Research article Special Issues

Novel inequalities for subadditive functions via tempered fractional integrals and their numerical investigations

  • Received: 29 January 2024 Revised: 25 March 2024 Accepted: 03 April 2024 Published: 09 April 2024
  • MSC : 26A33, 26A51, 26D07, 26D10, 26D15

  • In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities.

    Citation: Artion Kashuri, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Eman Al-Sarairah, Nejmeddine Chorfi. Novel inequalities for subadditive functions via tempered fractional integrals and their numerical investigations[J]. AIMS Mathematics, 2024, 9(5): 13195-13210. doi: 10.3934/math.2024643

    Related Papers:

    [1] Dong-Mei Li, Bing Chai . A dynamic model of hepatitis B virus with drug-resistant treatment. AIMS Mathematics, 2020, 5(5): 4734-4753. doi: 10.3934/math.2020303
    [2] Naveed Shahid, Muhammad Aziz-ur Rehman, Nauman Ahmed, Dumitru Baleanu, Muhammad Sajid Iqbal, Muhammad Rafiq . Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution. AIMS Mathematics, 2021, 6(8): 8294-8314. doi: 10.3934/math.2021480
    [3] Abdul Qadeer Khan, Fakhra Bibi, Saud Fahad Aldosary . Bifurcation analysis and chaos in a discrete Hepatitis B virus model. AIMS Mathematics, 2024, 9(7): 19597-19625. doi: 10.3934/math.2024956
    [4] Nauman Ahmed, Ali Raza, Ali Akgül, Zafar Iqbal, Muhammad Rafiq, Muhammad Ozair Ahmad, Fahd Jarad . New applications related to hepatitis C model. AIMS Mathematics, 2022, 7(6): 11362-11381. doi: 10.3934/math.2022634
    [5] Hui Miao . Global stability of a diffusive humoral immunity viral infection model with time delays and two modes of transmission. AIMS Mathematics, 2025, 10(6): 14122-14139. doi: 10.3934/math.2025636
    [6] Muhammad Farman, Ali Akgül, J. Alberto Conejero, Aamir Shehzad, Kottakkaran Sooppy Nisar, Dumitru Baleanu . Analytical study of a Hepatitis B epidemic model using a discrete generalized nonsingular kernel. AIMS Mathematics, 2024, 9(7): 16966-16997. doi: 10.3934/math.2024824
    [7] Zeeshan Afzal, Mansoor Alshehri . A comparative study of disease transmission in hearing-impaired populations using the SIR model. AIMS Mathematics, 2025, 10(5): 11290-11304. doi: 10.3934/math.2025512
    [8] Alireza Sayyidmousavi, Katrin Rohlf . Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics. AIMS Mathematics, 2019, 4(6): 1805-1823. doi: 10.3934/math.2019.6.1805
    [9] Liping Wang, Peng Wu, Mingshan Li, Lei Shi . Global dynamics analysis of a Zika transmission model with environment transmission route and spatial heterogeneity. AIMS Mathematics, 2022, 7(3): 4803-4832. doi: 10.3934/math.2022268
    [10] Ishtiaq Ali, Saeed Islam . Stability analysis of fractional two-dimensional reaction-diffusion model with applications in biological processes. AIMS Mathematics, 2025, 10(5): 11732-11756. doi: 10.3934/math.2025531
  • In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities.



    Assume that A represents the family of analytic functions in the open unit disc

    U={z:zC and |z|<1}.

    For f1, f2A, we say that f1 subordinate to f2 in U, indicated by

    f1(z)f2(z), zU,

    if there exists a Schwarz function w, defined by

    wB={w:wA, |w(z)|<1 and w(0)=0, zU},

    that satisfies the condition

    f1(z)=f2(w(z)), zU.

    Indeed, it is known that

    f1(z)f2(z)f1(0)=f2(0) and f1(U)f2(U).

    Moreover, if the function f2 is univalent in U then

    f1(z)f2(z)f1(0)=f2(0) and f1(U)f2(U).

    Let the class P be defined by

    P={hA:h(0)=1 and Re(h(z)>0}.

    The class of all functions in the normalized analytic function class A that are univalent in U is also denoted by the symbol S. The maximization of the non-linear functional |a3μa22| or other classes and subclasses of univalent functions has been the subject of a number of established results and these results are known as Fekete-Szegö problems, see [1]. If fS and of the form (1.1), then

    |a3μa22|{34μ, if μ0,1+2exp(2μμ1), if 0μ<1,4μ3, if μ1,}

    and the result |a3μa22| are sharp (see [1]). The Fekete-Szegö problems have a rich history in literature and for complex number μ.

    In the area of geometric function theory (GFT), the q-calculus and fractional q-calculus have been extensively employed by scholars who have developed and investigated a number of novel subclasses of analytic, univalent and bi-univalent functions. Jackson [2,3] first proposed the concept of the q-calculus operator and gave the definition of the q -difference operator Dq in 1909. In instance, Ismail et al. were the first to define a class of q-starlike functions in open unit disc U using Dq in [4]. The most significant usages of q -calculus in the perspective of GFT was basically furnished and the basic (or q-) hypergeometric functions were first used in GFT in a book chapter by Srivastava (see, for details, [5]). See the following articles [6,7,8,9,10] for more information about q-calculus operator theory in GFT.

    Now we review some fundamental definitions and ideas of the q -calculus, we utilize them to create some new subclasses in this paper.

    For a non-negative integer l, the q-number [l]q, (0<q<1), is defined by

    [l]q=1ql1q and [0]=0,

    and the q-number shift factorial is given by

    [l]q!=[1]q[2]q[3]q[l]q,[0]q!=1.

    For q1, then [l]! reduces to l!.

    The q-generalized Pochhammer symbol is defined by

    [l]k=Γq(l+k)Γq(l),  kN, lC.

    The q-gamma function Γq is defined by

    Γq(l)=(1q)lj=01qj+11qj+l.

    The q-generalized Pochhammer symbol is defined by

    [l]k=Γq(l+k)Γq(l),  kN, lC.

    Remark 1. For q1, then [l]q,k reduces to (l)k=Γ(l+k)Γ(l).

    Definition 1. Jackson [3] defined the q-integral of function f(z) as follows:

    f(z)dq(z)=n=0z(1q)f(qn(z))qn.

    Jackson [2] introduced the q-difference operator for analytic functions as follows:

    Definition 2. [2] For fA, the q-difference operator is defined as:

    Dqf(z)=f(qz)f(z)z(q1),  zU.

    Note that, for nN, zU and

    Dq(zn)=[n]qzn1,  Dq(n=1anzn)=n=1[n]qanzn1.

    Let Ap stand for the class of analytic functions with the form

    f(z)=zp+n=1an+pzn+p,  pN,zU (1.1)

    in the open unit disk U. More specifically, A1=A and

    f(z)=z+n=1an+1zn+1,  zU. (1.2)

    Consider the q-difference operator for fAp as follows:

    Definition 3. [11] For fAp, the q-difference operator is defined as:

    Dqf(z)=f(qz)f(z)z(q1),  zU.

    Note that, for nN, zU and

    Dq(zn+p)=[n+p]qzn+p1,  Dq(n=1an+pzn+p)=n=1[n+p]qan+pzn+p1.

    Let S(p) represents the class of p-valent starlike functions and every fS(p), if

    Re(zf(z)pf(z))>0,  zU,

    and K(p) represents the class of p-valent convex functions and every fK(p), if

    1p(1+Re(zf(z)f(z)))>0,  zU.

    These conditions are equivalent in terms of subordination as follows:

    S(p)={fAp:zf(z)pf(z)1+z1z}

    and

    K(p)={fAp:1p(1+zf(z)f(z))1+z1z}.

    The aforementioned two classes can be generalized as follows:

    S(p,φ)={fAp:zf(z)pf(z)φ(z)}

    and

    K(p,φ)={fAp:1p(1+zf(z)f(z))φ(z)},

    where φ(z) is a real part function that is positive and is normalized by the rule

    φ(0)=1 and φ(0)>0,

    and φ maps U onto a space that is symmetric with regard to the real axis and starlike with respect to 1. If p=1, then

    S(p,φ)=S(φ)

    and

    K(p,φ)=K(φ).

    These two classes S(φ) and K(φ) defined by Ma [12].

    A function fAp, is called p-valently starlike of order α (0α<1) with complex order bC{0}, if it satisfies the inequality

    Re{1+1b(zf(z)pf(z)1)>α, zU}.

    The class Sp(α,b) denotes the collection of all fAp functions that satisfy the aforementioned condition.

    A function fAp, is called p-valently convex function of order α (0α<1) with complex order bC{0}, if it satisfies the inequality

    Re{11b+1bp(1+zf(z)f(z))>α, zU}.

    The class Kp(α,b) denotes the collection of all functions fAp that satisfy the aforementioned condition.

    Note that

    fKp(α,b)1pzfSp(α,b).

    Kargar et al. [13] investigated the classes Sp(α,β) for fAp and defined as follows:

    fSp(α,β)α<Re(1pzf(z)f(z))<β,   (0α<1<β, zU).

    For 0α<1<β and bC{0}, then the function fAp belongs to the class Kb,p(α,β) if it satisfies the inequality

    α<Re(11b+1bp(1+zf(z)f(z)))<β,   (0α<1<β, zU).

    If p=1, then Kb,p(α,β)=Kb(α,β), studied by Kargar et al. in [13] and if β in above definition, Kb,p(α,β)=Kb,p(α,b).

    Recently, Bult [14] used the definition of subordination and defined new subclasses of p-valent starlike and convex functions associated with vertical strip domain as follows:

    Sp,b(α,β)={fAp:1+1b(1pzf(z)f(z)1)f(α,β;z)}

    and

    Kp,b(α,β)={fAp:11b+1bp(1+zf(z)f(z))f(α,β;z)},

    where

    f(α,β;z)=1+βαπilog(1e2πi1αβαz1z)

    and

    0α<1<β,bC{0}, zU.

    Bult [14] determined the coefficient bounds for functions belonging to these new classes.

    On the basis of the geometrical interpretation of their image domains, numerous subclasses of analytic functions have established using the concept of subordination. Right half plane [15], circular disc [16], oval and petal type domains [17], conic domain [18,19], leaf-like domain [20], generalized conic domains [21], and the most important one is shell-like curve [22,23,24,25] are some fascinating geometrical classes we obtain with this domain. The function

    h(z)=1+τ2z21τzτ2z2 (1.3)

    is essential for the shell-like shape, where

    τ=152.

    The image of unit circle under the function h gives the conchoid of Maclaurin's, due to the function

    h(eiφ)=52(32cosφ)+isinφ(4cosφ1)2(32cosφ)(1+cosφ), 0φ<2π.

    The function given in (1.3) has the following series representation:

    h(z)=1+n=1(un1+un+1)τnzn,

    where

    un=(1τ)nτn5,

    and un produces a Fibonacci series of coefficient constants that are more closely related to the Fibonacci numbers.

    Taking motivation from the idea of circular disc and shell-like curves, Malik et al. [26] defined new domain for analytic functions which is named as cardioid domain. A new class of analytic functions is defined associated with cardioid domain, for more detail, see [26].

    Definition 4. [26] Assume that CP(L,N) represents the class of functions p that are defined as

    p(z)¯p(L,N,z),

    where ¯p(L,N,z) is defined by

    ¯p(L,N,z)=2Lτ2z2+(L1)τz+22Nτ2z2+(N1)τz+2 (1.4)

    with 1<N<L1, τ=152 and zU.

    To understand the class CP(L,N), an explanation of the function ¯p(L,N,z) in geometric terms might be helpful in this instance. If we denote

    R¯p(L,N;eiθ)=u

    and

    I¯p(L,N;eiθ)=v,

    then the image ¯p(L,N,eiθ) of the unit circle is a cardioid like curve defined by

    {u=4+(L1)(N1)τ2+4LNτ4+2λcosθ+4(L+N)τ2cos2θ4+(N1)2τ2+4N2τ4+4(N1)(τ+Nτ3)cosθ+8Nτ2cos2θ,v=(LN)(ττ3)sinθ+2τ2sin2θ4+(N1)2τ2+4N2τ4+4(N1)(τ+Nτ3)cosθ+8Nτ2cos2θ,} (1.5)

    where

    λ=(L+N2)τ+(2LNLN)τ3, 1<N<L1, τ=152,

    and

    0θ2π.

    Moreover, we observe that

    ¯p(L,N,0)=1,

    and

    ¯p(L,N,1)=LN+9(L+N)+1+4(NL)5N2+18N+1.

    According to (1.5), the cusp of the cardioid-like curve is provided by

    γ(L,N)=¯p(L,N;e±iarccos(14))=2LN3(L+N)+2+(LN)52(N23N+1).

    The image of each inner circle is a nested cardioid-like curve if the open unit disc U is considered a collection of concentric circles with origin at the center. As a result, the open unit disc U is mapped onto a cardioid region by the function ¯p(L,N,z). This means that ¯p(L,N;U) is a cardioid domain. The above discussed cardioid like curve with different values of parameters can be seen in Figures 1 and 2.

    Figure 1.  The curve (1.5) with L = 0.8, N = 0.6 and the curve (1.5) with L = 0.5, N = -0.5.
    Figure 2.  The curve (1.5) with L = 0.6, N = 0.8 and the curve (1.5) with L = -0.5, N = 0.5.

    The relationship N<L links the parameters L and N. The cardioid-like curve is flipped by its voilation, as seen in the figures below.

    See Figure 3, if collection of concentric circles having origin as center. Thus, the function ¯p(L,N,z) maps the open unit disk U onto a cardioid region. See [26] for more details about cardioid region.

    Figure 3.  The open unit disk U.

    The operator theory of quantum calculus is the primary result of this research. Using standard uses in quantum calculus operator theory and the q -difference operator, we develop numerous novel q-analogous of the differential and integral operators. We construct a large number of new classes of q-starlike and q-convex functions using these operators and study some interesting characteristics of the corresponding analytic functions. In this paper, we gain inspiration from recent research by [14,26,27] and define two new classes of p-valent starlike, convex functions connected with the cardioid domain using the q-difference operator.

    Influenced by recent studies [14,26,27], we defined two new classes of p-valent starlike, convex functions related with cardioid domain.

    Definition 5. The function f of the form (1.1) related with cardioid domain, represented by Sp(L,N,q,b), is defined to be the functions f such that

    1+1b(1[p]qzDqf(z)f(z)1)¯p(L,N;z),

    where, bC{0} and ¯p(L,N;z) is given by (1.4).

    Definition 6. The function f of the form (1.1) related with cardioid domain, represented by Kp(L,N,q,b), is defined to be the functions f such that

    11b+1b[p]q(1+zD2qf(z)Dqf(z))¯p(L,N;z),

    where bC{0} and ¯p(L,N;z) is given by (1.4) and Kp(L,N,q,b) is the class of convex functions of order b related with cardioid domain.

    Special cases:

    (i) For q1,b=1 and p=1, in Definition 5, we have known class S(L,N) of starlike functions associated with cardioid domain proved by Zainab et al. in [27].

    (ii) For q1,L=1,N=1, b=1 and p=1 in Definition 5, then class Sp(L,N,q,b)=SL and this class is defined on starlike functions associated with Fibonacci numbers, introduced and studied by Sokół in [28].

    (iii) For q1, L=1, N=1, b=1 and p=1 in Definition 6 then class Kp(L,N,q,b)=K, and this family is referred to as a class of convex functions connected with Fibonacci numbers.

    There are four parts to this article. In Section 1, we briefly reviewed some basic concepts from geometric function theory, quantum calculus, and cardioid domain, studied the q-difference operator, and finally discussed this operator to define two new subclasses of multivalent q-starlike and q-convex functions. The established lemmas are presented in Section 2. Our main results and some known corollaries will be presented in Section 3, then some concluding remarks in Section 4.

    By utilizing the following lemmas, we will determine our main results.

    Lemma 1. [26] Let the function ¯p(L,N;z), defined by (1.4). Then,

    (i) For the disc |z|<τ2, the function ¯p(L,N;z) is univalent.

    (ii) If h(z)¯p(L,N;z), then Reh(z)>α, where

    α=2(L+N2)τ+2(2LNLN)τ3+16(L+N)τ2η4(N1)(τ+Nτ3)+32Nτ2η,

    where

    η=4+τ2N2τ24N2τ4(1Nτ2)χ(N)4τ(1+N2τ2),
    χ(N)=5(2Nτ2(N1)τ+2)(2Nτ2+(N1)τ+2),
    1<N<L1

    and

    τ=152.

    (iii) If

    ¯p(L,N;z)=1+n=1¯Qnzn,

    then

    ¯Qn={(LN)τ2,for n=1,(LN)(5N)τ222,for n=2,1N2τpn1Nτ2pn2,for n=3,4,5,, (2.1)

    where

    1<N<L1.

    (iv) Let h(z)¯p(L,N;z) and of the form h(z)=1+n=1hnzn. Then

    |h2vh21|(LN)|τ|4max{2,|τ(v(LN)+N5)|},  vC.

    Lemma 2. [29] Let hP, such that h(z)=1+n=1cnzn. Then

    |c2v2c21|max{2,2|v1|}={2,if  0v2,2|v1|,   elsewhere,} (2.2)

    and

    |cn|2 for n1. (2.3)

    Lemma 3. [30] Let hP, such that

    h(z)=1+n=1cnzn.

    Then for any complex number v

    |c2vc21|2max{1,|2v1|}

    and the result is sharp for

    h(z)=1+z21z2andh(z)=1+z1z.

    Lemma 4. [31] Let the function g given by

    g(z)=k=1bkzk

    be convex in U. Also let the function f given by

    f(z)=k=1akzk

    be analytic in U. If

    f(z)g(z),

    then

    |ak|<|b1|,  k=1,2,3,.

    For the recently described classes of multivalent q-starlike (Sp(L,N,q,b)) and multivalent q-convex (Kp(L,N,q,b)) functions, we get sharp estimates for the coefficients of Taylor series, Fekete-Szegő problems and coefficient inequalities.

    In the following theorems, we investigate the functions f(z) which can be used to find the sharpness of the results of this article.

    Theorem 5. A function fAp given by (1.1) is in the class Sp(L,N,q,b) if and only if there exists an analytic function S,

    S(z)¯p(L,N,z)=2Lτ2z2+(L1)τz+22Nτ2z2+(N1)τz+2,

    where, ¯p(L,N,0)=1, such that

    f(z)=zpexp{b[p]qz0S(t)1tdt},  zU. (3.1)

    Proof. Let fSp(L,N,q,b) and

    1+1b(1[p]qzDqf(z)f(z)1)=h(z)¯p(L,N,z).

    Then by integrating this equation we obtain (3.1). Conversely, if given by (3.1) with an analytic function S(z) such that S(z)¯p(L,N,z), then by logarithmic differentiation of (3.1) we obtain

    1+1b(1[p]qzDqf(z)f(z)1)=S(z).

    Therefore we have

    1+1b(1[p]qzDqf(z)f(z)1)¯p(L,N,z)

    and fSp(L,N,q,b).

    The initial coefficient bounds |ap+1| and |ap+2| for the functions fSp(L,N,b) are investigated in Theorem 6 using the Lemma 2.

    Theorem 6. Let fSp(L,N,q,b) be given by (1.1), 1N<L1. Then

    |ap+1|[p]q|b|(LN)τ2,|ap+2|[p]q|b|(LN)|τ|28(5N+[p]qb(LN)).

    These bounds are sharp.

    Proof. Let fSp(L,N,q,b), and of the form (1.1). Then

    1+1b(1[p]qzDqf(z)f(z)1)¯p(L,N;z), (3.2)

    where

    ¯p(L,N,z)=2Lτ2z2+(L1)τz+22Nτ2z2+(N1)τz+2.

    By applying the concept of subordination, there exists a function w with

    w(0)=0 and |w(z)|<1,

    such that

    1+1b(1[p]qzDqf(z)f(z)1)=¯p(L,N;w(z)). (3.3)

    Let

    w(z)=h(z)1h(z)+1=c1z+c2z2+c3z3+2+c1z+c2z2+=12c1z+12(c212c21)z2+12(c3c1c2+14c31)z3+. (3.4)

    Since

    ¯p(L,N;z)=1+n=1¯Qnzn,

    then

    ¯p(L,N;w(z))=1+¯Q1{12c1z+12(c212c21)z2}+¯Q2{12c1z+12(c212c21)z2}2+=1+¯Q1c12z+(12(c212c21)¯Q1+¯Q2c214)z2+. (3.5)

    Also consider the function

    ¯p(L,N;z)=2Lτ2z2+(L1)τz+22Nτ2z2+(N1)τz+2.

    Let τz=α0, then

    ¯p(L,N,z)=2Lα20+(L1)α0+22Nα20+(N1)α0+2=Lα20+(L1)2α0+1Nα20+(N1)2α0+1=(Lα20+(L1)2α0+1)[1+12(1N)α0+(N26N+14)α20+]=1+12(LN)α0+14(LN)(5N)α20+.

    This implies that

    ¯p(L,N;z)=1+12(LN)τz+14(LN)(5N)τ2z2+. (3.6)

    It is simple to observe from (3.5) that

    ¯p(L,N;w(z))=1+14(LN)τc1z+(14(LN)τ(c212c21)+(LN)(5N)τ2c2116)z2+. (3.7)

    Since fSp(L,N,b), then

    1+1b(1[p]qzDqf(z)f(z)1)=1+1b[p]qap+1z+1b[p]q(2ap+2a2p+1)z2+. (3.8)

    It is simple to show that by utilizing (3.3) and comparing the coefficients from (3.7) and (3.8), we get

    ap+1=b[p]q(LN)τc14. (3.9)

    Applying modulus on both side, we have

    |ap+1|[p]q|b|(LN)τ2.

    Now again comparing the coefficients from (3.7) and (3.8), we have

    2b[p]qap+2=14(LN)τ(c212c21)+(LN)(5N)τ2c2116+1[p]qba2p+1,
    |ap+2|=b[p]q(LN)τ8|c2v2c21|, (3.10)

    where

    v=1τ2(5N+[p]qb(LN)).

    It shows that v>2 which is satisfied by the relation L>N. Hence, by applying Lemma 2, we obtain the required result.

    Result is sharp for the function

    f(z)=zpexp(b[p]qz0¯p(L,N,t)1tdt)=zp+b[p]q(LN)τ2zp+1+b[p]q(LN)(5N)τ28zp+2+, (3.11)

    where ¯p(L,N,) defined in (1.4).

    Letting q1, b=1 and p=1 in Theorem 6, we get the known corollary proved in [32] for starlike functions connected with cardioid domain.

    Corollary 1. [32] Let fS(L,N) be given by (1.2), 1N<L1. Then

    |a2|(LN)|τ|2,|a3|(LN)|τ|28{L2N+5}.

    Fekete-Szegö problem |ap+2μa2p+1| for the functions fSp(L,N,b) are investigated in Theorem 7.

    Theorem 7. Let fSp(L,N,q,b) and of the form (1.1). Then

    |ap+2μa2p+1|[p]q|b|(LN)|τ|8max{2,|τ((LN)[p]qb+N5+2[p]qb(LN)μ)|}.

    This result is sharp.

    Proof. Since fSp(L,N,q,b), we have

    1+1b(1[p]qzDqf(z)f(z)1)=¯p(L,N;w(z)),  zU,

    where w is Schwarz function such that w(0) and |w(z)|<1 in U. Therefore

    1+1b(1[p]qzDqf(z)f(z)1)=h(z),1+1[p]qbzDqf(z)f(z)=1b+(1+h1z+h2z2+),zDqf(z)=[p]qbf(z)(1b+h1z+h2z2+),

    and after some simple calculation, we have

    [p]qzp+[p+1]qap+1zp+1+[p+2]qap+2zp+2+=[p]qb{zp+ap+1zp+1+ap+2zp+2+}(1b+h1z+h2z2+)=[p]q{zp+ap+1zp+1+ap+2zp+2+}(1+bh1z+bh2z2+).

    Comparing the coefficients of both sides, we get

    ap+1=[p]qbh1,   2ap+2=[p]qb(h1ap+1+h2).

    This implies that

    |ap+2μa2p+1|=[p]q|b|2|h2+(12μ)[p]qbh21|=[p]q|b|2|h2vh21|,

    where

    v=(2μ1)[p]qb.

    By using (iv) of Lemma 1 for

    v=(2μ1)[p]qb,

    we have the required result. The equality

    |ap+2μa2p+1|=[p]q|b|(LN)|τ|28|(LN)[p]qbN+52[p]qb(LN)μ|

    holds for f given in (3.11). Consider f0: UC defined as:

    f0(z)=zpexp([p]qbz0¯p(L,N;t2)1tdt)=zp+[p]qbτ2(LN)zp+2+, (3.12)

    where, ¯p(L,N;z) is defined in (1.4). Hence

    1+1b(1[p]qzDqf(z)f(z)1)=¯p(L,N;z2).

    This demonstrates f0Sp(L,N,q,b). Hence the equality

    |ap+2μa2p+1|=[p]q|b|(LN)|τ|2

    holds for the function f0 given in (3.12).

    Letting q1, b=1 and p=1 in Theorem 7, we get the known corollary proved in [32] for starlike functions associated with cardioid domain.

    Corollary 2. [32] Let fS(L,N) and of the form (1.2). Then

    |a3μa22|(LN)|τ|8max{2,|τ((L2N+5)+2(LN)μ)|}.

    This result is sharp.

    Coefficient inequality for the class Sp(L,N,b):

    Theorem 8. For function fAp, given by (1.1), if fSp(L,N,q,b), then

    |ap+n|n+1k=2([k2]q+[p]qqp|b((LN)τ2)|)[n]q!,  (  p,nN).

    Proof. Suppose fSp(L,N,q,b) and the function S(z) define by

    S(z)=1+1b(1[p]qzDqf(z)f(z)1). (3.13)

    Then by Definition 5, we have

    S(z)¯p(L,N;z),

    where, bC{0} and ¯p(L,N;z) is given by (1.4). Hence, applying the Lemma 4, we get

    |S(m)(0)m!|=|cm||¯Q1|,  mN, (3.14)

    where

    S(z)=1+c1z+c2z2+,

    and by (2.1), we have

    |¯Q1|=|(LN)τ2|. (3.15)

    Also from (3.13), we find

    zDqf(z)=[p]q{b[q(z)1]+1}f(z). (3.16)

    Since ap=1, in view of (3.16), we obtain

    [n+p]q[p]qap+n=[p]qb{cn+cn1ap+1++c1ap+n1}=b[p]qni=1ciap+ni. (3.17)

    Applying (3.14) into (3.17), we get

    qp[n]q|ap+n|[p]q|b||¯Q1|ni=1|ap+ni|,   p,nN.

    For n=1,2,3, we have

    |ap+1|[p]qqp|b¯Q1|,|ap+2|[p]q|b¯Q1|qp[2]q(1+|ap+1|)[p]q|b¯Q1|qp[2]q(1+[p]qqp|b¯Q1|)

    and

    |ap+3|[p]q|b¯Q1|qp[3]q(1+|ap+1|+|ap+2|)[p]q|b¯Q1|qp[3]q(1+[p]qqp|b¯Q1|+[p]q|b¯Q1|qp[2]q(1+[p]qqp|b¯Q1|))=[p]qqp|b¯Q1|((1+[p]qqp|b¯Q1|)([2]q+[p]qqp|b¯Q1|)[3]q[2]q),

    respectively. Applying the equality (3.15) and using the mathematical induction principle, we obtain

    |ap+n|n+1k=2([k2]q+[p]qqp|b¯Q1|)[n]q!=n+1k=2([k2]q+[p]qqp|b((LN)τ2)|)[n]q!.

    This evidently completes the proof of Theorem 8.

    The initial coefficient bounds |ap+1| and |ap+2| for the functions fKp(L,N,q,b) are investigated in Theorem 9 using the Lemma 2.

    Theorem 9. Let fKp(L,N,q,b) be given by (1.1), 1N<L1. Then

    |ap+1|[p]2q|b|(LN)τ2[p+1]q,|ap+2|b2[p]q(LN)τ8[p+2]q(5N+[p]qb2(LN)).

    These bounds are sharp.

    Proof. Let fKp(L,N,q,b), and be of the form (1.1). Then

    11b+1b[p]q(1+zD2qf(z)Dqf(z))¯p(L,N;z), (3.18)

    where

    ¯p(L,N,z)=2Lτ2z2+(L1)τz+22Nτ2z2+(N1)τz+2.

    By applying the concepts of subordination, there exists a function w with

    w(0)=0 and |w(z)|<1,

    such that

    11b+1b[p]q(1+zD2qf(z)Dqf(z))=¯p(L,N;w(z)). (3.19)

    Let

    w(z)=h(z)1h(z)+1=c1z+c2z2+c3z3+2+c1z+c2z2+=12c1z+12(c212c21)z2+12(c3c1c2+14c31)z3+. (3.20)

    Since

    ¯p(L,N;z)=1+n=1¯Qnzn,

    then

    ¯p(L,N;w(z))=1+¯Q1{12c1z+12(c212c21)z2}+¯Q2{12c1z+12(c212c21)z2}2+=1+¯Q1c12z+(12(c212c21)¯Q1+¯Q2c214)z2+. (3.21)

    Also consider the function

    ¯p(L,N;z)=2Lτ2z2+(L1)τz+22Nτ2z2+(N1)τz+2.

    Let τz=α0. Then

    ¯p(L,N,z)=2Lα20+(L1)α0+22Nα20+(N1)α0+2=Lα20+(L1)2α0+1Nα20+(N1)2α0+1=(Lα20+(L1)2α0+1)[1+12(1N)α0+(N26N+14)α20+]=1+12(LN)α0+14(LN)(5N)α20+.

    This implies that

    ¯p(L,N;z)=1+12(LN)τz+14(LN)(5N)τ2z2+. (3.22)

    It is simple to observe from (3.21) that

    ¯p(L,N;w(z))=1+14(LN)τc1z+(14(LN)τ(c212c21)+(LN)(5N)τ2c2116)z2+. (3.23)

    Since fKp(L,N,q,b), then

    11b+1b[p]q(1+zD2qf(z)Dqf(z))=1+[p+1]qb[p]qap+1z+1b[p]q(2[p+2]qap+2[p+1]2q[p]qa2p+1)z2+. (3.24)

    It is simple to show that by utilizing (3.19) and comparing the coefficients from (3.23) and (3.24), we get

    ap+1=b[p]2q(LN)τc14[p+1]q.

    Applying modulus on both side, we have

    |ap+1|[p]2q|b|(LN)τ2[p+1]q.

    Now again comparing the coefficients from (3.7) and (3.8), we have

    2[p+2]qb[p]qap+2=14(LN)τ(c212c21)+(LN)(5N)τ2c2116+(p+1)2bp2a2p+1,
    |ap+2|=b[p]q(LN)τ8[p+2]q|c2v2c21|,   

    where

    v=1τ2(5N+[p]qb2(LN)),

    it shows that v>2 which is satisfied by the relation L>N. Hence, by applying Lemma 2, we obtain the required result.

    Result is sharp for the function

    f(z)=zpexp(b[p]2qz0¯p(L,N,t)1tdt)=zp+b[p]2q(LN)τ2zp+1+b[p]2q(LN)(5N)τ28zp+2+, (3.25)

    where ¯p(L,N,) defined in (1.4).

    Fekete-Szegö problem |ap+2μa2p+1| for the functions fKp(L,N,q,b) are investigated in Theorem 10.

    Theorem 10. Let fKp(L,N,q,b) and be of the form (1.1). Then

    |ap+2μa2p+1|[p]q|b|(LN)|τ|4([p+1]q[p]q+1)[p+2]q×max{2,|τ((LN)[p]2qb+N5+[p]3qb[p+2]q([p+1]q[p]q+1)(LN)[p+1]2qμ)|}.

    This result is sharp.

    Proof. Since fKp(L,N,b), we have

    11b+1b[p]q(1+zD2qf(z)Dqf(z))=¯p(L,N;w(z)),   zU,

    where w is Schwarz function such that w(0) and |w(z)|<1 in U. Therefore

    11b+1b[p]q(1+zD2qf(z)Dqf(z))=h(z),1+zD2qf(z)Dqf(z)=(1b1)b[p]q+b[p]q(1+h1z+h2z2+),1+zD2qf(z)Dqf(z)=([p]q+b[p]qh1z+b[p]qh2z2+),1+zD2qf(z)Dqf(z)=[p]q[1+bh1z+bh2z2+],

    and after some simple calculation, we have

    [p]q(1+[p1]q)zp1+([p+1]q([p]q+1)ap+1zp+[p+2]q([p+1]q+1)ap+2zp+1+=[p]q{[p]qzp1+[p+1]qap+1zp+[p+2]qap+2zp+1+}(1+bh1z+bh2z2+)=[p]2qzp1+([p]q([p+1]qap+1+[p]2qbh1)zp+{[p]q[p+2]qap+2+[p]q[p+1]qbh1ap+1+[p]2qbh2}zp+1.

    Comparing the coefficients of both sides, we get

    ap+1=[p]2qbh1[p+1]q,   ap+2=[p]qb[p+2]q([p+1]q[p]q+1)([p+1]qh1ap+1+h2).

    This implies that

    ap+2μa2p+1=[p]q|b|([p+1]q[p]q+1)[p+2]q×(h2+(1[p+2]q[p]q([p+1]q[p]q+1)[p+1]2qμ)[p]2qbh21)=[p]q|b|([p+1]q[p]q+1)[p+2]q(h2vh21),

    where

    v=([p+2]q[p]q([p+1]q[p]q+1)[p+1]2qμ1)[p]2qb.

    By using (iv) of Lemma 1 for

    v=([p+2]q[p]q([p+1]q[p]q+1)[p+1]2qμ1)[p]2qb,

    we have the required result. The equality

    |ap+2μa2p+1|=[p]q|b|(LN)|τ|24([p+1]q[p]q+1)[p+2]q×|(LN)[p]2qbN+5[p]3qb([p+1]q[p]q+1)[p+2]q(LN)[p+1]2qμ|

    holds for f given in (3.25). Consider the function f0: UC be defined as:

    f0(z)=zpexp([p]2qbz0¯p(L,N;t2)1tdt)=zp+τ[p]2qb2(LN)zp+2+,

    where, ¯p(L,N;z) is defined in (1.4). Hence and

    11b+1b[p]q(1+zD2qf(z)Dqf(z))=¯p(L,N;z2).

    Theorem 11. Let fAp, be given by (1.1). If fKp(L,N,b), then

    |ap+n|[p]qn+1k=2([k2]q+[2]q[p]qqp|b((LN)τ2)|)[n]q!([p+n]q),   p,nN.

    Proof. We can obtain Theorem 11, by using the same technique of Theorem 8.

    In this article, we have used the ideas of cardioid domain, multivalent analytic functions, and q-calculus operator theory to define the new subfamilies of multivalent q-starlike and q-convex functions. In Section 1, we discussed some basic concepts from geometric functions, analytic functions, multivalent functions, q-calculus operator theory, and the idea of the cardioid domain. We also define two new classes of p-valent starlike, convex functions connected with the cardioid domain using the q -difference operator. The already known lemmas are presented in Section 2. In Section 3, for the class Sp(L,N,q,b), we investigated sharp coefficient bounds, Fekete-Szegö functional, and coefficient inequalities. Same type of results also studied for the class Sp(L,N,q,b). The research also demonstrated how the parameters, including some new discoveries, expand and enhance the results.

    For future studies, researchers can use a number of ordinary differential and q-analogous of difference and integral operators and can define a number of new subclasses of multivalent functions. By applying the ideas of this article, many new results can be found. The idea presented in this article can be implemented on papers [33,34,35], and researchers can discuss the new properties of multivalent functions associated with the cardioid domain.

    The authors declare that they did not employ any artificial intelligence in the execution of this work.

    The authors extend their appreciation to the Arab Open University for funding this work through research fund No. (AOURG-2023-007).

    All the authors claim to have no conflicts of interest.



    [1] E. Hille, R. S. Phillips, Functional analysis and semigroups, American Mathematical Society, 1996.
    [2] R. A. Rosenbaum, Sub-additive functions, Duke Math. J., 17 (1950), 227–247. https://doi.org/10.1215/S0012-7094-50-01721-2 doi: 10.1215/S0012-7094-50-01721-2
    [3] F. M. Dannan, Submultiplicative and subadditive functions and integral inequalities of Bellman-Bihari type, J. Math. Anal. Appl., 120 (1986), 631–646. https://doi.org/10.1016/0022-247X(86)90185-X doi: 10.1016/0022-247X(86)90185-X
    [4] R. G. Laatsch, Subadditive functions of one real variable, Oklahoma State University, 1962.
    [5] J. Matkowski, On subadditive functions and Ψ-additive mappings, Open Math., 1 (2003), 435–440.
    [6] S. K. Sahoo, E. Al-Sarairah, P. O. Mohammed, M. Tariq, K. Nonlaopon, Modified inequalities on center-radius order interval-valued functions pertaining to Riemann-Liouville fractional integrals, Axioms, 11 (2022), 1–18. https://doi.org/10.3390/axioms11120732 doi: 10.3390/axioms11120732
    [7] J. Matkowski, T. Swiatkowski, On subadditive functions, Proc. Amer. Math. Soc., 119 (1993), 187–197.
    [8] M. A. Ali, M. Z. Sarikaya, H. Budak, Fractional Hermite-Hadamard type inequalities for subadditive functions, Filomat, 36 (2022), 3715–3729. https://doi.org/10.2298/FIL2211715A doi: 10.2298/FIL2211715A
    [9] H. Kadakal, Hermite-Hadamard type inequalities for subadditive functions, AIMS Math., 5 (2020), 930–939. https://doi.org/10.3934/math.2020064 doi: 10.3934/math.2020064
    [10] M. Kadakal, İ. İşcan, Exponential type convexity and some related inequalities, J. Inequal. Appl., 2020 (2020), 1–9. https://doi.org/10.1186/s13660-020-02349-1 doi: 10.1186/s13660-020-02349-1
    [11] M. Alomari, M. Darus, U. S. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comput. Math. Appl., 59 (2010), 225–232. https://doi.org/10.1016/j.camwa.2009.08.002 doi: 10.1016/j.camwa.2009.08.002
    [12] X. M. Zhang, Y. M. Chu, X. H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its applications, J. Inequal. Appl., 2010 (2010), 1–11. https://doi.org/10.1155/2010/507560 doi: 10.1155/2010/507560
    [13] S. S. Dragomir, J. Pećarič, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335–341.
    [14] H. M. Srivastava, S. K. Sahoo, P. O. Mohammed, A. Kashuri, N. Chorfi, Results on Minkowski-type inequalities for weighted fractional integral operators, Symmetry, 15 (2023), 1–26. https://doi.org/10.3390/sym15081522 doi: 10.3390/sym15081522
    [15] B. Y. Xi, F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacet. J. Math. Stat., 42 (2013), 243–257.
    [16] S. Mehmood, P. O. Mohammed, A. Kashuri, N. Chorfi, S. A. Mahmood, M. A. Yousif, Some new fractional inequalities defined using cr-Log-h-convex functions and applications, Symmetry, 16 (2024), 1–12. https://doi.org/10.3390/sym16040407 doi: 10.3390/sym16040407
    [17] P. O. Mohammed, T. Abdeljawad, S. D. Zeng, A. Kashuri, Fractional Hermite-Hadamard integral inequalities for a new class of convex functions, Symmetry, 12 (2020), 1–12. https://doi.org/10.3390/sym12091485 doi: 10.3390/sym12091485
    [18] L. L. Zhang, Y. Peng, T. S. Du, On multiplicative Hermite-Hadamard- and Newton-type inequalities for multiplicatively (P,m)-convex functions, J. Math. Anal. Appl., 534 (2024), 128117. https://doi.org/10.1016/j.jmaa.2024.128117 doi: 10.1016/j.jmaa.2024.128117
    [19] M. Z. Sarikaya, M. A. Ali, Hermite-Hadamard type inequalities and related inequalities for subadditive functions, Miskolc Math. Notes, 22 (2021), 929–937. https://doi.org/10.18514/MMN.2021.3154 doi: 10.18514/MMN.2021.3154
    [20] P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 1–17. https://doi.org/10.3390/sym12040595 doi: 10.3390/sym12040595
    [21] Y. Cao, J. F. Cao, P. Z. Tan, T. S. Du, Some parameterized inequalities arising from the tempered fractional integrals involving the (μ,η)-incomplete gamma functions, J. Math. Inequal., 16 (2022), 1091–1121. https://doi.org/10.7153/jmi-2022-16-73 doi: 10.7153/jmi-2022-16-73
  • This article has been cited by:

    1. Rahat Zarin, Kamel Guedri, Sobhy M. Ibrahim, Alhanouf Alburaikan, Hamiden Abd El-Wahed Khalifa, Zeeshan Nadar, Nonlinear dynamics of measles and HIV co-infection using a reaction-diffusion model and artificial neural networks, 2025, 0924-090X, 10.1007/s11071-025-11432-5
    2. Manal Alqhtani, Rahat Zarin, Aurang Zeb, Amir Khan, Akram A. Naji, Khaled M. Saad, Spatiotemporal dynamics of HIV–measles co-infection: a coupled reaction–diffusion model with cross-infection and hospitalization effects, 2025, 140, 2190-5444, 10.1140/epjp/s13360-025-06468-w
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1326) PDF downloads(40) Cited by(3)

Figures and Tables

Figures(4)  /  Tables(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog