1, we obtain the following result.

    Corollary 2. Let the function f given by (1.1) belong to the class Lq,λΣ(η,β,m,α). Suppose also that

    0η<1,0β1,λ>1,α>0,mN0and0<q<1.

    If ak=0 for 2kn1, then

    |an|2(1η)(m+n)α[λ+1]q,n1n[1+(n1)β][n]q!(n+1)α+1.

    If we take

    ψn=ρn1(n1)!eρ(ρ>0)

    in Theorem 1, we obtain the following special case.

    Corollary 3. Let the function f given by (1.1) belong to the class Mq,λΣ(η,β,ρ). Suppose also that

    0η<1,0β1,λ>1,ρ>0and0<q<1.

    If ak=0 for 2kn1, then

    |an|2(1η)(n1)![λ+1]q,n1n[1+(n1)β][n]q!ρn1eρ+1.

    Upon setting

    ψn=(m+n2n1) θn1(1θ)m(mN;0θ1)

    in Theorem 1, we are led to the following result for the above-defined class Wq,λΣ(η,β,m,θ).

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

    Wq,λΣ(η,β,m,θ)
    (0η<1;0β1;λ>1;0<q<1;mN;0θ1).

    If ak=0 for 2kn1, then

    |an|2(1η)[λ+1]q,n1n[1+(n1)β][n]q!(m+n2n1)θn1(1θ)m+1.

    In particular, if we let g(z)=f(z), we obtain the class Sq,λΣ(η,β,Υ), which is a subclass of Hq,λΣ(η,β,Υ). We then give the next theorem, which involves the coefficients of this subclass of the analytic and bi-starlike functions in Δ.

    Theorem 2. Let the function f given by (1.1) belong to the class Sq,λΣ(η,β,Υ). Suppose also that

    γ1,η0,λ>1,0β<1and0<q<1.

    Then

    |a2|{2(1η)[λ+1]q(1+β) [2]q!ψ2(0η<1(1+β)2 ([2]q!)2[λ+2]q ψ222(1+2ββ2) [3]q![λ+1]q ψ3)2(1η)[λ+1]q,2(1+2ββ2) [3]q!ψ3(1(1+β)2 ([2]q!)2[λ+2]q ψ222(1+2ββ2) [3]q![λ+1]qψ3η<1) (2.9)

    and

    |a3|{2(1η)[λ+1]q,2(1+2ββ2) [3]q!ψ3(0η<1(1+β)2 ([2]q!)2[λ+2]q ψ222(1+2ββ2) [3]q![λ+1]q ψ3)(1η)(1+2β)([λ+1]q,2[3]q!ψ3+2(1η)[λ+1]2q([2]q!)2 ψ22)(1(1+β)2 ([2]q!)2[λ+2]q ψ222(1+2ββ2) [3]q![λ+1]q ψ3η<1). (2.10)

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

    (1+β) [2]q![λ+1]qψ2a2=(1η)c1, (2.11)
    [2(1+2β) a3(1+β)2a22][3]q![λ+1]q,2ψ3=(1η)c2, (2.12)
    (1+β) [2]q![λ+1]qψ2a2=(1η)d1 (2.13)

    and

    [2(1+2β) a3+(3+6ββ2)a22][3]q![λ+1]q,2ψ3=(1η)d2. (2.14)

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

    |a2|=(1η)[λ+1]q|c1|(1+β)[2]q!ψ2=(1β)[λ+1]q|d1|(1+γ+2η)[2]q!ψ22(1η)[λ+1]q(1+β)[2]q!ψ2. (2.15)

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

    2(1+2ββ2) [3]q![λ+1]q,2ψ3a22=(1β)(c2+d2).

    Thus, by using the Carathéodory Lemma, we obtain

    |a2|2(1β)[λ+1]q,2(1+2ββ2) [3]q!ψ3. (2.16)

    From (2.15) and (2.16), we obtain the desired estimate on the coefficient |a2| as asserted in (2.9).

    In order to find the bound on the coefficient |a3|, we subtract (2.14) from (2.12), so that

    4(1+2β) [3]q![λ+1]q,2ψ3(a3a22)=(1η)(c2d2),

    that is,

    a3=a22+(1η)(c2d2)[λ+1]q,24(1+2β)[3]q!ψ3. (2.17)

    Now, upon substituting the value of a22 from (2.16) into (2.17) and using the Carathéodory Lemma, we find that

    |a3|2(1β)[λ+1]q,2(1+2ββ2) [3]q!ψ3. (2.18)

    Moreover, upon substituting the value of a22 from (2.11) into (2.12), we have

    a3=(1η)2(1+2β)([λ+1]q,2 c2[3]q!ψ3+(1η)[λ+1]2qc21([2]q!)2ψ22).

    Applying the Carathéodory Lemma, we obtain

    |a3|(1η)(1+2β)([λ+1]q,2 [3]q!ψ3+2(1η)[λ+1]2q([2]q!)2ψ22). (2.19)

    Finally, by combining (2.18) and (2.19), we have the desired estimate on the coefficient |a3| 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<p1) 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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