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The Faber polynomial expansion method and the Taylor-Maclaurin coefficient estimates of Bi-Close-to-Convex functions connected with the q-convolution

  • In this paper, we introduce a new class of analytic and bi-close-to-convex functions connected with q-convolution, which are defined in the open unit disk. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this subclass by using the Faber polynomial expansion method. Several corollaries and consequences of our main results are also briefly indicated.

    Citation: H. M. Srivastava, Sheza M. El-Deeb. The Faber polynomial expansion method and the Taylor-Maclaurin coefficient estimates of Bi-Close-to-Convex functions connected with the q-convolution[J]. AIMS Mathematics, 2020, 5(6): 7087-7106. doi: 10.3934/math.2020454

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  • In this paper, we introduce a new class of analytic and bi-close-to-convex functions connected with q-convolution, which are defined in the open unit disk. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this subclass by using the Faber polynomial expansion method. Several corollaries and consequences of our main results are also briefly indicated.


    In convex function theory, the classical Hermite-Hadamard inequality is one of the most well-known inequalities with geometrical interpretation, and it has a wide range of applications, see [1,2].

    Let S:KR+ be a convex function on a convex set K and ρ,ςK with ρς. Then,

    S(ρ+ς2)1ςρςρS(ϖ)dϖS(ρ)+S(ς)2. (1)

    In [3], Fejér looked at the key extensions of HH-inequality which is known as Hermite-Hadamard-Fejér inequality (HH-Fejér inequality).

    Let S:KR+ be a convex function on a convex set K and ρ,ς K with ρς. Then,

    S(ρ+ς2)1ςρD(ϖ)dϖςρS(ϖ)D(ϖ)dϖS(ρ)+S(ς)2ςρD(ϖ)dϖ. (2)

    If D(ϖ)=1, then we obtain (1) from (2). We should remark that Hermite-Hadamard inequality is a refinement of the idea of convexity, and it can be simply deduced from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and there have been a lot of improvements and generalizations examined. Sarikaya [4] proved the Hadamard type inequality for coordinated convex functions such that

    Let G:ΔR+ be a coordinate convex function on Δ=[ς,ρ]×[μ,ν]. If G is double fractional integrable, then following inequalities hold:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]+Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (3)

    If α=1, then we obtain the following Dragomir inequality [5] on coordinates:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (4)

    For more details related to inequalities, see [6,7,8,9] and reference therein.

    Interval analysis, on the other hand, is a well-known example of set-valued analysis, which is the study of sets in the context of mathematical analysis and general topology. It was created as a way of dealing with the interval uncertainty that can be found in many mathematical or computer models of deterministic real-world phenomena. Archimede's method, which is used to calculate the circumference of a circle, is an old example of an interval enclosure. Moore [10], who is credited with being the first user of intervals in computational mathematics, published the first book on interval analysis in 1966. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a helpful technique in a variety of fields that are interested in ambiguous data because of its applicability. Computer graphics, experimental and computational physics, error analysis, robotics, and many more fields have applications.

    Furthermore, in recent years, numerous major inequalities (Hermite-Hadamard, Ostrowski and others) have been addressed for interval-valued functions. Chalco-Cano et al. used the Hukuhara derivative for interval-valued functions to construct Ostrowski type inequalities for interval-valued functions in [11,12,13,14]. For interval-valued functions, Román-Flores et al. developed Minkowski and Beckenbach's inequality in [15]. For fuzzy interval-valued function, Khan et al. [16,17,18] derived some new versions of Hermite-Hadamard type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. [19,20] discussed some novel types of Hermite-Hadamard type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities. Recently, Khan et al. [21] introduced the new class of convexity in fuzzy-interval calculus which is known as coordinated convex fuzzy-interval-valued functions and with the support of these classes, some Hermite-Hadamard type inequalities are obtained via newly defined fuzzy-interval double integrals. We encourage readers to [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54] for other related results.

    The following is an overview of the paper's structure. Section 2 recalls some preliminary notions and definitions. Moreover, some properties of introduced coordinated LR-convex IVF are also discussed. Section 3 presents some Hermite-Hadamard type inequalities for coordinated LR-convex IVF. With the help of this class, some fractional integral inequalities are also derived for the coordinated LR-convex IVF and for the product of two coordinated LR-convex IVFs. The fourth section, Conclusions and Future Work, brings us to a close.

    Let R be the set of real numbers and RI be the space of all closed and bounded intervals of R, such that URI is defined by

    U=[U,U]={yR|UyU},(U,UR). (5)

    If U=U, then U is said to be degenerate. If U0, then [U,U] is called positive interval. The set of all positive interval is denoted by R+I and defined as R+I={[U,U]:[U,U]RIandU0}.

    Let ϱR and ϱU be defined by

    ϱ.U={[ϱU,ϱU]ifϱ>0,{0}ifϱ=0,[ϱU,ϱU]ifϱ<0. (6)

    Then, the Minkowski difference DU, addition U+D and U×D for U,DRI are defined by

    [D,D][U,U]=[DU,DU],[D,D]+[U,U]=[D+U,D+U], (7)

    and

    [D,D]×[U,U]=[min{DU,DU,DU,DU},max{DU,DU,DU,DU}].

    The inclusion "⊇" means that

    UD if and only if, [U,U][D,D], and if and only if

    UD,DU. (8)

    Remark 1. [36] (ⅰ) The relation "≤p" is defined on RI by

    [D,D]p[U,U]ifandonlyifDU,DU, (9)

    for all [D,D],[U,U]RI, and it is a pseudo order relation. The relation [D,D]p[U,U] coincident to [D,D][U,U] on RI when it is "≤p"

    (ⅱ) It can be easily seen that "p" looks like "left and right" on the real line R, so we call "p" is "left and right" (or "LR" order, in short).

    For [D,D],[U,U]RI, the Hausdorff-Pompeiu distance between intervals [D,D] and [U,U] is defined by

    d([D,D],[U,U])=max{|DU|,|DU|}. (10)

    It is familiar fact that (RI,d) is a complete metric space.

    Theorem 1. [10] If G:[μ,ν]RRI is an I-V-F given by (x) [G(x),G(x)], then G is Riemann integrable over [μ,ν] if and only if, G and G both are Riemann integrable over [μ,ν] such that

    (IR)νμG(x)dx=[(R)νμG(x)dx,(R)νμG(x)dx]. (11)

    The collection of all Riemann integrable real valued functions and Riemann integrable I-V-F is denoted by R[μ,ν] and TR[μ,ν], respectively.

    Definition 1. [31,33] Let G:[μ,ν]RI be interval-valued function and GTR[μ,ν]. Then interval Riemann-Liouville-type integrals of G are defined as

    Iαμ+G(y)=1Γ(α)yμ(yt)α1G(t)dt(y>μ), (12)
    IανG(y)=1Γ(α)νy(ty)α1G(t)dt(y<ν), (13)

    where α>0 and Γ is the gamma function.

    Theorem 2. [20] Let G:[ς,ρ]RI+ be a LR-convex I-V.F such that G(y)=[G(y),G(y)] for all y[ς,ρ]. If GL([ς,ρ],R+I), then

    G(ς+ρ2)pΓ(α+1)2(ρς)α[Iας+G(ρ)+IαρG(ς)]pG(ς)+G(ρ)2. (14)

    Theorem 3. [20] Let G,S:[ς,ρ]R+I be two LR-convex I-V.Fs such that G(x)=[G(x),G(x)] and S(x)=[S(x),S(x)] for all x[ς,ρ]. If G×SL([ς,ρ],R+I) is fuzzy Riemann integrable, then

    Γ(α+1)2(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    p(12α(α+1)(α+2))M(ς,ρ)+(α(α+1)(α+2))N(ς,ρ), (15)

    and

    G(ς+ρ2)×S(ς+ρ2)
    pΓ(α+1)4(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    +12(12α(α+1)(α+2))M(ς,ρ)+12(α(α+1)(α+2))N(ς,ρ), (16)

    where M(ς,ρ)=G(ς)×S(ς)+G(ρ)×S(ρ), N(ς,ρ)=G(ς)×S(ρ)+G(ρ)×S(ς),

    and M(ς,ρ)=[M(ς,ρ),M(ς,ρ)] and N(ς,ρ)=[N(ς,ρ),N(ς,ρ)].

    Note that, the Theorem 1 is also true for interval double integrals. The collection of all double integrable I-V-F is denoted TOΔ, respectively.

    Theorem 4. [35] Let Δ=[ς,ρ]×[μ,ν]. If G:ΔRI is interval-valued doubl integrable (ID-integrable) on Δ. Then, we have

    (ID)ρςνμG(x,y)dydx=(IR)ρς(IR)νμG(x,y)dydx.

    Definition 2. [36] Let G:ΔR+I and GTOΔ. The interval Riemann-Liouville-type integrals Iα,βμ+,ς+,Iα,βμ+,ρ, Iα,βν,ς+,Iα,βν,ρ of G order α,β>0 are defined by

    Iα,βμ+,ς+G(x,y)=1Γ(α)Γ(β)xμyς(xt)α1(ys)β1G(t,s)dsdt(x>μ,y>ς), (17)
    Iα,βμ+,ρG(x,y)=1Γ(α)Γ(β)xμρy(xt)α1(sy)β1G(t,s)dsdt(x>μ,y<ρ), (18)
    Iα,βν,ς+G(x,y)=1Γ(α)Γ(β)νxyς(tx)α1(ys)β1G(t,s)dsdt(x<ν,y>ς), (19)
    Iα,βν,ρG(x,y)=1Γ(α)Γ(β)νxρy(tx)α1(sy)β1G(t,s)dsdt(x<ν,y<ρ). (20)

    Definition 3. [38] The I-V.F G:ΔR+I is said to be coordinated LR-convex I-V.F on Δ if

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ), (21)

    for all (μ,ν),(ς,ρ)Δ, and τ,s[0,1]. If inequality (21) is reversed, then G is called coordinate LR-concave I-V.F on Δ.

    Lemma 1. [38] Let G:ΔR+I be an coordinated I-V.F on Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if there exist two coordinated LR-convex I-V.Fs Gx:[ς,ρ]R+I, Gx(w)=G(x,w) and Gy:[μ,ν]R+I, Gy(z)=G(z,y).

    Theorem 5. [38] Let G:ΔR+I be a I-V.F on Δ such that

    G(x,ϖ)=[G(x,ϖ),G(x,ϖ)], (22)

    for all (x,ϖ)Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if, G(x,ϖ) and G(x,ϖ) are coordinated convex functions.

    Example 1. We consider the I-V.Fs G:[0,1]×[0,1]R+I defined by,

    G(x)(σ)={σ2(6+ex)(6+eϖ),σ[0,2(6+ex)(6+eϖ)]4(6+ex)(6+eϖ)σ2(6+ex)(6+eϖ),σ(2(6+ex)(6+eϖ),4(6+ex)(6+eϖ)]0,otherwise,

    Then, for each θ[0,1], we have G(x)=[2θ(6+ex)(6+eϖ),(4+2θ)(6+ex)(6+eϖ)]. Since end point functions G((x,ϖ),θ), G((x,ϖ),θ) are coordinate concave functions for each θ[0,1]. Hence S(x,ϖ) is coordinate LR-concave I-V.F.

    From Lemma 1, we can easily note that each LR-convex I-V.F is coordinated LR-convex I-V.F. But the converse is not true.

    Remark 2. If one takes G(x,ϖ)=G(x,ϖ), then G is known as coordinated function if G satisfies the coming inequality

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid which is defined by Dragomir [5]

    Let one takes G(x,ϖ)G(x,ϖ), where G(x,ϖ) is affine function and G(x,ϖ) is a concave function. If coming inequality,

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid, then G is named as coordinated IVF which is defined by Zhao et al. [37, Definition 2 and Example 2]

    In this section, we shall continue with the following fractional HH-inequality for coordinated LR-convex I-V.Fs, and we also give fractional HH-Fejér inequality for coordinated LR-convex I-V.F through fuzzy order relation.

    Theorem 6. Let G:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] for all (x,y)Δ. If GTOΔ, then following inequalities holds:

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (23)

    If G(x) coordinated LR-concave I-V.F, then

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (24)

    Proof. Let G:[μ,ν]R+I be a coordinated LR-convex I-V.F. Then, by hypothesis, we have

    4G(μ+ν2,ς+ρ2)pG(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Theorem 5, we have

    4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ),4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Lemma 1, we have

    2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ),2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (25)

    and

    2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y),2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y). (26)

    From (25) and (26), we have

    2[G(x,ς+ρ2),G(x,ς+ρ2)]
    p[G(x,τς+(1τ)ρ),G(x,τς+(1τ)ρ)]
    +[G(x,(1τ)ς+τρ),G(x,(1τ)ς+τρ)],

    and

    2[G(μ+ν2,y),G(μ+ν2,y)]
    p[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)]
    +[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)],

    It follows that

    G(x,ς+ρ2)pG(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (27)

    and

    G(μ+ν2,y)pG(τμ+(1τ)ν,y)+G(τμ+(1τ)ν,y). (28)

    Since G(x,.) and G(.,y), both are coordinated LR-convex-IVFs, then from inequality (14), inequalities (27) and (28) we have

    Gx(ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+Gx(ρ)+IβρGx(ς)]pGx(ς)+Gx(ρ)2. (29)

    and

    Gy(μ+ν2)pΓ(α+1)2(νμ)α[Iαμ+Gy(ν)+IανGy(μ)]pGy(μ)+Gy(ν)2 (30)

    Since Gx(w)=G(x,w), the inequality (29) can be written as

    G(x,ς+ρ2)pΓ(β+1)2(ρς)β[Iας+G(x,ρ)+IαρG(x,ς)]pG(x,ς)+G(x,ρ)2. (31)

    That is

    G(x,ς+ρ2)pβ2(ρς)β[ρς(ρs)β1G(x,s)ds+ρς(sς)β1G(x,s)ds]pG(x,ς)+G(x,ρ)2.

    Multiplying double inequality (31) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pνμρς(νx)α1(ρs)β1G(x,s)dsdx+νμρς(νx)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(νx)α1G(x,ς)dx+νμ(νx)α1G(x,ρ)dx]. (32)

    Again multiplying double inequality (31) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pαβ4(νμ)α(ρς)βνμρς(xμ)α1(ρs)β1G(x,s)dsdx
    +αβ4(νμ)α(ρς)βνμρς(xμ)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(xμ)α1G(x,ς)dx+νμ(xμ)α1G(x,d)dx]. (33)

    From (32), we have

    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(ν,ς)]
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)]. (34)

    From (33), we have

    Γ(α+1)2(νμ)α[IανG(μ,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)4(νμ)α[IανG(μ,ς)+IανG(μ,ρ)]. (35)

    Similarly, since Gy(z)=G(z,y) then, from (34) and (35), (30) we have

    Γ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(μ,ρ)]
    pΓ(β+1)4(ρς)β[Iβς+G(μ,ρ)+Iβς+G(ν,ρ)], (36)

    and

    Γ(β+1)2(ρς)α[IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)+Iα,βν,ρG(μ,ς)]
    pΓ(β+1)4(ρς)β[IβρG(μ,ς)+IβρG(ν,ς)]. (37)

    After adding the inequalities (46), (35), (36) and (37), we will obtain as resultant second, third and fourth inequalities of (23).

    Now, from left part of inequality (14), we have

    G(μ+ν2,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (38)

    and

    G(μ+ν2,ς+ρ2)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]. (39)

    Summing the inequalities (38) and (39), we obtain the following inequality:

    G(μ+ν2,ς+ρ2)
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (40)

    this is the first inequality of (23).

    Now, from right part of inequality (14), we have

    Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)]pG(μ,ς)+G(μ,ρ)2, (41)
    Γ(β+1)2(ρς)β[Iβς+G(ν,ρ)+IβρG(ν,ς)]pG(ν,ς)+G(ν,ρ)2, (42)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)]pG(μ,ς)+G(ν,ς)2, (43)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ρ)+IανG(μ,ρ)]pG(μ,ρ)+G(ν,ρ)2. (44)

    Summing inequalities (41), (42), (43) and (44), and then taking multiplication of the resultant with 14, we have

    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)+Iαμ+G(ν,ρ)+IανG(μ,ρ)]
    +Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)+Iβς+G(ν,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(μ,ρ)+G(ν,ς)+G(ν,ρ)4. (45)

    This is last inequality of (23) and the result has been proven.

    Remark 3. If one to take α=1 and β=1, then from (23), we achieve the coming inequality, see [38]:

    G(μ+ν2,ς+ρ2)
    p12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]p1(νμ)(ρς)νμρςG(x,y)dydxp14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (46)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then from Remark 2 and (24), we acquire the coming inequality, see [31]:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (47)

    Let one takes α=1 and β=1, G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then Remark 2 and from (24), we acquire the coming inequality, see [37]:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx
    14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (48)

    Example 2. We consider the I-V-Fs G:[0,1]×[0,1]R+I defined by,

    G(x)=[2,6](6+ex)(6+ey).

    Since end point functions G(x,y), G(x,y) are convex functions on coordinate, then G(x,y) is convex I-V-F on coordinate. Then for α=1 and β=1, we have

    G(μ+ν2,ς+ρ2)=[2(5+e12)2,6(6+e12)2],
    Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    =[4(6+e12)(5+e),12(6+e12)(5+e)],
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    =[2(5+e)2,6(5+e)2],
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    =[(5+e)(13+e),3(5+e)(13+e)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4=[(6+e)(20+e)+492,6((6+e)(20+e)+49)2].

    That is

    [2(5+e12)2,6(6+e12)2]p[4(6+e12)(5+e),12(6+e12)(5+e)]
    p[2(5+e)2,6(5+e)2]
    p[(5+e)(13+e),3(5+e)(13+e)]
    p[(6+e)(20+e)+492,3((6+e)(20+e)+49)].

    Hence, Theorem 3.1 has been verified

    Next both results obtain Hermite-Hadamard type inequalities for the product of two coordinate LR-convex I-V.Fs

    Theorem 7. Let G,S:ΔR+I be a coordinate LR-convex I-V.Fs on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (49)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (50)

    Where

    K(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς)+G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ),
    L(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ς)˜+G(ν,ρ)×S(μ,ρ)+G(ν,ς)×S(μ,ς)+G(μ,ρ)×S(ν,ρ),
    M(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ)+G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς),
    N(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)+G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς).

    and K(μ,ν,ς,ρ), ˜L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are defined as follows:

    K(μ,ν,ς,ρ)=[K(μ,ν,ς,ρ),K(μ,ν,ς,ρ)],
    L(μ,ν,ς,ρ)=[L(μ,ν,ς,ρ),L(μ,ν,ς,ρ)],
    M(μ,ν,ς,ρ)=[M(μ,ν,ς,ρ),M(μ,ν,ς,ρ)],
    N(μ,ν,ς,ρ)=[N(μ,ν,ς,ρ),N(μ,ν,ς,ρ)].

    Proof. Let G and S both are coordinated LR-convex I-V.Fs on [μ,ν]×[ς,ρ]. Then

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    and

    S(τμ+(1τ)ν,sς+(1s)ρ)
    pτsS(μ,ς)+τ(1s)S(μ,ρ)+(1τ)sS(ν,ς)+(1τ)(1s)S(ν,ρ).

    Since G and S both are coordinated LR-convex I-V.Fs, then by Lemma 1, there exist

    Gx:[ς,ρ]R+I,Gx(y)=G(x,y),Sx:[ς,ρ]R+I,Sx(y)=S(x,y),

    Since Gx, and Sx are I-V.Fs, then by inequality (15), we have

    Γ(β+1)2(ρς)β[Iβς+Gx(ρ)×Sx(ρ)+IβρGx(ς)×Sx(ς)]
    p(12β(β+1)(β+2))(Gx(ς)×Sx(ς)+Gx(ρ)×Sx(ρ))
    +(β(β+1)(β+2))(Gx(ς)×Sx(ρ)+Gx(ρ)×Sx(ς)).

    That is

    β2(ρς)β[ρς(ρy)β1G(x,y)×S(x,y)ρy+ρς(yς)β1G(x,y)×S(x,y)ρy]
    p(12β(β+1)(β+2))(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))
    +(β(β+1)(β+2))(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς)). (51)

    Multiplying double inequality (51) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we get

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (52)

    Again, multiplying double inequality (51) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we gain

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (53)

    Summing (52) and (53), we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    +Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)). (54)

    Now, again with the help of integral inequality (15) for first two integrals on the right-hand side of (54), we have the following relation

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)). (55)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)). (56)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)). (57)

    And

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)). (58)

    From (55)–(58), inequality (54) we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ).

    Hence, the result has been proven.

    Remark 4. If one to take α=1 and β=1, then from (49), we achieve the coming inequality, see [38]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    p19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (59)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [36]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (60)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [37]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (61)

    If G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (49), we acquire the coming inequality, see [39]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (62)

    Theorem 8. Let G,S:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)
    +[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)
    +[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (63)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (64)

    Where K(μ,ν,ς,ρ), L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are given in Theorem 7.

    Proof. Since G,S:ΔR+I be two LR-convex I-V.Fs, then from inequality (16), we have

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+(α(α+1)(α+2))(G(μ,ς+ρ2)×S(μ,ς+ρ2)+G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(G(μ,ς+ρ2)×S(ν,ς+ρ2)+G(ν,ς+ρ2)×S(μ,ς+ρ2)), (65)

    and

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pβ2(ρς)β[ρς(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ς)+G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ρ)+G(μ+ν2,ρ)×S(μ+ν2,ς)), (66)

    Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(α(α+1)(α+2))(2G(μ,ς+ρ2)×S(μ,ς+ρ2)+2G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(2G(μ,ς+ρ2)×S(ν,ς+ρ2)+2G(ν,ς+ρ2)×S(μ,ς+ρ2))+(β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ς)+2G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ρ)+2G(μ+ν2,ρ)×S(μ+ν2,ς)). (67)

    Again, with the help of integral inequality (16) and Lemma 1 for each integral on the right-hand side of (67), we have

    α2(νμ)ανμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(νx)α1(ρy)β1G(x,y)dydx+νμρς(νx)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (68)
    α2(νμ)ανμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(xμ)α1(ρy)β1G(x,y)dydx+νμρς(xμ)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (69)
    β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(ν,ρ)). (70)
    β2(ρς)β[ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ρG(ν,ς)×S(ν,ς)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(ν,ς)). (71)

    And

    2G(μ+ν2,ς)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)), (72)
    2G(μ+ν2,ρ)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)), (73)
    2G(μ+ν2,ς)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)), (74)
    2G(μ+ν2,ρ)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)]
    +α(α+1)(α+2)(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)), (75)
    2G(μ,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(μ,ρ)+IβρG(μ,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(μ,ς)×S(μ,ς)+G(μ,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(μ,ρ)+G(μ,ρ)×S(μ,ς)), (76)
    2G(ν,ς+ρ2)×Sϕ(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(ν,ρ)+IβρG(ν,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(ν,ς)×S(ν,ς)+G(ν,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(ν,ρ)+G(ν,ρ)×S(ν,ς)), (77)
    2G(μ,ς+ρ2)×S(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(ν,ρ)+IβρG(μ,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(μ,ς)×S(ν,ς)+G(μ,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(ν,ρ)+G(μ,ρ)×S(ν,ς)), (78)

    and

    2G(ν,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(μ,ρ)+IβρG(ν,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(ν,ς)×S(μ,ς)+G(ν,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(μ,ρ)+G(ν,ρ)×S(μ,ς)), (79)

    From inequalities (68) to (79), inequality (67) we have

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)2(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+(2α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))]+2(12α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))]+2(β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))]+2(12β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))]
    +2α(α+1)(α+2)β(β+1)(β+2)K(μ,ν,ς,ρ)++(12α(α+1)(α+2))2β(β+1)(β+2)L(μ,ν,ς,ρ)
    +2α(α+1)(α+2)(12β(β+1)(β+2))M(μ,ν,ς,ρ)+2(12α(α+1)(α+2))(12β(β+1)(β+2))N(μ,ν,ς,ρ). (80)

    Again, with the help of integral inequality (15) and Lemma 1, for each integral on the right-hand side of (80), we have

    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))p(12β(β+1)(β+2))K(μ,ν,ς,ρ)+β(β+1)(β+2)M(μ,ν,ς,ρ). (81)
    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))p(12β(β+1)(β+2))L(μ,ν,ς,ρ)+β(β+1)(β+2)N(μ,ν,ς,ρ). (82)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))p(12α(α+1)(α+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)L(μ,ν,ς,ρ). (83)
    Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))p(12α(α+1)(α+2))M(μ,ν,ς,ρ)+α(α+1)(α+2)N(μ,ν,ς,ρ). (84)

    From (77) to (84), (80) we have

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (85)

    This concludes the proof of Theorem 8 result has been proven.

    Remark 5. If we take α=1 and β=1, then from (63), we achieve the coming inequality, see [38]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)p1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (86)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64), we acquire the coming inequality, see [37]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (87)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64) we acquire the coming inequality, see [36]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (88)

    If we take G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (63), we acquire the coming inequality, see [39]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (89)

    In this study, with the help of coordinated LR-convexity for interval-valued functions, several novel Hermite-Hadamard type inequalities are presented. It is also demonstrated that the conclusions reached in this study represent a possible extension of previously published equivalent results. Similar inequalities may be discovered in the future using various forms of convexities. This is a novel and intriguing topic, and future study will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.

    The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. All authors read and approved the final manuscript. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.



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