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Research article

Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system

  • Received: 28 December 2021 Revised: 22 February 2022 Accepted: 03 March 2022 Published: 21 March 2022
  • MSC : 35Q53, 37J15, 37J35

  • The gl3(C) rational Gaudin model governed by 3×3 Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.

    Citation: Xue Geng, Liang Guan, Dianlou Du. Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system[J]. AIMS Mathematics, 2022, 7(6): 9989-10008. doi: 10.3934/math.2022557

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  • The gl3(C) rational Gaudin model governed by 3×3 Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.



    In convex functions theory, Hermite-Hadamard inequality is very important which was discovered by C. Hermite and J. Hadamard independently (see, also [1], and [2,p.137])

    F(π1+π22)1π2π1π2π1F(ϰ)dϰF(π1)+F(π2)2, (1.1)

    whereF is a convex function. In the case of concave mappings, the above inequality is satisfied in reverse order.

    Over the last twenty years, numerous studies have focused on obtaining trapezoid and midpoint type inequalities which give bounds for the right-hand side and left-hand side of the inequality (1.1), respectively. For example, the authors first obtained trapezoid and midpoint type inequalities for convex functions in [3] and in [4], respectively. In [5], Sarikaya et al. obtained the inequalities (1.1) for Riemann-Liouville fractional integrals and the authors also proved some corresponding trapezoid type inequalities for fractional integrals. Iqbal et al. presented some fractional midpoint type inequalities for convex functions in [6]. Sarikaya and Ertuğral [7] introduced the notions of generalized fractional integrals and proved some Hermite-Hadamard type inequalities for convex functions. In [8], Budak et al. used the generalized fractional integrals to prove Hermite-Hadamard type inequalities for twice differentiable convex functions. After that, the authors used generalized fractional integrals and proved the different variants of integral inequalities in [9,10,11,12,13,14].

    On the other hand, İşcan [15] defined the following class of functions called harmonically convex functions:

    If the mapping F:IR{0}R satisfies the inequality

    F(1σϰ+1σy)σF(ϰ)+(1σ)F(y),

    for all ϰ,yI and σ[0,1], then F is called harmonically convex function. In the case of harmonically concave mappings, the above inequality is satisfied in reverse order.

    It is worth noting that the harmonic feature has been important in a variety of disciplines in pure and applied sciences. The authors explore the significance of the harmonic mean in Asian stock company [16]. Harmonic methods are used in electric circuit theory, which is interesting. The overall resistance of a set of parallel resistors is just half of the entire resistors' harmonic mean. If r1 and r2 are the resistances of two parallel resistors, the total resistance may be calculated using the following formula:

    rσ=r1r2r1+r2=12H(r1,r2),

    which is the half of the harmonic mean.

    The harmonic mean, according to Noor [17], is also important in the creation of parallel algorithms for solving nonlinear problems. Several researchers have proposed iterative approaches for solving linear and nonlinear systems of equations using harmonic means and harmonically convex functions.

    Several research articles have recently been published on various generalizations of integral inequalities using various approaches. For example, İşcan established some new Hermite-Hadamard type inequalities for harmonically convex functions and trapezoid type inequalities for this class of functions in [15]. In [18], İ şcan and Wu established Hermite-Hadamard type inequalities for harmonically convex functions via Riemann-Liouville fractional integrals. They also proved some fractional trapezoid type inequalities for mapping whose derivatives in absolute value are harmonically convex. İşcan proved Ostrowski type integral inequalities for harmonically s-convex functions in [19] and in [20], Chen gave an extension of fractional Hermite-Hadamard type inequalities for harmonically convex functions. Kunt et al. [21] and Set et al. [22] used the Riemann-Liouville fractional integrals and proved Hermite-Hadamard type inequalities for harmonically convex functions. In [23], Șanlı proved several fractional midpoint type inequalities utilizing differentiable convex functions. The authors used the generalized fractional integrals and proved Hermite-Hadamard type inequalities for harmonically convex functions in [24,25]. Mohsen et al. [26] used the h- harmonically convexity to prove some new Ostrowski type inequalities and in [27], Akhtar et al. proved a new variant of Ostrowski inequalities for harmonically convex functions. In the literature there are several papers on the inequalities for harmonically convex functions. For some recent developments in integral inequalities and harmonically convexity, one can consult [28,29,30].

    Inspired by the ongoing studies, we use the generalized fractional integrals to develop some new Ostrowski type inequalities for differentiable harmonically convex functions. We also show that the newly developed inequalities are extensions of some previously known inequalities.

    The following is the structure of this paper: Section 2 provides a brief overview of the fractional calculus as well as other related studies in this field. In Section 3, we establish Ostrowski type inequalities for differentiable functions. The relationship between the findings reported here and similar findings in the literature are also taken into account. We discuss the special cases of newly established inequalities in Section 4 and obtain several new Ostrowski type inequalities. We give some applications to special means of real numbers in Section 5. Section 6 concludes with some recommendations for future research.

    In this section, we recall some basic concepts of fractional integrals and related integral inequalities.

    Definition 2.1. [7] The left and right-sided generalized fractional integrals given as follows:

    π1+IφF(ϰ)=ϰπ1φ(ϰσ)ϰσF(σ)dσ,  ϰ>π1, (2.1)
    π2IφF(ϰ)=π2ϰφ(σϰ)σϰF(σ)dσ,  ϰ<π2, (2.2)

    where the function φ:[0,)[0,) satisfies 10φ(σ)σdσ<. For the details about the genrarlized fractional integrals, one can consult [7].

    The most important feature of the generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, Katugampola fractional integrals, conformable fractional integral, Hadamard fractional integrals, etc. Few important special cases of the integral operators (2.1) and (2.2) are mentioned below.

    i) Taking φ(σ)=σ, the operators (2.1) and (2.2) reduces to the classical Riemann integrals as follows:

    Iπ+1F(ϰ)=ϰπ1F(σ)dσ,ϰ>π1,
    Iπ2F(ϰ)=π2ϰF(σ)dσ,  ϰ<π2.

    ii) Taking φ(σ)=σαΓ(α), the operators (2.1) and (2.2) reduces to the well-known Riemann–Liouville fractional integrals as follows:

    Jαπ+1F(ϰ)=1Γ(α)ϰπ1(ϰσ)α1F(σ)dσ,  ϰ>π1,
    Jαπ2F(ϰ)=1Γ(α)π2ϰ(σϰ)α1F(σ)dσ,  ϰ<π2.

    iii) Taking φ(σ)=σαkkΓk(α), the operators (2.1) and (2.2) reduces to the well-known k–Riemann–Liouville fractional integrals as follows:

    Jα,kπ1+F(ϰ)=1kΓk(α)ϰπ1(ϰσ)αk1F(σ)dσ,  ϰ>π1,
    Jα,kπ2F(ϰ)=1kΓk(α)π2ϰ(σϰ)αk1F(σ)dσ,  ϰ<π2,

    where

    Γk(α)=0σα1eσkkdσ,   R(α)>0

    and

    Γk(α)=kαk1Γ(αk),  R(α)>0;k>0.

    Recently, Zhao et al. used the generalized fractional integrals and proved the following Hermite-Hadamard type inequalities.

    Theorem 2.2. [25] For any harmonically convex mapping, the following inequalityholds:

    F(2π1π2π1+π2)12Φ(1){Jα1π1(Fg)(1π2)+Jα1π2+(Fg)(1π1)}F(π1)+F(π2)2, (2.3)

    where g(ϰ)=1ϰ and Φ(σ)=σ0φ(π2π1π1π2s)sds<+.

    Remark 2.3. It is obvious that if we set φ(σ)=σ in inequality (2.3), then we obtain the following inequality of Hermite-Hadamard type inequality (see, [15]):

    F(2π1π2π1+π2)π1π2π2π1π2π1F(ϰ)dϰF(π1)+F(π2)2.

    Remark 2.4. It is obvious that if we set φ(σ)=σαΓ(α) in inequality (2.3), then we obtain the following inequality of Hermite-Hadamard type inequality for Riemann-Liouville fractional integrals (see, [18]):

    F(2π1π2π1+π2)Γ(α+1)2(π1π2π2π1)α{Jα1π1(Fg)(1π2)+Jα1π2+(Fg)(1π1)}F(π1)+F(π2)2.

    Remark 2.5. It is obvious that if we set φ(σ)=σαkkΓk(α) in inequality (2.3), then we obtain the following inequality of Hermite-Hadamard type inequality for k-Riemann Liouville fractional integrals (see, [25]):

    F(2π1π2π1+π2)kΓk(α+k)2(π1π2π2π1)αk{Jα,k1π1(Fg)(1π2)+Jα,k1π2+(Fg)(1π1)}F(π1)+F(π2)2.

    In this section, we prove some new Ostrowski type inequalities for differentiable harmonically convex functions via the generalized fractional integrals. For brevity, we give the following special functions:

    (1) The Beta function:

    β(ϰ,y)=Γ(ϰ)Γ(y)Γ(ϰ+y)=10σϰ1(1σ)y1dσ, ϰ,y>0.

    (2) The hypergeometric function:

    2F1(π1,π2;c;z)=1β(π2,cπ2)10σπ21(1σ)cπ21(1zσ)αdσ, c>π2>0, |z|<1.

    Lemma 3.1. Let F:I=[π1,π2](0,+)R be a differentiable function on I such that FL([π1,π2]). Then, the following generalized fractional integrals identity holds for all ϰ(π1,π2):

    π1ϰ(ϰπ1)10Δ(σ)(σπ1+(1σ)ϰ)2F(π1ϰσπ1+(1σ)ϰ)dσϰπ2(π2ϰ)10Λ(σ)(σπ2+(1σ)ϰ)2F(π2ϰσπ2+(1σ)ϰ)dσ=(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)], (3.1)

    where the mappings Δ and Λ are defined as:

    Δ(σ)=σ0φ(ϰπ1π1ϰs)sds<+,

    and

    Λ(σ)=σ0φ(π2ϰπ2ϰs)sds<+.

    Proof. Consider

    π1ϰ(ϰπ1)10Δ(σ)(σπ1+(1σ)ϰ)2F(π1ϰσπ1+(1σ)ϰ)dσϰπ2(π2ϰ)10Λ(σ)(σπ2+(1σ)ϰ)2F(π2ϰσπ2+(1σ)ϰ)dσ=I1I2. (3.2)

    From fundamentals of integrations, we have

    I1=π1ϰ(ϰπ1)10Δ(σ)(σπ1+(1σ)ϰ)2F(π1ϰσπ1+(1σ)ϰ)dσ=10Δ(σ)dF(π1ϰσπ1+(1σ)ϰ)dσ=Δ(1)F(ϰ)10φ((ϰπ1)π1ϰσ)σF(π1ϰσπ1+(1σ)ϰ)dσ=Δ(1)F(ϰ)1ϰ+Iφ(Fg)(1π1).

    Similarly, we have

    I2=ϰπ2(π2ϰ)10Λ(σ)(σπ2+(1σ)ϰ)2F(π2ϰσπ2+(1σ)ϰ)dσ=Λ(1)F(ϰ)+1ϰIφ(Fg)(1π2).

    Thus, we obtain the required identity (3.1) by using the calculated values of I1 and I2 in (3.2).

    Remark 3.2. If we set φ(σ)=σ in Lemma 3.1, then we obtain the following equality:

    π1π2π2π1{(ϰπ1)210σ(σπ1+(1σ)ϰ)2F(π1ϰσπ1+(1σ)ϰ)dσ(π2ϰ)210σ(σπ2+(1σ)ϰ)2F(π2ϰσπ2+(1σ)ϰ)dσ}=F(ϰ)π1π2π2π1π2π1F(u)u2du.

    This is proved by İşcan in [19,Lemma 2.1].

    Remark 3.3. In Lemma 3.1, if we set φ(σ)=σαΓ(α), then we have the following equality for Riemann-Liouville fractional integrals:

    (ϰπ1)α+1(π1ϰ)α110σα(σπ1+(1σ)ϰ)2F(π1ϰσπ1+(1σ)ϰ)dσ(π2ϰ)α+1(ϰπ2)α110σα(σπ2+(1σ)ϰ)2F(π2ϰσπ2+(1σ)ϰ)dσ=[(ϰπ1π1ϰ)α+(π2ϰπ2ϰ)α]F(ϰ)Γ(α+1)[1ϰ+Jα(Fg)(1π1)+1ϰJα(Fg)(1π2)].

    This is proved by İşcan in [31].

    Corollary 3.4. In Lemma 3.1, if we set φ(σ)=σαkkΓ(α), then we have thefollowing new equality for k-Riemann-Liouville fractional integrals:

    (ϰπ1)α+kk(π1ϰ)αkk10σαk(σπ1+(1σ)ϰ)2F(π1ϰσπ1+(1σ)ϰ)dσ(π2ϰ)α+kk(ϰπ2)αkk10σαk(σπ2+(1σ)ϰ)2F(π2ϰσπ2+(1σ)ϰ)dσ=[(ϰπ1π1ϰ)αk+(π2ϰπ2ϰ)αk]F(ϰ)Γk(α+k)[1ϰ+Jα,k(Fg)(1π1)+1ϰJα,k(Fg)(1π2)].

    Theorem 3.5. We assume that the conditions of Lemma 3.1 are valid. If |F|q is harmonicallyconvex on [π1,π2] for some q1, then the followinginequality holds for the generalized fractional integrals:

    |(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)]|π1ϰ(ϰπ1)Θ11q1(Θ2|F(ϰ)|q+Θ3|F(π1)|q)1q+π2ϰ(π2ϰ)Θ11q4(Θ5|F(ϰ)|q+Θ6|F(π2)|q)1q,

    where

    Θ1=10Δ(σ)(σπ1+(1σ)ϰ)2dσ,Θ2=10σΔ(σ)(σπ1+(1σ)ϰ)2dσ,Θ3=10(1σ)Δ(σ)(σπ1+(1σ)ϰ)2dσ,Θ4=10Λ(σ)(σπ2+(1σ)ϰ)2dσ,Θ5=10σΛ(σ)(σπ2+(1σ)ϰ)2dσ,

    and

    Θ6=10(1σ)Λ(σ)(σπ2+(1σ)ϰ)2dσ.

    Proof. Taking absolute in Lemma 3.1 and then applying the well known power mean inequality, we have

    |(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)]|π1ϰ(ϰπ1)10Δ(σ)(σπ1+(1σ)ϰ)2|F(π1ϰσπ1+(1σ)ϰ)|dσ+π2ϰ(π2ϰ)10Λ(σ)(σπ2+(1σ)ϰ)2|F(π2ϰσπ2+(1σ)ϰ)|dσπ1ϰ(ϰπ1)(10Δ(σ)(σπ1+(1σ)ϰ)2dσ)11q×(10Δ(σ)(σπ1+(1σ)ϰ)2|F(π1ϰσπ1+(1σ)ϰ)|qdσ)1q+π2ϰ(π2ϰ)(10Λ(σ)(σπ2+(1σ)ϰ)2dσ)11q×(10Λ(σ)(σπ2+(1σ)ϰ)2|F(π2ϰσπ2+(1σ)ϰ)|qdσ)1q.

    Now from harmonically convexity of |F|q, we have

    (10Δ(σ)(σπ1+(1σ)ϰ)2dσ)11q(10Δ(σ)(σπ1+(1σ)ϰ)2|F(π1ϰσπ1+(1σ)ϰ)|qdσ)1qΘ11q1(|F(ϰ)|q10σΔ(σ)(σπ1+(1σ)ϰ)2dσ+|F(π1)|q10(1σ)Δ(σ)(σπ1+(1σ)ϰ)2dσ)1q=Θ11q1(Θ2|F(ϰ)|q+Θ3|F(π1)|q)1q.

    and

    (10Λ(σ)(σπ2+(1σ)ϰ)2dσ)11q(10Λ(σ)(σπ2+(1σ)ϰ)2|F(π2ϰσπ2+(1σ)ϰ)|qdσ)1qΘ11q4(|F(ϰ)|q10σΛ(σ)(σπ2+(1σ)ϰ)2dσ+|F(π2)|q10(1σ)Λ(σ)(σπ2+(1σ)ϰ)2dσ)1q=Θ11q4(Θ5|F(ϰ)|q+Θ6|F(π2)|q)1q.

    Thus, the proof is completed.

    Remark 3.6. In Theorem 3.5, if we assume φ(σ)=σ, then we have the following Ostrowski type inequalities:

    |F(ϰ)π1π2π2π1π2π1F(u)u2du|π1π2π2π1{χ11q1(π1,ϰ)(ϰπ1)2(χ2(π1,ϰ,1,1)|F(ϰ)|q+χ3(π1,ϰ,1,1)|F(π1)|q)1q+χ11q1(π2,ϰ)(π2ϰ)2(χ4(π2,ϰ,1,1)|F(ϰ)|q+χ5(π2,ϰ,1,1)|F(π2)|q)1q},

    where

    χ1(υ,ϰ)=1ϰν[1υlnϰlnυϰυ],χ2(π1,ϰ,υ,μ)=β(μ+2,1)ϰ2υ2F1(2υ,μ+2;μ+3;1π1ϰ),χ3(π1,ϰ,υ,μ)=β(μ+1,1)ϰ2υ2F1(2υ,μ+1;μ+3;1π1ϰ),χ4(π2,ϰ,υ,μ)=β(1,μ+2)π2υ22F1(2υ,1;μ+3;1ϰπ2),χ5(π2,ϰ,υ,μ)=β(2,μ+1)π2υ22F1(2υ,2;μ+3;1ϰπ2).

    This is proved by İşcan in [19,Theorem 2.4 for s=1].

    Corollary 3.7. In Theorem 3.5, if we set |F(ϰ)|M, ϰ[π1,π2], then we obtain the following Ostrowski typeinequality for generalized fractional integrals:

    |(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)]|M{π1ϰ(ϰπ1)Θ11q1(Θ2+Θ3)1q+π2ϰ(π2ϰ)Θ11q4(Θ5+Θ6)1q}.

    Remark 3.8. In Theorem 3.5, if we set φ(σ)=σαΓ(α), then we obtain the following Ostrowski type inequality for Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)α+(π2ϰπ2ϰ)α]F(ϰ)+Γ(α+1)[1ϰ+Jα(Fg)(1π1)+1ϰJα(Fg)(1π2)]|(ϰπ1)α+1(π1ϰ)α1Ω11q1(π1,ϰ,α)(Ω3(π1,ϰ,α)|F(ϰ)|q+Ω4(π1,ϰ,α)|F(π1)|q)1q+(π2ϰ)α+1(π2ϰ)α1Ω11q2(π2,ϰ,α)(Ω5(π2,ϰ,α)|F(ϰ)|q+Ω6(π2,ϰ,α)|F(π2)|q)1q,

    where

    Ω1(π1,ϰ,α)=ϰ22F1(2,α+1;α+2;1π1ϰ),Ω2(π2,ϰ,α)=π222F1(2,1;α+2;1ϰπ2)Ω3(π1,ϰ,α)=β(α+2,1)ϰ22F1(2,α+2;α+3;1π1ϰ),Ω4(π1,ϰ,α)=Ω1(π1,ϰ,α)Ω3(π1,ϰ,α),Ω5(π2,ϰ,α)=β(1,α+2)π222F1(2,1;α+3;1ϰπ2),Ω6(π2,ϰ,α)=Ω2(π2,ϰ,α)Ω5(π2,ϰ,α).

    This is proved by İşcan in [31].

    Corollary 3.9. In Theorem 3.5, if we set φ(σ)=σαkkΓk(α), then we obtain the followingnew Ostrowski type inequality for k-Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)αk+(π2ϰπ2ϰ)αk]F(ϰ)+Γk(α+k)[1ϰ+Jα,k(Fg)(1π1)+1ϰJα,k(Fg)(1π2)]|(ϰπ1)α+kk(π1ϰ)αkkΩ11q1(π1,ϰ,αk)(Ω3(π1,ϰ,αk)|F(ϰ)|q+Ω4(π1,ϰ,αk)|F(π1)|q)1q+(π2ϰ)α+kk(π2ϰ)αkkΩ11q2(π2,ϰ,αk)(Ω5(π2,ϰ,αk)|F(ϰ)|q+Ω6(π2,ϰ,αk)|F(π2)|q)1q,

    where

    Ω1(π1,ϰ,αk)=ϰ22F1(2,αk+1;αk+2;1π1ϰ),Ω2(π2,ϰ,αk)=π222F1(2,1;αk+2;1ϰπ2)Ω3(π1,ϰ,αk)=β(αk+2,1)ϰ22F1(2,αk+2;αk+3;1π1ϰ),Ω4(π1,ϰ,αk)=Ω1(π1,ϰ,αk)Ω3(π1,ϰ,αk),Ω5(π2,ϰ,αk)=β(1,αk+2)π222F1(2,1;αk+3;1ϰπ2),Ω6(π2,ϰ,αk)=Ω2(π2,ϰ,αk)Ω5(π2,ϰ,αk).

    Theorem 3.10. We assume that the conditions of Lemma 3.1 are valid. If |F|q is harmonicallyconvex on [π1,π2] for some q>1, then the followinginequality holds for the generalized fractional integrals:

    |(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)]|π1ϰ(ϰπ1)Θ1p7(|F(ϰ)|q+|F(π1)|q2)1q+π2ϰ(π2ϰ)Θ1p8(|F(ϰ)|q+|F(π2)|q2)1q,

    where 1p+1q=1 and

    Θ7=10(Δ(σ)(σπ1+(1σ)ϰ)2)pdσ,Θ8=10(Λ(σ)(σπ2+(1σ)ϰ)2)pdσ.

    Proof. From Lemma 3.1 and applying well-known Hölder's inequality, we have

    |(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)]|π1ϰ(ϰπ1)10Δ(σ)(σπ1+(1σ)ϰ)2|F(π1ϰσπ1+(1σ)ϰ)|dσ+π2ϰ(π2ϰ)10Λ(σ)(σπ2+(1σ)ϰ)2|F(π2ϰσπ2+(1σ)ϰ)|dσπ1ϰ(ϰπ1)(10(Δ(σ)(σπ1+(1σ)ϰ)2)pdσ)1p(10|F(π1ϰσπ1+(1σ)ϰ)|qdσ)1q+π2ϰ(π2ϰ)(10(Λ(σ)(σπ2+(1σ)ϰ)2)pdσ)1p(10|F(π2ϰσπ2+(1σ)ϰ)|qdσ)1q.

    Now from harmonically convexity of |F|q, we have

    (10(Δ(σ)(σπ1+(1σ)ϰ)2)pdσ)1p(10|F(π1ϰσπ1+(1σ)ϰ)|qdσ)1qΘ1p7(|F(ϰ)|q10σdσ+|F(π1)|q10(1σ)dσ)1q=Θ1p7(|F(ϰ)|q+|F(π1)|q2)1q.

    and

    (10(Λ(σ)(σπ2+(1σ)ϰ)2)pdσ)1p(10|F(π2ϰσπ2+(1σ)ϰ)|qdσ)1qΘ1p8(|F(ϰ)|q10σdσ+|F(π2)|q10(1σ)dσ)1q=Θ1p8(|F(ϰ)|q+|F(π2)|q2)1q.

    Thus, the proof is completed.

    Remark 3.11. In Theorem 3.10, if we set φ(σ)=σ, then we obtain the following Ostrowski type inequality:

    |F(ϰ)π1π2π2π1π2π1F(u)u2du|π1π2π2π1{(χ2(π1,ϰ,υ,μ))1p(ϰπ1)2(|F(ϰ)|q+|F(π1)|q2)1q+(χ4(π2,ϰ,υ,μ))1p(π2ϰ)2(|F(ϰ)|q+|F(π2)|q2)1q},

    where

    χ2(π1,ϰ,υ,μ)=β(μ+1,1)ϰ2υ2F1(2υ,μ+1;μ+2;1π1ϰ),χ4(π2,ϰ,υ,μ)=β(1,μ+1)π2υ22F1(2υ,1;μ+2;1ϰπ2).

    This is proved by İşcan in [19,Theorem 2.6 for s=1].

    Corollary 3.12. In Theorem 3.5, if we set |F(ϰ)|M, ϰ[π1,π2], then we obtain the following Ostrowski typeinequality for generalized fractional integrals:

    |(Δ(1)+Λ(1))F(ϰ)[1ϰ+Iφ(Fg)(1π1)+1ϰIφ(Fg)(1π2)]|M{π1ϰ(ϰπ1)Θ1p7+π2ϰ(π2ϰ)Θ1p8}.

    Remark 3.13. In Theorem 3.10, if we set φ(σ)=σαΓ(α), then we obtain the following Ostrowski type inequalities for Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)α+(π2ϰπ2ϰ)α]F(ϰ)+Γ(α+1)[1ϰ+Jα(Fg)(1π1)+1ϰJα(Fg)(1π2)]|(ϰπ1)α+1(π1ϰ)α1Ω1p7(π1,ϰ,α,p)(|F(ϰ)|q+|F(π1)|q2)1q+(π2ϰ)α+1(π2ϰ)α1Ω1p7(π2,ϰ,α,p)(|F(ϰ)|q+|F(π2)|q2)1q,

    where

    Ω7(υ,ϰ,α,p)=ϰ2αp+12F1(2p,αp+1;αp+2;1υϰ).

    This is proved by İşcan in [31].

    Corollary 3.14. In Theorem 3.5, if we set φ(σ)=σαkkΓk(α), then we obtain the followingnew Ostrowski type inequality for k-Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)αk+(π2ϰπ2ϰ)αk]F(ϰ)+Γk(α+k)[1ϰ+Jα,k(Fg)(1π1)+1ϰJα,k(Fg)(1π2)]|(ϰπ1)α+kk(π1ϰ)αkkΩ1p7(π1,ϰ,αk,p)(|F(ϰ)|q+|F(π1)|q2)1q+(π2ϰ)α+kk(π2ϰ)αkkΩ1p7(π2,ϰ,αk,p)(|F(ϰ)|q+|F(π2)|q2)1q,

    where

    Ω7(υ,ϰ,αk,p)=kϰ2αp+k2F1(2p,αp+kk;αp+2kk;1υϰ).

    In this section, we discuss more special cases of the results proved in the last section.

    Remark 4.1. In Corollary 3.7, if we set φ(σ)=σ, then we obtain the following Ostrowski type inequality:

    |F(ϰ)π1π2π2π1π2π1F(u)u2du|Mπ1π2π2π1{χ11q1(π1,ϰ)(ϰπ1)2(χ2(π1,ϰ,1,1)+χ3(π1,ϰ,1,1))1q+χ11q1(π2,ϰ)(π2ϰ)2(χ4(π2,ϰ,1,1)+χ5(π2,ϰ,1,1))1q}. (4.1)

    This is proved by İşcan in [19,Corollary 2.3 for s=1].

    Remark 4.2. In Corollary 3.7, if we set φ(σ)=σαΓ(α), then we obtain the following Ostrowski type inequality for Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)α+(π2ϰπ2ϰ)α]F(ϰ)+Γ(α+1)[1ϰ+Jα(Fg)(1π1)+1ϰJα(Fg)(1π2)]|M[(ϰπ1)α+1(π1ϰ)α1Ω11q1(π1,ϰ,α)(Ω3(π1,ϰ,α)+Ω4(π1,ϰ,α))1q+(π2ϰ)α+1(π2ϰ)α1Ω11q2(π2,ϰ,α)(Ω5(π2,ϰ,α)+Ω6(π2,ϰ,α))1q].

    This is proved by İşcan in [31].

    Remark 4.3. In Corollary 3.7, if we set φ(σ)=σαkkΓk(α), then we obtain the following new Ostrowski type inequality for k-Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)αk+(π2ϰπ2ϰ)αk]F(ϰ)+Γk(α+k)[1ϰ+Jα,k(Fg)(1π1)+1ϰJα,k(Fg)(1π2)]|M[(ϰπ1)α+kk(π1ϰ)αkkΩ11q1(π1,ϰ,αk)(Ω3(π1,ϰ,αk)+Ω4(π1,ϰ,αk))1q+(π2ϰ)α+kk(π2ϰ)αkkΩ11q2(π2,ϰ,αk)(Ω5(π2,ϰ,αk)+Ω6(π2,ϰ,αk))1q].

    Remark 4.4. In Corollary 3.12, if we set φ(σ)=σ, then we obtain the following Ostrowski type inequality:

    |F(ϰ)π1π2π2π1π2π1F(u)u2du|Mπ1π2π2π1{(χ2(π1,ϰ,υ,μ))1p(ϰπ1)2+(χ4(π2,ϰ,υ,μ))1p(π2ϰ)2}. (4.2)

    This is proved by İşcan in [19,Corollary 2.5 for s=1].

    Remark 4.5. In Corollary 3.12, if we set φ(σ)=σαΓ(α), then we obtain the following Ostrowski type inequality for Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)α+(π2ϰπ2ϰ)α]F(ϰ)+Γ(α+1)[1ϰ+Jα(Fg)(1π1)+1ϰJα(Fg)(1π2)]|M[(ϰπ1)α+1(π1ϰ)α1Ω1p7(π1,ϰ,α,p)+(π2ϰ)α+1(π2ϰ)α1Ω1p7(π2,ϰ,α,p)].

    This is proved by İşcan in [31].

    Remark 4.6. In Corollary 3.12, if we set φ(σ)=σαkkΓk(α), then we obtain the following new Ostrowski type inequality for k-Riemann-Liouville fractional integrals:

    |[(ϰπ1π1ϰ)αk+(π2ϰπ2ϰ)αk]F(ϰ)+Γk(α+k)[1ϰ+Jα,k(Fg)(1π1)+1ϰJα,k(Fg)(1π2)]|M[(ϰπ1)α+kk(π1ϰ)αkkΩ1p7(π1,ϰ,αk,p)+(π2ϰ)α+kk(π2ϰ)αkkΩ1p7(π2,ϰ,αk,p)].

    Remark 4.7. If we set q=1 in Theorem 3.5 and Corollaries 3.7–3.14, then we obtain some new Ostrowski type inequlities for the harmonically convexity of |F|. Moreover, for different choices of φ in the generalized fractional integrals, one can obtain several Ostrowski type inequalities via Katugampola fractional integrals, conformable fractional integral, Hadamard fractional integrals, etc.

    For arbitrary positive numbers π1,π2(π1π2), we consider the means as follows:

    (1) The arithmatic mean

    A=A(π1,π2)=π1+π22.

    (2) The geometric mean

    G=G(π1,π2)=π1π2.

    (3) The harmonic means

    H=H(π1,π2)=2π1π2π1+π2.

    (4) The logarithmic mean

    L=L(π1,π2)=π2π1lnπ2lnπ1.

    (5) The generalize logarithmic mean

    Lp=Lp(π1,π2)=[π2π1(π2π1)(p+1)]1p,pR{1,0}.

    (6) The identric mean

    I=I(π1,π2)={1e(π2π1)1π2π1, if π1π2,π1,              if π1=π2,π1,π2>0.

    These means are often employed in numerical approximations and other fields. However, the following straightforward relationship has been stated in the literature.

    HGLIA.

    Proposition 5.1. For π1,π2(0,) with π1<π2, then the following inequality holds:

    |A(π1,π2)G2(π1,π2)L(π1,π2)|MG2(π1,π2)(π2π1)4×{χ11q1(π1,A(π1,π2))(χ2(π1,A(π1,π2),1,1)+χ3(π1,A(π1,π2),1,1))1q+χ11q1(π2,A(π1,π2))(χ4(π2,A(π1,π2),1,1)+χ5(π2,A(π1,π2),1,1))1q}.

    Proof. The inequality (4.1) with ϰ=π1+π22 for mapping F:(0,)R, F(ϰ)=ϰ leads to this conclusion.

    Proposition 5.2. For π1,π2(0,) with π1<π2, then the following inequality holds:

    |H(π1,π2)G2(π1,π2)L(π1,π2)|MG2(π1,π2)π2π1×{χ11q1(π1,H(π1,π2))(G2(π1,π2)π212A(π1,π2))2(χ2(π1,H(π1,π2),1,1)+χ3(π1,H(π1,π2),1,1))1q+χ11q1(π2,H(π1,π2))(π22G2(π1,π2)2A(π1,π2))2(χ4(π2,H(π1,π2),1,1)+χ5(π2,H(π1,π2),1,1))1q}.

    Proof. The inequality (4.1) with ϰ=2π1π2π1+π2 for mapping F:(0,)R, F(ϰ)=ϰ leads to this conclusion.

    Proposition 5.3. For π1,π2(0,) with π1<π2, then the following inequality holds:

    |Ap+2(π1,π2)G2(π1,π2)Lpp(π1,π2)|MG2(π1,π2)(π2π1)4×{χ11q1(π1,A(π1,π2))(χ2(π1,A(π1,π2),1,1)+χ3(π1,A(π1,π2),1,1))1q+χ11q1(π2,A(π1,π2))(χ4(π2,A(π1,π2),1,1)+χ5(π2,A(π1,π2),1,1))1q}.

    Proof. The inequality (4.1) with ϰ=π1+π22 for mapping F:(0,)R, F(ϰ)=ϰp+2,p(1,){0} leads to this conclusion.

    Proposition 5.4. For π1,π2(0,) with π1<π2, then the following inequality holds:

    |Hp+2(π1,π2)G2(π1,π2)Lpp(π1,π2)|MG2(π1,π2)π2π1×{χ11q1(π1,H(π1,π2))(G2(π1,π2)π212A(π1,π2))2(χ2(π1,H(π1,π2),1,1)+χ3(π1,H(π1,π2),1,1))1q+χ11q1(π2,H(π1,π2))(π22G2(π1,π2)2A(π1,π2))2(χ4(π2,H(π1,π2),1,1)+χ5(π2,H(π1,π2),1,1))1q}.

    Proof. The inequality (4.1) with ϰ=2π1π2π1+π2 for mapping F:(0,)R, F(ϰ)=ϰp+2, p(1,){0} leads to this conclusion.

    Proposition 5.5. For π1,π2(0,) with π1<π2, then the following inequality holds:

    |A2(π1,π2)ln(A(π1,π2))G2(π1,π2)ln(I(π1,π2))|MG2(π1,π2)(π2π1)4×{χ11q1(π1,A(π1,π2))(χ2(π1,A(π1,π2),1,1)+χ3(π1,A(π1,π2),1,1))1q+χ11q1(π2,A(π1,π2))(χ4(π2,A(π1,π2),1,1)+χ5(π2,A(π1,π2),1,1))1q}.

    Proof. The inequality (4.1) with ϰ=π1+π22 for mapping F:(0,)R, F(ϰ)=ϰ2lnϰ, leads to this conclusion.

    Proposition 5.6. For π1,π2(0,) with π1<π2, then the following inequality holds:

    |H2(π1,π2)ln(H(π1,π2))G2(π1,π2)ln(I(π1,π2))|MG2(π1,π2)π2π1×{χ11q1(π1,H(π1,π2))(G2(π1,π2)π212A(π1,π2))2(χ2(π1,H(π1,π2),1,1)+χ3(π1,H(π1,π2),1,1))1q+χ11q1(π2,H(π1,π2))(π22G2(π1,π2)2A(π1,π2))2(χ4(π2,H(π1,π2),1,1)+χ5(π2,H(π1,π2),1,1))1q}.

    Proof. The inequality (4.1) with ϰ=2π1π2π1+π2 for mapping F:(0,)R, F(ϰ)=ϰ2lnϰ leads to this conclusion.

    In this paper, we have proved several new Ostrowski type inequalities for differentiable harmonically convex functions via the generalized fractional integrals. Moreover, we have proved that the established inequalities are the extensions of some existing inequalities in the literature. It is an interesting and new problem that the upcoming researchers can offer similar inequalities for different type of harmonically and co-ordinated harmonically convexity.

    This research was funded by King Mongkut's University of Technology North Bangkok. Contract No. KMUTNB-63-KNOW-22.

    The authors declare no conflict of interest.



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