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Research article

On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation

  • The Kuramoto-Sinelshchikov equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking in account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the initial-boundary value problem for this equation, under appropriate boundary conditions.

    Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation[J]. Mathematics in Engineering, 2021, 3(4): 1-43. doi: 10.3934/mine.2021036

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  • The Kuramoto-Sinelshchikov equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking in account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the initial-boundary value problem for this equation, under appropriate boundary conditions.



    It is well known fact that the concept of interval analysis fell into oblivion for long time until the 1950's: Moore [1], Warmus [2] and Sunaga [3]. The literature of interval analysis can be tracked back to the computation of lower and upper bounds for π by Archimedes in the following way 31071<π<317. The first monograph was published by Moore in 1960 [4], this field has attracted much attention in the theoretical and applied research. This research field has yielded important results over the past 50 years.

    Recently, several classical integral inequalities have been generalized to the context of set-valued and fuzzy-set-valued functions by means of inclusion and pseudo order relation. In light of this, Sadowska [5] arrived at the following conclusion for an IVF:

    Let F:[u,ν]RK+I be a convex interval-valued function (convex-IVF) given by F(ω)=[F(ω),F(ω)] for all ω[u,ν], where F(ω) and F(ω) are convex and concave functions, respectively. If F is interval Riemann integrable (in sort, IR-integrable), then

    F(u+ν2)1νu(IR)νuF(ω)dωF(u)+F(ν)2. (1)

    Note that, the inclusion relation (Eq 1) is reversed when Fconcave-IVF is. Following that, many scholars used inclusion relations and various integral operators to establish a close relationship between inequality and IVFs. Recently, Costa [6] obtained Jensen's type inequality for FIVF. Costa and Roman-Flores [7,8] introduced different types of inequalities for FIVF and IVF, and discussed their properties. Roman-Flores et al. [9] derived Gronwall for IVFs. Moreover, Chalco-Cano et al. [10,11] presented Ostrowski-type inequalities for IVFs by using the generalized Hukuhara derivative and provided applications in numerical integration in IVF. Nikodem et al. [12], and Matkowski and Nikodem [13] presented the new versions of Jense's inequality for strongly convex and convex functions. Zhao et al. [14,15] derived Chebyshev, Jensen's and H-H type inequalities for IVFs. Recently, Zhang et al. [16] generalized the Jense's inequalities and defined new version of Jensen's inequalities for set-valued and fuzzy-set-valued functions through pseudo order relation. After that, for convex-IVF, Budek [17] established interval-valued fractional Riemann-Liouville HHinequality by means of inclusion relation. For more useful details, see [18,19,20,21,22,23,24] and the references therein.

    Recently, Khan et al. [25] introduced the new class of convex fuzzy mappings is known as (h1,h2)-convex FIVFs by means of FOR and presented the following new version of H-H type inequality for (h1,h2)-convex FIVF involving fuzzy-interval Riemann integrals:

    LetF:[u,ν]F0 be a (h1,h2)-convex FIVF with h1,h2:[0,1]R+ and h1(12)h2(12)0. Then, from θ-levels, we get the collection of IVFs Fθ:[u,ν]RK+C are given by Fθ(ω)=[F(ω,θ),F(ω,θ)] for all ω[u,ν] and for all θ[0,1]. If F is fuzzy-interval Riemann integrable (in sort, FR-integrable), then

    12h1(12)h2(12)F(u+ν2)1νu(FR)νuF(ω)dω[F(u) +F(ν)]10h1(τ)h2(1τ)dτ. (2)

    If h1(τ)=τ and h2(τ)1, then from (2), we get following the result for convex FIVF:

    F(u+ν2)1νu(FR)νuF(ω)dωF(u) +F(ν)2. (3)

    A one step forward, Khan et al. introduced new classes of convex and generalized convex FIVF, and derived new fractional H-H type and H-H type inequalities for convex FIVF [26], h-convex FIVF [27], (h1,h2)-preinvex FIVF [28], log-s-convex FIVFs in the second sense [29], LR-log-h-convex IVFs [30], harmonically convex FIVFs [31], coordinated convex FIVFs [32] and the references therein. We refer to the readers for further analysis of literature on the applications and properties of fuzzy-interval, and inequalities and generalized convex fuzzy mappings, see [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49] and the references therein.

    This study is organized as follows: Section 2 presents preliminary notions and results in interval space, the space of fuzzy intervals and convex analysis. Moreover, the new concept of p-convex fuzzy-IVF is also introduced. Section 3 obtains fuzzy-interval HH-inequalities for p-convex fuzzy-IVFs via fuzzy Riemann integrals. In addition, some interesting examples are also given to verify our results. Section 4 derives discrete Jensen's and Schur's type inequalities for p-convex fuzzy-IVFS. Section 5 gives conclusions and future plans.

    In this section, some preliminary notions, elementary concepts and results are introduced as a pre-work, including operations, orders, and distance between interval and fuzzy numbers, Riemannian integrals, and fuzzy Riemann integrals. Some new definitions and results are also given which will be helpful to prove our main results.

    Let R be the set of real numbers and KC be the space of all closed and bounded intervals of R, and ϖKC be defined by

    ϖ=[ϖ,ϖ]={ωR|ϖωϖ},(ϖ,ϖR).

    If ϖ=ϖ, then ϖ is said to be degenerate. If ϖ0, then [ϖ,ϖ] is called positive interval. The set of all positive interval is denoted by K+C and defined as K+C={[ϖ,ϖ]:[ϖ,ϖ]KC  and  ϖ0}.

    Let ϱR and ϱϖ be defined by

    ϱ.ϖ={[ϱϖ,ϱϖ],if  ϱ>0,{0},if  ϱ=0,[ϱϖ,ϱϖ],if  ϱ<0. (4)

    Then the Minkowski difference ξϖ, addition ϖ+ξ and ϖ×ξ for ϖ,ξKC are defined by

    [ξ,ξ][ϖ,ϖ]=[ξϖ,ξϖ],[ξ,ξ]+[ϖ,ϖ]=[ξ+ϖ,ξ+ϖ], (5)

    and

    [ξ,ξ]×[ϖ,ϖ]=[min{ξϖ,ξϖ,ξϖ,ξϖ},max{ξϖ,ξϖ,ξϖ,ξϖ}].

    The inclusion "⊆" means that

    ξϖif and only if,[ξ,ξ][ϖ,ϖ],if and only if  ϖξ,ξϖ. (6)

    Remark 2.1. [33] The relation "≤I" defined on KC by

    [ξ,ξ]I[ϖ,ϖ]  if and only if   ξϖ,ξϖ, (7)

    for all [ξ,ξ],[ϖ,ϖ]KC, it is an order relation. For given [ξ,ξ],[ϖ,ϖ]KC, we say that [ξ,ξ]I[ϖ,ϖ] if and only if ξϖ,ξϖ.

    For [ξ,ξ],[ϖ,ϖ]KC, the Hausdorff–Pompeiu distance between intervals [ξ,ξ] and [ϖ,ϖ] is defined by

    d([ξ,ξ],[ϖ,ϖ])=max{|ξϖ|,|ξϖ|}. (8)

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

    A fuzzy subset T of R is characterize by a mapping ξ:R[0,1] called the membership function, for each fuzzy set and θ(0,1], then θ-level sets of ξ is denoted and defined as follows ξθ={uR|ξ(u)θ}. If θ=0, then supp(ξ)={ωR|ξ(ω)>0} is called support of ξ. By [ξ]0 we define the closure of supp(ξ).

    Let F(R) be the collection of all fuzzy sets and ξF(R) be a fuzzy set. Then, we define the following:

    (1) ξ is said to be normal if there exists ωR and ξ(ω)=1;

    (2) ξ is said to be upper semi continuous on R if for given ωR, there exist ϵ>0 there exist δ>0 such that ξ(ω)ξ(y)<ϵ for all yR with |ωy|<δ;

    (3) ξ is said to be fuzzy convex if ξθ is convex for every θ[0,1];

    (4) ξ is compactly supported if supp(ξ) is compact.

    A fuzzy set is called a fuzzy number or fuzzy interval if it has properties (1)–(4). We denote by F0 the family of all fuzzy intervals.

    Let ξF0 be a fuzzy-interval, if and only if, θ-levels [ξ]θ is a nonempty compact convex set ofR. From these definitions, we have

    [ξ]θ=[ξ(θ),ξ(θ)],

    where

    ξ(θ)=inf{ωR|ξ(ω)θ},ξ(θ)=sup{ωR|ξ(ω)θ}.

    Proposition 2.2. [7] If ξ,ϖF0, then relation "≼" defined on F0 by

    ξϖ  if and only if, [ξ]θI[ϖ]θ, for all  θ[0,1], (9)

    this relation is known as partial order relation.

    For ξ,ϖF0 and ϱR, the sum ξ +ϖ, product ξ ×ϖ, scalar product ϱ.ξ and sum with scalar are defined by:

    Then, for all θ[0,1], we have

    [ξ +ϖ]θ=[ξ]θ+[ϖ]θ, (10)
    [ξ ×ϖ]θ=[ξ]θ×[ϖ]θ, (11)
    [ϱξ]θ=ϱ[ξ]θ, (12)
    [ϱ +ξ]θ=ϱ+[ξ]θ. (13)

    For ψF0 such that ξ=ϖ +ψ, then by this result we have existence of Hukuhara difference of ξ and ϖ, and we say that ψ is the H-difference of ξ and ϖ, and denoted by ξ ϖ.

    Definition 2.3. [16] A fuzzy-interval-valued mapF:KRF0 is called FIVF. For each θ(0,1], θ-levels define the family of IVFs Fθ:KRKC are given by Fθ(ω)=[F(ω,θ),F(ω,θ)] for all ωK. Here, for each θ(0,1], the end point real functions F(.,θ),F(.,θ):KR are called lower and upper functions of F.

    Definition 2.5. [34] Let F:[u,ν]RF0 be a FIVF. Then, fuzzy Riemann integral of F over [u,ν], denoted by (FR)νuF(ω)dω, it is given level-wise by

    [(FR)νuF(ω)dω]θ=(IR)νuFθ(ω)dω={νuF(ω,θ)dω:F(ω,θ)R([u,ν],θ)}, (14)

    for all θ(0,1], where R([u,ν],θ) denotes the collection of Riemannian integrable functions of IVFs. F is FR-integrable over [u,ν] if (FR)νuF(ω)dωF0. Note that, if both end point functions are Lebesgue-integrable, then F is fuzzy Aumann-integrable function over [u,ν] [16,34].

    Theorem 2.6. Let F:[u,ν]RF0 be a FIVF and for all θ(0,1], θ-levels define the family of IVFs Fθ:[u,ν]RKC are given by Fθ(ω)=[F(ω,θ),F(ω,θ)] for all ω[u,ν]. Then, F is fuzzy Riemann integrable (FR-integrable) over [u,ν] if and only if, F(ω,θ) and F(ω,θ) both are Riemann integrable (R-integrable) over [u,ν]. Moreover, if F is FR-integrable over [u,ν], then

    [(FR)νuF(ω)dω]θ=[(R)νuF(ω,θ)dω,(R)νuF(ω,θ)dω]=(IR)νuFθ(ω)dω, (15)

    for all θ(0,1], where IR represent interval Riemann integration of Fθ(ω). For all θ(0,1], FR([u,ν],θ) denotes the collection of all FR-integrable FIVFs over [u,ν].

    Definition 2.7. Let [u,ν] be a p-convex interval. Then, FIVF F:[u,ν]F0 is said to be p-convex on[u,ν] if

    F([ηxp+(1η)yp]1p)ηF(x) +(1η)F(y), (16)

    for allx,y[u,ν],η[0,1], where F(ω) 0, for all ω[u,ν]. If inequality (16) is reversed, then F is said to be p-concave FIVF on [u,ν]. The set of all p-convex (LR-p-concave) FIVFs is denoted by

    SXF([u,ν],F0,p),(SVF([u,ν],F0,p)).

    Remark 2.8. The p-convex FIVFs have some very nice properties similar to convex FIVF:

    If F is p-convex FIVF, then YF is also p-convex for Y0.

    If F and T both are p-convex FIVFs, then max(F(ω),T(ω)) is also p-convex FIVF.

    We now discuss some new and known special cases of p-convex FIVFs:

    If p1, then p-convex FIVF becomes convex FIVF, see [35], that is

    F(ηx+(1η)y)ηF(x) +(1η)F(y),x,y[u,ν],η[0,1]. (17)

    In Theorem 2.9, we will try to establish relation between the p-convex FIVFs and endpoint functions F(ω,θ), F(ω,θ) because through endpoint functions, FIVFs can be easily handled.

    Theorem 2.9. Let [u,ν] be convex set, and F:[u,ν]F0 be a FIVF. Then, θ-levels define the family of IVFs Fθ:[u,ν]RKC+KC are given by

    Fθ(ω)=[F(ω,θ),F(ω,θ)],ω[u,ν], (18)

    for all ω[u,ν] and for all θ[0,1]. Then, F is p-convex on [u,ν], if and only if, for all θ[0,1], F(ω,θ) and F(ω,θ) both are p-convex functions.

    Proof. Assume that for each θ[0,1], F(ω,θ) and F(ω,θ) are p-convex functions on [u,ν]. Then, from (16) we have

    F([ηxp+(1η)yp]1p,θ)ηF(x,θ)+(1η)F(y,θ),x,y[u,ν],η[0,1],

    and

    F([ηxp+(1η)yp]1p,θ)ηF(x,θ)+(1η)F(y,θ),x,y[u,ν],η[0,1].

    Then by (18), (10) and (12), we obtain

    Fθ([ηxp+(1η)yp]1p)=[F([ηxp+(1η)yp]1p,θ),F([ηxp+(1η)yp]1p,θ)],
    I[ηF(x,θ),ηF(x,θ)]+[(1η)F(y,θ),(1η)F(y,θ)],

    that is

    F([ηxp+(1η)yp]1p)ηF(x) +(1η)F(y),x,y[u,ν],η[0,1].

    Hence, F is p-convex FIVF on [u,ν].

    Conversely, let F be p-convex FIVF on [u,ν]. Then, for all x,y[u,ν] and η[0,1], we have

    F([ηxp+(1η)yp]1p)ηF(x) +(1η)F(y).

    Therefore, from (18), we have

    Fθ([ηxp+(1η)yp]1p)=[F([ηxp+(1η)yp]1p,θ),F([ηxp+(1η)yp]1p,θ)].

    Again, from (18), (10) and (12), we obtain

    ηFθ(x) +(1η)Fθ(y)=[ηF(x,θ),ηF(x,θ)]+[(1η)F(y,θ),(1η)F(y,θ)],

    for all x,y[u,ν] and η[0,1]. Then, by p-convexity of F, we have for all x,y[u,ν] and η[0,1]such that

    F([ηxp+(1η)yp]1p,θ)ηF(x,θ)+(1η)F(y,θ),

    and

    F([ηxp+(1η)yp]1p,θ)ηF(x,θ)+(1η)F(y,θ),

    for each θ[0,1]. Hence, the result follows.

    Example 2.10. We consider the FIVF F:[0,1]F0 defined by,

    F(ω)(λ)={λ2ω2,λ[0,2ω2]4ω2λ2ω2,λ(2ω2,4ω2]0,otherwise,

    then, for each θ[0,1], we have Fθ(ω)=[2θω2,(42θ)ω2]. Since end point functions F(ω,θ), F(ω,θ) are convex functions for each θ[0,1]. Hence F(ω) is convex FIVF.

    Remark 2.11. If F(ω,θ)=F(ω,θ), then Definition 2.7 reduces to the definition of classical p-convex function, [43].

    If F(ω,θ)=F(ω,θ) and p1, then Definition 2.7 reduces to the definition of classical convex function.

    In this section, we will prove two types of inequalities. First one is Hermite-Hadamard and their variant forms, and the second one is Hermite-Hadamard-Fejér inequalities for p-convex FIVFs where the integrands are FIVFs. We will verify these inequalities with the help of nontrivial examples.

    Theorem 3.1. Let FSXF([u,ν],F0,p). Then, θ-levels define the family of IVFs Fθ:[u,ν]RKC+ are given by Fθ(ω)=[F(ω,θ),F(ω,θ)] for all ω[u,ν] and for all θ[0,1]. If FFR([u,ν],θ), then

    F([up+νp2]1p)pνpup(FR)νuωp1F(ω)dωF(u) +F(ν)2. (19)

    If F(ω) is p-concave FIVF, then

    F([up+νp2]1p)pνpup(FR)νuωp1F(ω)dωF(u) +F(ν)2. (20)

    Proof. Let F:[u,ν]F0 be a p-convex FIVF. Then, by hypothesis, we have

    2F([up+νp2]1p)F([ηup+(1η)vp]1p) +F([(1η)up+ηνp]1p).

    Therefore, for every θ[0,1], we have

    2F([up+νp2]1p,θ)F([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ),2F([up+νp2]1p,θ)F([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ).

    Then

    210F([up+νp2]1p,θ)dη10F([ηup+(1η)νp]1p,θ)dη+10F([(1η)up+ηνp]1p,θ)dη,210F([up+νp2]1p,θ)dη10F([ηup+(1η)νp]1p,θ)dη+10F([(1η)up+ηνp]1p,θ)dη.

    It follows that

    F([up+νp2]1p,θ)pνpupνuωp1F(ω,θ)dω,F([up+νp2]1p,θ)pνpupνuωp1F(ω,θ)dω.

    That is

    [F([up+νp2]1p,θ),F([up+νp2]1p,θ)]Ipνpup[νuωp1F(ω,θ)dω,νuωp1F(ω,θ)dω].

    Thus,

    F([up+νp2]1p)pνpup(FR)νuωp1F(ω)dω. (21)

    In a similar way as above, we have

    pνpup(FR)νuωp1F(ω)dωF(u) +F(ν)2. (22)

    Combining (21) and (22), we have

    F([up+νp2]1p)pνpup(FR)νuωp1F(ω)dωF(u) +F(ν)2.

    Hence, the required result.

    Remark 3.2. If p=1, then Theorem 3.1, reduces to the result for convex FIVF, see [25]:

    F(u+ν2)1νu(FR)νuF(ω)dωF(u) +F(ν)2.

    If F(ω,θ)=F(ω,θ) withθ=1, then Theorem 3.1, reduces to the result for p-convex function [43]:

    F([up+νp2]1p)pνpup(R)νuωp1F(ω)dωF(u)+F(ν)2.

    If F(ω,θ)=F(ω,θ) with θ=1 and p=1, then Theorem 3.1, reduces to the result for classical convex function:

    F(u+ν2)1νu(R)νuF(ω)dωF(u)+F(ν)2.

    Example 3.3. Let p be an odd number and the FIVF F:[u,ν]=[2,3]F0 defined by,

    F(ω)(λ)={λ(2ωp2)λ[0,2ωp2],2(2ωp2)λ(2ωp2)λ(2ωp2,2(2ωp2)],0otherwise. (23)

    Then, for each θ[0,1], we have Fθ(ω)=[θ(2ωp2),(2θ)(2ωp2)]. Since end point functions F(ω,θ)=θ(2ωp2), F(ω,θ)=(2θ)(2ωp2) are p-convex functions for each θ[0,1]. Then, F(ω) is p-convex FIVF.

    We now computing the following

    F([up+νp2]1p,θ)=4102θ,F([up+νp2]1p,θ)=4102(2θ),
    pνpupνuωp1F(ω,θ)dω=θ32(2ωp2)dω=2150θ,pνpupνuωp1F(ω,θ)dω=(2θ)32(2ωp2)dω=2150(2θ),
    F(u,θ)+F(ν,θ)2=4232θ,F(u,θ)+F(ν,θ)2=4232(2θ),

    for all θ[0,1]. That means

    [4102θ,4102(2θ)]I[2150θ,2150(2θ)]I[4232θ,4232(2θ)],

    for all θ[0,1], and the Theorem 3.1 has been verified.

    To prove some related inequalities for the above theorem, we obtain following inequality for p-convex FIVFs

    Theorem 3.4. Let FSXF([u,ν],F0,p). Then, θ-levels define the family of IVFs Fθ:[u,ν]RKC+ are given by Fθ(ω)=[F(ω,θ),F(ω,θ)] for all ω[u,ν] and for all θ[0,1]. If FFR([u,ν],θ), then

    F([up+νp2]1p)2pνpup(FR)νuωp1F(ω)dω1F(u) +F(ν)2,

    where

    1=F(u) +F(ν)2 +F([up+νp2]1p)2,2=F([3up+νp4]1p) +F([up+3νp4]1p)2,and1=[1,1],2=[2,2].

    Proof. Take [up,up+νp2], we have

    2F([ηup+(1η)up+νp2]1p2+[(1η)up+ηup+νp2]1p2)F([ηup+(1η)up+νp2]1p) +F([(1η)up+ηup+νp2]1p).

    Therefore, for every θ[0,1], we have

    2F([ηup+(1η)up+νp2]1p2+[(1η)up+ηup+νp2]1p2,θ)F([ηup+(1η)up+νp2]1p,θ)+F([(1η)up+ηup+νp2]1p,θ),
    2F([ηup+(1η)up+νp2]1p2+[(1η)up+ηup+νp2]1p2,θ)F([ηup+(1η)up+νp2]1p,θ)+F([(1η)up+ηup+νp2]1p,θ).

    In consequence, we obtain

    F([3up+νp4]1p,θ)2pνpupup+νp2uF(ω,θ)dω,F([3up+νp4]1p,θ)2pνpupup+νp2uF(ω,θ)dω.

    That is

    [F([3up+νp4]1p,θ),F([3up+νp4]1p,θ)]2pνpup[up+νp2uF(ω,θ)dω,up+νp2uF(ω,θ)dω].

    It follows that

    F([3up+νp4]1p)2pνpup(FR)up+νp2uF(ω)dω. (24)

    In a similar way as above, we have

    F([up+3νp4]1p)2pνpup(FR)νup+νp2F(ω)dω. (25)

    Combining (24) and (25), we have

    [F([3up+νp4]1p) +F([up+3νp4]1p)]2pνpup(FR)νuF(ω)dω.

    By using Theorem 3.1, we have

    F([up+νp2]1p)=F([12.3up+νp4+12.up+3νp4]1p).

    Therefore, for every θ[0,1], we have

    F([up+νp2]1p,θ)=F([12.3up+νp4+12.up+3νp4]1p,θ),F([up+νp2]1p,θ)=F([12.3up+νp4+12.up+3νp4]1p,θ),
    [12F([3up+νp4]1p,θ)+12F([up+3νp4]1p,θ)],[12F([3up+νp4]1p,θ)+12F([up+3νp4]1p,θ)],
    =2,=2,
    pνpupνuF(ω,θ)dω,pνpupνuF(ω,θ)dω,
    12[F(u,θ)+F(ν,θ)2+F([up+νp2]1p,θ)],12[F(u,θ)+F(ν,θ)2+F([up+νp2]1p,θ)],
    =1,=1,
    12[F(u,θ)+F(ν,θ)2+F(u,θ)+F(ν,θ)2],12[F(u,θ)+F(ν,θ)2+F(u,θ)+F(ν,θ)2],
    =F(u,θ)+F(ν,θ)2,=F(u,θ)+F(ν,θ)2,

    that is

    F([up+νp2]1p)2pνpup(FR)νuF(ω)dω1F(u) +F(ν)2,

    hence, the result follows.

    Example 3.5. Let p be an odd number and the FIVF F:[u,ν]=[2,3]F0 defined by, Fθ(ω)=[θ(2ωp2),(2θ)(2ωp2)], as in Example 3.3, then F(ω) is p-convex FIVF and satisfying (38). We have F(ω,θ)=θ(2ωp2) and F(ω,θ)=(2θ)(2ωp2). We now computing the following

    [F(u,θ)+F(ν,θ)2]=4232θ,[F(u,θ)+F(ν,θ)2]=4232(2θ),
    1=F(u,θ)+F(ν,θ)2+F([up+νp2]1p,θ)2=823104θ,1=F(u,θ)+F(ν,θ)2+F([up+νp2]1p,θ)2=823104(2θ),
    2=12[F([3up+νp4]1p,θ)+F([up+3νp4]1p,θ)]=5114θ,2=12[F([3up+νp4]1p,θ)+F([up+3νp4]1p,θ)]=5114(2θ),
    F([up+νp2]1p,θ)=4102θ,F([up+νp2]1p,θ)=4102(2θ).

    Then, we obtain that

    4102θ5114θ2150θ823104θ4232θ,4102(2θ)5114(2θ)2150(2θ)823104(2θ)4232(2θ).

    Hence, Theorem 3.4 is verified.

    From Theorem 3.6 and Theorem 3.7, we now obtain some H-H inequalities for the product of p-convex FIVFs. These inequalities are refinements of some known inequalities [42,43].

    Theorem 3.6. Let F,JSXF([u,ν],F0,p). Then, θ-levels Fθ,Jθ:[u,ν]RKC+ are defined by Fθ(ω)=[F(ω,θ),F(ω,θ)] and Jθ(ω)=[J(ω,θ),J(ω,θ)] for all ω[u,ν] and for all θ[0,1]. If F,J and FJFR([u,ν],θ), then

    pνpup(FR)νuωp1F(ω) ×J(ω)dωM(u,ν)3 +N(u,ν)6.

    Where M(u,ν)=F(u) ×J(u) +F(ν) ×J(ν), N(u,ν)=F(u) ×J(ν) +F(ν) ×J(u), and Mθ(u,ν)=[M((u,ν),θ),M((u,ν),θ)] and Nθ(u,ν)=[N((u,ν),θ),N((u,ν),θ)].

    Proof. The proof is similar to the proof of Theorem 3.3 [46].

    Example 3.7. Let p be an odd number, and p-convex FIVFs F,J:[u,ν]=[2,3]F0 are, respectively defined by, Fθ(ω)=[θ(2ωp2),(2θ)(2ωp2)], as in Example 3.3 and Jθ(ω)=[θωp,(2θ)ωp]. Since F(ω) and J(ω) both are p-convex FIVFs and F(ω,θ)=θ(2ωp2), F(ω,θ)=(2θ)(2ωp2), and J(ω,θ)=θωp, J(ω,θ)=(2θ)ωp, then we computing the following

    pνpupνuωp1F(ω,θ)×J(ω,θ)dω=θ2,pνpupνuωp1F(ω,θ)×J(ω,θ)dω=(2θ)2,
    M((u,ν),θ)=(102233)θ23,M((u,ν),θ)=(102233)(2θ)23,
    N((u,ν),θ)=(103223)θ26N((u,ν),θ)=(103223)(2θ)26,

    for each θ[0,1], that means

    θ2(307283)θ26,(2θ)2(307283)(2θ)26.

    Hence, Theorem 3.6 is demonstrated.

    Theorem 3.8. Let F,JSXF([u,ν],F0,p). Then, θ-levels define the family of IVFs Fθ,Jθ:[u,ν]RKC+ are given by Fθ(ω)=[F(ω,θ),F(ω,θ)] and Jθ(ω)=[J(ω,θ),J(ω,θ)] for all ω[u,ν] and for all θ[0,1]. If F ×JFR([u,ν],θ), then

    2F([up+νp2]1p) ×J([up+νp2]1p)pνpup(FR)νuωp1F(ω) ×J(ω)dω +M(u,ν)6 +N(u,ν)3.

    WhereM(u,ν)=F(u) ×J(u) +F(ν) ×J(ν), N(u,ν)=F(u) ×J(ν) +F(ν) ×J(u), and Mθ(u,ν)=[M((u,ν),θ),M((u,ν),θ)] and Nθ(u,ν)=[N((u,ν),θ),N((u,ν),θ)].

    Proof. By hypothesis, for each θ[0,1], we have

    F([up+νp2]1p,θ)×J([up+νp2]1p,θ)F([up+νp2]1p,θ)×J([up+νp2]1p,θ)
    14[F([ηup+(1η)νp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([ηup+(1η)νp]1p,θ)×J([(1η)up+ηνp]1p,θ)]
    +14[F([(1η)up+ηνp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ)×J([(1η)up+ηνp]1p,θ)]
    ,14[F([ηup+(1η)νp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([ηup+(1η)νp]1p,θ)×J([(1η)up+ηνp]1p,θ)]+14[F([(1η)up+ηνp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ)×J([(1η)up+ηνp]1p,θ)],
    14[F([ηup+(1η)νp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ)×J([(1η)up+ηνp]1p,θ)]+14[(ηF(u,θ)+(1η)F(ν,θ))×((1η)J(u,θ)+ηJ(ν,θ))+((1η)F(u,θ)+ηF(ν,θ))×(ηJ(u,θ)+(1η)J(ν,θ))],14[F([ηup+(1η)νp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ)×J([(1η)up+ηνp]1p,θ)]+14[(ηF(u,θ)+(1η)F(ν,θ))×((1η)J(u,θ)+ηJ(ν,θ))+((1η)F(u,θ)+ηF(ν,θ))×(ηJ(u,θ)+(1η)J(ν,θ))],
    =14[F([ηup+(1η)νp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ)×J([(1η)up+ηνp]1p,θ)]+12[{η2+(1η)2}N((u,ν),θ)+{η(1η)+(1η)η}M((u,ν),θ)],=14[F([ηup+(1η)νp]1p,θ)×J([ηup+(1η)νp]1p,θ)+F([(1η)up+ηνp]1p,θ)×J([(1η)up+ηνp]1p,θ)]+12[{η2+(1η)2}N((u,ν),θ)+{η(1η)+(1η)η}M((u,ν),θ)],

    R-Integrating over [0,1], we have

    2F([up+νp2]1p,θ)×J([up+νp2]1p,θ)pνpupνuωp1F(ω,θ)×J(ω,θ)dω+M((u,ν),θ)6+N((u,ν),θ)3,2F([up+νp2]1p,θ)×J([up+νp2]1p,θ)pνpupνuωp1F(ω,θ)×J(ω,θ)dω+M((u,ν),θ)6+N((u,ν),θ)3,

    that is

    2F([up+νp2]1p) ×J([up+νp2]1p)pνpup(FR)νuωp1F(ω) ×J(ω)dω +M(u,ν)6 +N(u,ν)3.

    Hence, the required result.

    Example 3.9. Let p be an odd number, and p-convex FIVFs F,J:[u,ν]=[2,3]F0 are, respectively defined by, Fθ(ω)=[θ(2ωp2),(2θ)(2ωp2)], as in Example 3.3 and Jθ(ω)=[θωp,(2θ)ωp]. Since F(ω) and J(ω) both are p-convex FIVFs and F(ω,θ)=θ(2ωp2), F(ω,θ)=(2θ)(2ωp2), and J(ω,θ)=θωp, J(ω,θ)=(2θ)ωp, then we computing the following

    2F([up+νp2]1p,θ)×J([up+νp2]1p,θ)=205102θ2,2F([up+νp2]1p,θ)×J([up+νp2]1p,θ)=205102(2θ)2,
    M((u,ν),θ)6=(102233)θ26,M((u,ν),θ)6=(102233)(2θ)26,
    \begin{array}{c}\frac{{\mathcal{N}}_{\mathcal{*}}\left(\left(\mathit{u}, \nu \right), \theta \right)}{3} = \left(10-3\sqrt{2}-2\sqrt{3}\right)\frac{{\theta }^{\begin{array}{c}\\ 2\end{array}}}{3}, \\ \frac{{\mathcal{N}}^{\mathcal{*}}\left(\left(\mathit{u}, \nu \right), \theta \right)}{3} = \left(10-3\sqrt{2}-2\sqrt{3}\right)\frac{{(2-\theta )}^{2}}{3}, \end{array}

    for each \theta \in \left[0, 1\right], that means

    \begin{array}{c}\frac{20-5\sqrt{10}}{2}{\theta }^{2}\le \left(30-8\sqrt{2}-7\sqrt{3}\right)\frac{{\theta }^{2}}{6}, \\ \frac{20-5\sqrt{10}}{2}{(2-\theta )}^{2}\le \left(30-8\sqrt{2}-7\sqrt{3}\right)\frac{{(2-\theta )}^{2}}{6}, \end{array}

    hence, Theorem 3.8 is verified.

    We now give H-H Fejér inequalities for p -convex FIVFs. Firstly, we obtain the second H-H Fejér inequality for p -convex FIVF.

    Theorem 3.10. Let \mathcal{F}\in SXF\left(\left[\mathit{u}, \nu \right], {\mathbb{F}}_{0}, p\right) . Then, \theta -levels define the family of IVFs {\mathcal{F}}_{\theta }:\left[\mathit{u}, \nu \right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+} are given by {\mathcal{F}}_{\theta }\left(\omega \right) = \left[{\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right)\right] for all \omega \in \left[\mathit{u}, \nu \right] and for all \theta \in \left[0, 1\right] . If \mathcal{F}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathit{u}, \nu \right], \theta \right)} and \Omega :\left[\mathit{u}, \nu \right]\to \mathbb{R}, \Omega \left(\omega \right)\ge 0, symmetric with respect to {\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, then

    \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}\left(FR\right){\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\mathcal{F}\left(\omega \right)\Omega \left(\omega \right)d\omega \preccurlyeq \left[\mathcal{F}\left(\mathit{u}\right)\stackrel{~}{+}\mathcal{F}\left(\nu \right)\right]{\int }_{0}^{1}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta . (26)

    If \mathcal{F} is p -concave FIVF, then inequality (26) is reversed.

    Proof. Let \mathcal{F} be a p -convex FIVF. Then, for each \theta \in \left[0, 1\right], we have

    {\mathcal{F}}_{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)\\ \le \left(\eta {\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)+\left(1-\eta \right){\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right)\right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right), (27)
    {\mathcal{F}}^{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)\\ \le \left(\eta {\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)+\left(1-\eta \right){\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)

    And

    \begin{array}{c}\\ {\mathcal{F}}_{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\\ \le \left(\left(1-\eta \right){\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)+\eta {\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right)\right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right), \\ {\mathcal{F}}^{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\\ \le \left(\left(1-\eta \right){\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)+\eta {\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right).\end{array} (28)

    After adding (27) and (28), and integrating over \left[0, 1\right], we get

    \begin{array}{c}\\ {\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ +{\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ \le {\int }_{0}^{1}\left[\begin{array}{c}{\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)\left\{\eta \Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)+\left(1-\eta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\right\}\\ +{\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right)\left\{\left(1-\eta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)+\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\right\}\end{array}\right]d\eta , \\ {\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ +{\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ \le {\int }_{0}^{1}\left[\begin{array}{c}{\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)\left\{\eta \Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)+\left(1-\eta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\right\}\\ +{\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\left\{\left(1-\eta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)+\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\right\}\end{array}\right]d\eta , \end{array}
    \begin{array}{c}\\ = 2{\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right){\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta +2{\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right){\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta , \\ = 2{\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right){\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta +2{\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right){\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta .\end{array}

    Since \Omega is symmetric, then

    \begin{array}{c}\\ = 2\left[{\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta , \\ = 2\left[{\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta .\end{array} (29)

    Since

    \begin{array}{c}\\ {\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = {\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \\ {\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = {\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega .\end{array} (30)

    Then, from (29), we have

    \begin{array}{c}\\ \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \le \left[{\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta , \\ \\ \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \le \left[{\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta , \end{array}

    that is

    \left[\frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \right]
    {\le }_{p}\left[{\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta ,

    hence

    \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}\left(FR\right){\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\mathcal{F}\left(\omega \right)\Omega \left(\omega \right)d\omega \preccurlyeq \left[\mathcal{F}\left(\mathit{u}\right)\stackrel{~}{+}\mathcal{F}\left(\nu \right)\right]{\int }_{0}^{1}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta .

    Next, we construct first H-H Fejér inequality for p -convex FIVF, which generalizes first H-H Fejér inequalities for convex function [44].

    Theorem 3.11. Let \mathcal{F}\in SXF\left(\left[\mathit{u}, \nu \right], {\mathbb{F}}_{0}, p\right) . Then, \theta -levels define the family of IVFs {\mathcal{F}}_{\theta }:\left[\mathit{u}, \nu \right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+} are given by {\mathcal{F}}_{\theta }\left(\omega \right) = \left[{\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right)\right] for all \omega \in \left[\mathit{u}, \nu \right] and for all \theta \in \left[0, 1\right] . If \mathcal{F}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathit{u}, \nu \right], \theta \right)} and \Omega :\left[\mathit{u}, \nu \right]\to \mathbb{R}, \Omega \left(\omega \right)\ge 0, symmetric with respect to {\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, and {\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\Omega \left(\omega \right)d\omega > 0 , then

    \mathcal{F}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}\right)\preccurlyeq \frac{1}{{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\Omega \left(\omega \right)d\omega }\left(FR\right){\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\mathcal{F}\left(\omega \right)\Omega \left(\omega \right)d\omega . (31)

    If \mathcal{F} is p -concave FIVF, then inequality (31) is reversed.

    Proof. Since \mathcal{F} is a convex, then for \theta \in \left[0, 1\right], we have

    \begin{array}{c}\\ {\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\le \frac{1}{2}\left({\mathcal{F}}_{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)+{\mathcal{F}}_{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\right), \\ {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\le \frac{1}{2}\left({\mathcal{F}}^{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)+{\mathcal{F}}^{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\right), \end{array} (32)

    Since \Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right) = \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right) , then by multiplying (32) by \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right) and integrate it with respect to \eta over \left[0, 1\right], we obtain

    \begin{array}{c}\\ {\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right){\int }_{0}^{1}\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ \le \frac{1}{2}\left(\begin{array}{c}{\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ +{\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)d\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}\right), \\ {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right){\int }_{0}^{1}\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ \le \frac{1}{2}\left(\begin{array}{c}{\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ +{\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \end{array}\right).\end{array} (33)

    Since

    \begin{array}{c}\\ {\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = {\int }_{0}^{1}{\mathcal{F}}_{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , \\ \\ {\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = {\int }_{0}^{1}{\mathcal{F}}^{\mathcal{*}}\left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}, \theta \right)\Omega \left({\left[\eta {\mathit{u}}^{p}+\left(1-\eta \right){\nu }^{p}\right]}^{\frac{1}{p}}\right)d\eta \\ = \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega .\\ \end{array} (34)

    Then, from (34) we have

    \begin{array}{c}\\ {\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\le \frac{1}{{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\Omega \left(\omega \right)d\omega }{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , \\ {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\le \frac{1}{{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\Omega \left(\omega \right)d\omega }{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , \end{array}

    from which, we have

    \begin{array}{c}\\ \left[{\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\right]\\ {\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_{I}\frac{1}{{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\Omega \left(\omega \right)d\omega }\left[{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , {\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \right], \\ \end{array}

    that is

    \mathcal{F}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}\right)\preccurlyeq \frac{1}{{\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\Omega \left(\omega \right)d\omega }\left(FR\right){\int }_{\mathit{u}}^{\nu }{\omega }^{p-1}\mathcal{F}\left(\omega \right)\Omega \left(\omega \right)d\omega ,

    this completes the proof.

    Remark 3.12. If in the Theorem 3.10 and Theorem 3.11, p = 1 , then we obtain the appropriate theorems for convex fuzzy-IVFs [26].

    If in the Theorem 3.10 and Theorem 3.11, {\mathcal{T}}_{\mathcal{*}}\left(\omega, \gamma \right) = {\mathcal{T}}^{\mathcal{*}}\left(\omega, \gamma \right) with \gamma = 1 , then we obtain the appropriate theorems for p -convex function [43].

    If in the Theorem 3.10 and Theorem 3.11, {\mathcal{T}}_{\mathcal{*}}\left(\omega, \gamma \right) = {\mathcal{T}}^{\mathcal{*}}\left(\omega, \gamma \right) with \gamma = 1 and p = 1 , then we obtain the appropriate theorems for convex function [44].

    If \Omega \left(\omega \right) = 1, then combining Theorem 3.10 and Theorem 3.11, we get Theorem 3.1.

    Example 3.13. We consider the FIVF \mathcal{F}:\left[1, 4\right]\to {\mathbb{F}}_{0} defined by,

    \mathcal{F}\left(\omega \right)\left(\lambda \right) = \left\{\begin{array}{c}\frac{\lambda -{e}^{{\omega }^{p}}}{{e}^{{\omega }^{p}}},& \lambda \in \left[{e}^{{\omega }^{p}}, 2{e}^{{\omega }^{p}}\right], \\ \frac{4{e}^{{\omega }^{p}}-\lambda }{2{e}^{{\omega }^{p}}},& \lambda \in \left(2{e}^{{\omega }^{p}}, 4{e}^{{\omega }^{p}}\right], \\ 0, &otherwise, \end{array}\right. (35)

    then, for each \theta \in \left[0, 1\right], we have {\mathcal{F}}_{\theta }\left(\omega \right) = \left[(1+\theta){e}^{{\omega }^{p}}, 2(2-\theta){e}^{{\omega }^{p}}\right] . Since end point functions {\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right) are p -convex functions, for each \theta \in [0, 1] , then \mathcal{F}\left(\omega \right) is p -convex FIVF. If

    \Omega \left(\omega \right) = \left\{\begin{array}{c}{\omega }^{p}-1, \lambda \in \left[1, \frac{5}{2}\right], \\ 4-{\omega }^{p}, \lambda \in \left(\frac{5}{2}, 4\right], \end{array}\right. (36)

    where p = 1 . Then, we have

    \begin{array}{c}\\ \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{1}^{4}{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega = \frac{1}{3}{\int }_{1}^{4}{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \\ = \frac{1}{3}{\int }_{1}^{\frac{5}{2}}{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega +\frac{1}{3}{\int }_{\frac{5}{2}}^{4}{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , \\ = \frac{1}{3}\left(1+\theta \right){\int }_{1}^{\frac{5}{2}}{e}^{\omega }\left(\omega -1\right)d\omega +\frac{1}{3}\left(1+\theta \right){\int }_{\frac{5}{2}}^{4}{e}^{\omega }\left(4-\omega \right)d\omega \approx 11\left(1+\theta \right), \\ \frac{p}{{\nu }^{p}-{\mathit{u}}^{p}}{\int }_{1}^{4}{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega = \frac{1}{3}{\int }_{1}^{4}{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \\ = \frac{1}{3}{\int }_{1}^{\frac{5}{2}}{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega +\frac{1}{3}{\int }_{\frac{5}{2}}^{4}{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega , \\ \begin{array}{c}\\ = \frac{2}{3}\left(2-\theta \right){\int }_{1}^{\frac{5}{2}}{e}^{\omega }\left(\omega -1\right)d\omega +\frac{2}{3}\left(2-\theta \right){\int }_{\frac{5}{2}}^{4}{e}^{\omega }\left(4-\omega \right)d\omega \approx 22\left(2-\theta \right), \\ \end{array}\end{array} (37)

    and

    \begin{array}{c}\\ \left[{\mathcal{F}}_{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}_{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta \\ \\ \left[{\mathcal{F}}^{\mathcal{*}}\left(\mathit{u}, \theta \right)+{\mathcal{F}}^{\mathcal{*}}\left(\nu , \theta \right)\right]{\int }_{0}^{1}\begin{array}{c}\eta \Omega \left({\left[\left(1-\eta \right){\mathit{u}}^{p}+\eta {\nu }^{p}\right]}^{\frac{1}{p}}\right)\end{array}d\eta \end{array}
    \begin{array}{c} = \left(1+\theta \right)\left[e+{e}^{4}\right]\left[{\int }_{0}^{\frac{1}{2}}3{\eta }^{2}d\omega +{\int }_{\frac{1}{2}}^{1}\eta \left(3-3\eta \right)d\eta \right]\approx \frac{43}{2}\left(1+\theta \right).\\ \\ = 2\left(2-\theta \right)\left[e+{e}^{4}\right]\left[{\int }_{0}^{\frac{1}{2}}3{\eta }^{2}d\omega +{\int }_{\frac{1}{2}}^{1}\eta \left(3-3\eta \right)d\eta \right]\approx 43\left(2-\theta \right).\end{array} (38)

    From (37) and (38), we have

    \left[11\left(1+\theta \right), 22\left(2-\theta \right)\right]{\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_{I}\left[\frac{43}{2}\left(1+\theta \right), 43\left(2-\theta \right)\right], for each \theta \in \left[0, 1\right].

    Hence, Theorem 3.10 is verified.

    For Theorem 3.11, we have

    \begin{array}{c}{\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\approx \frac{61}{5}\left(1+\theta \right), \\ {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{{\mathit{u}}^{p}+{\nu }^{p}}{2}\right]}^{\frac{1}{p}}, \theta \right)\approx \frac{122}{5}\left(2-\theta \right), \\ \end{array} (39)
    {\int }_{\mathit{u}}^{\nu }\Omega \left(\omega \right)d\omega = {\int }_{1}^{\frac{5}{2}}\left(\omega -1\right)d\omega {\int }_{\frac{5}{2}}^{4}\left(4-\omega \right)d\omega = \frac{9}{4},
    \begin{array}{c}\\ \frac{p}{{\int }_{\mathit{u}}^{\nu }\Omega \left(\omega \right)d\omega }{\int }_{1}^{4}{\omega }^{p-1}{\mathcal{F}}_{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \approx \frac{73}{5}\left(1+\theta \right), \\ \frac{p}{{\int }_{\mathit{u}}^{\nu }\Omega \left(\omega \right)d\omega }{\int }_{1}^{4}{\omega }^{p-1}{\mathcal{F}}^{\mathcal{*}}\left(\omega , \theta \right)\Omega \left(\omega \right)d\omega \approx \frac{293}{10}\left(2-\theta \right).\\ \end{array} (40)

    From (39) and (40), we have

    \left[\frac{61}{5}\left(1+\theta \right), \frac{122}{5}\left(2-\theta \right)\right]{\begin{array}{c}\begin{array}{c}\le \end{array}\end{array}}_{I}\left[\frac{73}{5}\left(1+\theta \right), \frac{293}{10}\left(2-\theta \right)\right] ,

    hence, Theorem 3.11 is demonstrated.

    In this section, we propose the concept of discrete Jensen's and Schur's type inequality for p -convex FIVF. Some refinements of discrete Jensen's type inequality are also obtained. We begin by presenting the discrete Jensen's type inequality for p -convex FIVF in the following result.

    Theorem 4.1. (Discrete Jense's type inequality for p -convex FIVF) Let {\eta }_{j}\in {\mathbb{R}}^{+} , {\mathit{u}}_{j}\in \left[\mathit{u}, \nu \right], \left(j = 1, 2, 3, \dots, k, k\ge 2\right) and \mathcal{F}\in SXF\left(\left[\mathit{u}, \nu \right], {\mathbb{F}}_{0}, p\right) and for all \theta \in \left[0, 1\right] , \theta -levels define the family of IVFs {\mathcal{F}}_{\theta }:\left[\mathit{u}, \nu \right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+} are given by {\mathcal{F}}_{\theta }\left(\omega \right) = \left[{\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right)\right] for all \omega \in \left[\mathit{u}, \nu \right] . Then,

    \mathcal{F}\left({\left[\frac{1}{{W}_{k}}\sum _{j = 1}^{k}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}\right)\preccurlyeq {\sum }_{j}^{k}\frac{{\eta }_{j}}{{W}_{k}}\mathcal{F}\left({\mathit{u}}_{j}\right), (41)

    where {W}_{k} = \sum _{j = 1}^{k}{\eta }_{j}. If \mathcal{F} is p -concave, then inequality (41) is reversed.

    Proof. When k = 2 then, inequality (41) is true. Consider inequality (19) is true for k = n-1, then

    \mathcal{F}\left({\left[\frac{1}{{W}_{n-1}}\sum \limits_{j = 1}^{n-1}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}\right)\preccurlyeq {\sum }_{j = 1}^{n-1}\frac{{\eta }_{j}}{{W}_{n-1}}\mathcal{F}\left({\mathit{u}}_{j}\right).

    Now, let us prove that inequality (41) holds for k = n.

    \mathcal{F}\left({\left[\frac{1}{{W}_{n}}\sum\limits _{j = 1}^{n}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}\right) = \mathcal{F}\left({\left[\frac{1}{{W}_{n}}\sum\limits _{j = 1}^{n-2}{\eta }_{j}{{\mathit{u}}_{j}}^{p}+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}(\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n-1}}^{p}+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n}}^{p}\right]}^{\frac{1}{p}}\right).

    Therefore, for each \theta \in \left[0, 1\right], we have

    \begin{array}{c}{\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{1}{{W}_{n}}\sum _{j = 1}^{n}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}, \theta \right)\\ {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{1}{{W}_{n}}\sum _{j = 1}^{n}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}, \theta \right)\end{array}
    \begin{array}{c} = {\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{1}{{W}_{n}}\sum _{j = 1}^{n-2}{\eta }_{j}{{\mathit{u}}_{j}}^{p}+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}(\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n-1}}^{p}+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n}}^{p}\right]}^{\frac{1}{p}}, \theta \right), \\ = {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{1}{{W}_{n}}\sum _{j = 1}^{n-2}{\eta }_{j}{{\mathit{u}}_{j}}^{p}+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}(\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n-1}}^{p}+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n}}^{p}\right]}^{\frac{1}{p}}, \theta \right), \end{array}
    \begin{array}{c}\le {\sum }_{j = 1}^{n-2}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n-1}}^{p}+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n}}^{p}\right]}^{\frac{1}{p}}, \theta \right), \\ \le {\sum }_{j = 1}^{n-2}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n-1}}^{p}+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{{\mathit{u}}_{n}}^{p}\right]}^{\frac{1}{p}}, \theta \right), \end{array}
    \begin{array}{c}\le {\sum }_{j = 1}^{n-2}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}\left[\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{n-1}, \theta \right)+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{n}, \theta \right)\right], \\ \le {\sum }_{j = 1}^{n-2}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)+\frac{{\eta }_{n-1}+{\eta }_{n}}{{W}_{n}}\left[\frac{{\eta }_{n-1}}{{\eta }_{n-1}+{\eta }_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{n-1}, \theta \right)+\frac{{\eta }_{n}}{{\eta }_{n-1}+{\eta }_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{n}, \theta \right)\right], \end{array}
    \begin{array}{c}\le {\sum }_{j = 1}^{n-2}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)+\left[\frac{{\eta }_{n-1}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{n-1}, \theta \right)+\frac{{\eta }_{n}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{n}, \theta \right)\right], \\ \le {\sum }_{j = 1}^{n-2}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)+\left[\frac{{\eta }_{n-1}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{n-1}, \theta \right)+\frac{{\eta }_{n}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{n}, \theta \right)\right], \end{array}
    \begin{array}{c} = {\sum }_{j = 1}^{n}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right), \\ = {\sum }_{j = 1}^{n}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right).\end{array}

    From which, we have

    \left[{\mathcal{F}}_{\mathcal{*}}\left({\left[\frac{1}{{W}_{n}}\sum \limits_{j = 1}^{n}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left({\left[\frac{1}{{W}_{n}}\sum\limits _{j = 1}^{n}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}, \theta \right)\right]{\le }_{I}\left[{\sum }_{j = 1}^{n}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right), {\sum }_{j = 1}^{n}\frac{{\eta }_{j}}{{W}_{n}}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\right],

    that is,

    \mathcal{F}\left({\left[\frac{1}{{W}_{n}}\sum \limits_{j = 1}^{n}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}\right)\preccurlyeq {\sum }_{j = 1}^{n}\frac{{\eta }_{j}}{{W}_{n}}\mathcal{F}\left({\mathit{u}}_{j}\right),

    and the result follows.

    If {\eta }_{1} = {\eta }_{2} = {\eta }_{3} = \dots = {\eta }_{k} = 1, then Theorem 4.1 reduces to the following result:

    Corollary 4.2. Let {\mathit{u}}_{j}\in \left[\mathit{u}, \nu \right], \left(j = 1, 2, 3, \dots, k, k\ge 2\right) and \mathcal{F}\in SXF\left(\left[\mathit{u}, \nu \right], {\mathbb{F}}_{0}, p\right) . Then, \theta -levels define the family of IVFs {\mathcal{F}}_{\theta }:\left[\mathit{u}, \nu \right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+} are given by {\mathcal{F}}_{\theta }\left(\omega \right) = \left[{\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right)\right] for all \omega \in \left[\mathit{u}, \nu \right] and for all \theta \in \left[0, 1\right] . Then,

    \mathcal{F}\left({\left[\frac{1}{{W}_{k}}\sum _{j = 1}^{k}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}\right)\preccurlyeq {\sum }_{J = 1}^{k}\frac{1}{k}\mathcal{F}\left({\mathit{u}}_{j}\right). (42)

    If \mathcal{F} is a p -concave, then inequality (42) is reversed.

    The next Theorem 4.3 gives the Schur's type inequality for p -convex FIVFs.

    Theorem 4.3. (Discrete Schur's type inequality for p -convex FIVF) Let \mathcal{F}\in SXF\left(\left[\mathit{u}, \nu \right], {\mathbb{F}}_{0}, p\right) . Then, \theta -levels define the family of IVFs {\mathcal{F}}_{\theta }:\left[\mathit{u}, \nu \right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+} are given by {\mathcal{F}}_{\theta }\left(\omega \right) = \left[{\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right)\right] for all \omega \in \left[\mathit{u}, \nu \right] and for all \theta \in \left[0, 1\right] . If {\mathit{u}}_{1}, {\mathit{u}}_{2}, {\mathit{u}}_{3}\in \left[\mathit{u}, \nu \right] , such that {\mathit{u}}_{1} < {\mathit{u}}_{2} < {\mathit{u}}_{3} and {{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p} , {{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}, {{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\in \left[0, 1\right] , then we have

    \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right)\mathcal{F}\left({\mathit{u}}_{2}\right)\preccurlyeq \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right)\mathcal{F}\left({\mathit{u}}_{1}\right)+\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right)\mathcal{F}\left({\mathit{u}}_{3}\right). (43)

    If \mathcal{F} is a p -concave, then inequality (43) is reversed.

    Proof. Let {\mathit{u}}_{1}, {\mathit{u}}_{2}, {\mathit{u}}_{3}\in \left[\mathit{u}, \nu \right] and {{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p} > 0. Consider \eta = \frac{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}} , then {{\mathit{u}}_{2}}^{p} = \eta {{\mathit{u}}_{1}}^{p}+\left(1-\eta \right){{\mathit{u}}_{3}}^{p}. Since \mathcal{F} is a p -convex FIVF, then by hypothesis, we have

    \mathcal{F}\left({\mathit{u}}_{2}\right)\preccurlyeq \left(\frac{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}}\right)\mathcal{F}\left({\mathit{u}}_{1}\right)+\left(\frac{{{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}}\right)\mathcal{F}\left({\mathit{u}}_{3}\right).

    Therefore, for each \theta \in \left[0, 1\right], we have

    \begin{array}{c}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{2}, \theta \right)\le \left(\frac{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\left(\frac{{{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right), \\ {\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{2}, \theta \right)\le \left(\frac{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\left(\frac{{{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}}{{{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right), \end{array} (44)
    \begin{array}{c} = \frac{\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right)}{\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right)}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\frac{\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right)}{\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right)}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right), \\ = \frac{\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right)}{\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right)}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\frac{\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right)}{\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right)}{\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right).\end{array} (45)

    From (45), we have

    \begin{array}{c}\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{2}, \theta \right)\le \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right), \\ \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{2}, \theta \right)\le \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right), \end{array}

    that is

    \left[\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{2}, \theta \right), \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{2}, \theta \right)\right]{\le }_{I}\left[\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right), \left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{1}, \theta \right)+\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{3}, \theta \right)\right],

    hence

    \left({{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{1}}^{p}\right)\mathcal{F}\left({\mathit{u}}_{2}\right)\preccurlyeq {{\mathit{u}}_{3}}^{p}-{{\mathit{u}}_{2}}^{p}\mathcal{F}\left({\mathit{u}}_{1}\right)+\left({{\mathit{u}}_{2}}^{p}-{{\mathit{u}}_{1}}^{p}\right)\mathcal{F}\left({\mathit{u}}_{3}\right).

    A refinement of Jense's type inequality for p -convex FIVF is given in the following theorem.

    Theorem 4.4. Let {\eta }_{j}\in {\mathbb{R}}^{+} , {\mathit{u}}_{j}\in \left[\mathit{u}, \nu \right], \left(j = 1, 2, 3, \dots, k, k\ge 2\right) and \mathcal{F}\in SXF\left(\left[\mathit{u}, \nu \right], {\mathbb{F}}_{0}, p\right) . Then, \theta -levels define the family of IVFs {\mathcal{F}}_{\theta }:\left[\mathit{u}, \nu \right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+} are given by {\mathcal{F}}_{\theta }\left(\omega \right) = \left[{\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(\omega, \theta \right)\right] for all \omega \in \left[\mathit{u}, \nu \right] and for all \theta \in \left[0, 1\right] . If \left(L, U\right)\subseteq [\mathit{u}, \nu] , then

    {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left({\mathit{u}}_{j}\right)\preccurlyeq {\sum }_{j = 1}^{k}\left(\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(U, \theta \right)\right), (46)

    where {W}_{k} = \sum _{j = 1}^{k}{\eta }_{j}. If \mathcal{F} is p -concave, then inequality (46) is reversed.

    Proof. Consider = {\mathit{u}}_{1}, {\mathit{u}}_{j} = {\mathit{u}}_{2}, \left(j = 1, 2, 3, \dots, k\right) , U = {\mathit{u}}_{3} . Then, by hypothesis and inequality (44), we have

    \mathcal{F}\left({\mathit{u}}_{j}\right)\le \left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\mathcal{F}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\mathcal{F}\left(U, \theta \right).

    Therefore, for each \theta \in \left[0, 1\right] , we have

    \begin{array}{c}{\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\le \left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right){\mathcal{F}}_{\mathcal{*}}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right){\mathcal{F}}_{\mathcal{*}}\left(U, \theta \right), \\ {\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\le \left(\frac{U-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right){\mathcal{F}}^{\mathcal{*}}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right){\mathcal{F}}^{\mathcal{*}}\left(U, \theta \right).\end{array}

    Above inequality can be written as,

    \begin{array}{c}\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\le \left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left(U, \theta \right), \\ \left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\le \left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left(U, \theta \right).\end{array} (47)

    Taking sum of all inequalities (47) for j = 1, 2, 3, \dots, k, we have

    \begin{array}{c}{\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\le {\sum }_{j = 1}^{k}\left(\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left(U, \theta \right)\right), \\ {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\le {\sum }_{j = 1}^{k}\left(\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left(L, \theta \right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left(U, \theta \right)\right).\end{array}

    that is

    {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left({\mathit{u}}_{j}\right) = \left[{\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right), {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left({\mathit{u}}_{j}, \theta \right)\right]
    {\le }_{I}\left[{\sum }_{j = 1}^{k}\left(\begin{array}{c}\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left(L, \theta \right)\\ +\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}_{\mathcal{*}}\left(U, \theta \right)\end{array}\right), {\sum }_{j = 1}^{k}\left(\begin{array}{c}\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left(L, \theta \right)\\ +\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right){\mathcal{F}}^{\mathcal{*}}\left(U, \theta \right)\end{array}\right)\right],
    {\le }_{I}{\sum }_{j = 1}^{k}\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\left[{\mathcal{F}}_{\mathcal{*}}\left(L, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(L, \theta \right)\right]+{\sum }_{j = 1}^{k}\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\left[{\mathcal{F}}_{\mathcal{*}}\left(U, \theta \right), {\mathcal{F}}^{\mathcal{*}}\left(U, \theta \right)\right].
    = {\sum }_{j = 1}^{k}\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(L, \theta \right)+{\sum }_{j = 1}^{k}\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(U, \theta \right).

    Thus,

    {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left({\mathit{u}}_{j}\right)\preccurlyeq {\sum }_{j = 1}^{k}\left(\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(L\right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(U\right)\right),

    this completes the proof.

    We now consider some special cases of Theorem 4.1 and 4.4.

    If {\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right) = {\mathcal{F}}_{\mathcal{*}}\left(\omega, \theta \right) with \theta = 1 , then Theorem 3.1 and 3.4 reduce to the following results:

    Corollary 4.5. [42] (Jense's inequality for p -convex function) Let {\eta }_{j}\in {\mathbb{R}}^{+} , {\mathit{u}}_{j}\in \left[\mathit{u}, \nu \right], \left(j = 1, 2, 3, \dots, k, k\ge 2\right) and let \mathcal{F}:\left[\mathit{u}, \nu \right]\to {\mathbb{R}}^{+} be a non-negative real-valued function. If \mathcal{F} is a p -convex function, then

    \mathcal{F}\left({\left[\frac{1}{{W}_{k}}\sum _{j = 1}^{k}{\eta }_{j}{{\mathit{u}}_{j}}^{p}\right]}^{\frac{1}{p}}\right)\le {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left({\mathit{u}}_{j}\right), (48)

    where {W}_{k} = \sum _{j = 1}^{k}{\eta }_{j}. If \mathcal{F} is p -concave function, then inequality (48) is reversed.

    Corollary 4.6. Let {\eta }_{j}\in {\mathbb{R}}^{+} , {\mathit{u}}_{j}\in \left[\mathit{u}, \nu \right], \left(j = 1, 2, 3, \dots, k, k\ge 2\right) and \mathcal{F}:\left[\mathit{u}, \nu \right]\to {\mathbb{R}}^{+} be an non-negative real-valued function. If \mathcal{F} is a p -convex function and {\mathit{u}}_{1}, {\mathit{u}}_{2}, \dots, {\mathit{u}}_{j}\in \left(L, U\right)\subseteq [\mathit{u}, \nu] then,

    {\sum }_{j = 1}^{k}\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left({\mathit{u}}_{j}\right)\le {\sum }_{j = 1}^{k}\left(\left(\frac{{U}^{p}-{{\mathit{u}}_{j}}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(L\right)+\left(\frac{{{\mathit{u}}_{j}}^{p}-{L}^{p}}{{U}^{p}-{L}^{p}}\right)\left(\frac{{\eta }_{j}}{{W}_{k}}\right)\mathcal{F}\left(U\right)\right), (49)

    where {W}_{k} = \sum _{j = 1}^{k}{\eta }_{j}. If \mathcal{F} is a p -concave function, then inequality (49) is reversed.

    In this we defined the p -convex (concave, affine) class for fuzzy-IVFs. We obtained some HH -inequalities for p -convex fuzzy-IVFs via fuzzy Riemann integrals. Moreover, we derived some novel discrete Jensen's and Schur's type inequalitities for p -convex fuzzy-IVFs. With the help of examples, we showed that our results include a wide class of new and known inequalities for p convex fuzzy-IVFs and their variant forms as special cases. In future, we try to explore these concepts and to investigate Jensen's and HH -inequalities for IVF and fuzzy-IVFs on time scale. In future, we will explore this by using fuzzy Katugampola fractional integrals for p -convex fuzzy-IVFs. We hope that the concepts and techniques of this paper may be starting point for further research in this area.

    The work was supported by Taif University Researches Supporting Project number (TURSP-2020/318), Taif University, Taif, Saudi Arabia.

    The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments and work was supported by Taif University Researches Supporting Project number (TURSP-2020/318), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.



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