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Weighted Ostrowski type inequalities via Montgomery identity involving double integrals on time scales

  • These authors contributed equally to this work and are co-first authors
  • In this paper, the Montgomery identity is generalized for double integrals on time scales by employing a novel analytical approach to develop the generalized Ostrowski type integral inequalities involving double integrals. Some inimitable cases are discussed for different parameters and parametric functions. Moreover, applications to some particular time scales are also presented.

    Citation: Atiqe Ur Rahman, Khuram Ali Khan, Ammara Nosheen, Muhammad Saeed, Thongchai Botmart, Nehad Ali Shah. Weighted Ostrowski type inequalities via Montgomery identity involving double integrals on time scales[J]. AIMS Mathematics, 2022, 7(9): 16657-16672. doi: 10.3934/math.2022913

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  • In this paper, the Montgomery identity is generalized for double integrals on time scales by employing a novel analytical approach to develop the generalized Ostrowski type integral inequalities involving double integrals. Some inimitable cases are discussed for different parameters and parametric functions. Moreover, applications to some particular time scales are also presented.



    In 1938, A. M. Ostrowski introduced the subsequent appealing integral-based inequality to relate the value of a function Ψ with its integral from α to β:

    Consider Ψ:[α,β]R is continuous on [α,β] and differentiable in (α,β), such that Ψ:(α,β)R is bounded in (α,β), i.e. Ψ:=supˆa(α,β)|Ψ(ˆa)|<. Subsequently for any ˆa[α,β], we have:

    |(βα)Ψ(ˆa)βαΨ(s)ds|[(βα)24+(ˆaα+β2)2]Ψ. (1.1)

    If the fraction 14 cannot be substituted by any smaller value then it leads to sharpness of the above inequality.

    Afterward, several researchers have brought in few findings through the extensions and generalizations of Inequality (1.1). Such inequalities can be employed to guesstimate the inaccuracy of approximation in integration while investigating the steadiness and consistency of statistical calculation [1].

    Stefan Hilger [2] developed the calculus of measure chains in 1988. His Ph.D. supervisor Bernd Aulbach described core contributions of this theory of Unification, Extension and Discretization. The concept of time scales is a novelty in applied sciences as well as in mathematics as it enlightens a number of indefinite points about differential equations and solutions of some fractional order differential equations, which have been proved to be inadequate for their solution. Additionally, it has enlarged its contribution to the literature with its applications in areas such as engineering, biostatistics, mathematical biology, functional spaces, optimization theory and dynamic inequalities. The theory of time scales has attracted a great attention of researchers for resolving many problems in analysis. Some dynamical inequalities on time scales can be found in [3,4,5], where researchers established convex function-based inequalities, new Hardy-type inequalities and inequalities for product of different kinds of convex functions respectively using various analytical and theoretical approach on time scales. Furthermore Hu & Wang [6] discussed dynamic inequalities on time scales with applications in permanence of predator-prey system and Saker [7] employed some dynamic inequalities of Opial-type on time scales to prove numerous results related to the spacing between successive zeros of a solution of a second order dynamic equation with a damping term. Bohner and Matthews ([8,Theorem 3.5]) characterized the following relation on Γ as a generalization of the Ostrowski Inequality (1.1) :

    Let α,β,ˆu,ˆvΓ,α<β, and Ψ:[α,β]Γ=[α,β]ΓR is differentiable. Then we have the inequality:

    |Ψ(ˆv)1βαβαΨσ(ˆu)Δ(ˆu)|Mβα(Ω2(ˆv,α)+Ω2(ˆv,β)),

    where M=supα<ˆv<β|ΨΔ(ˆv)|< and [α,β]Γ is a closed interval under Γ. If its R.H.S of Inequality (1.1) cannot be substituted by any smaller value then it leads to sharpness of the above inequality.

    Liu & Ngô ([9,Lemma 3.2]) proved the following identity: Let α,β,ˆu,ˆvΓ,α<β and Ψ:[α,β]ΓR is differentiable. Then the following relation for all η[0,1] holds.

    (1η)Ψ(ˆv)+ηΨ(α)+Ψ(β)2=1βαβαΨσ(ˆu)Δˆu+1βαβαK(ˆv,ˆu)ΨΔ(ˆv)Δˆu, (1.2)

    where,

    K(ˆu,ˆv)={ˆuα+ηβα2,ˆu[α,ˆv),ˆuβηβα2,ˆu[ˆv,β].

    By making use of (1.2), they have expanded the Inequality (1.1) by considering parameter η and developed the following Ostrowski type inequality involving parameter η ([9,Theorem 3.1]):

    Let α,β,ˆu,ˆvΓ,α<β, and that Ψ:[α,β]ΓR is differentiable function. Then the following relation holds true:

    |(1η)Ψ(ˆv)+ηΨ(α)+Ψ(β)21βαβαΨσ(ˆu)Δˆu|Mβα[Ω2(α,α+ηβα2)+Ω2(ˆv,α+ηβα2)+Ω2(ˆv,βηβα2)+Ω2(β,βηβα2)] (1.3)

    for all η[0,1], such that α+ηβα2 and βηβα2 are in Γ where ˆv[α+ηβα2,βηβα2]Γ, and M=supα<ˆv<β|ΨΔ(ˆv)|<. Its sharpness is conditioned with

    η2α(βα)+η24(βα)2α+ηβα2αˆuΔˆu.

    Here [α,β]Γ represents a closed interval on Γ.

    Xu & Fang [10,Lemma 1] investigated the following identity: Let α,β,ˆu,ˆvΓ,α<β,Ψ:[α,β]ΓR is differentiable, and ζ:[0,1][0,1] is given, then we have

    Hζ(η)2Ψ(ˆv)+ζ(η)Ψ(α)+GΨ(β)2=1βαβαΨσ(ˆu)Δˆu+1βαβαK(ˆu,ˆv)ΨΔ(ˆu)Δˆu, (1.4)

    where

    K(ˆu,ˆv)={ˆuα1,ˆu[α,ˆv),ˆuα2,ˆu[ˆv,β].

    By making use of (1.4), they proved Ostrowski Inequality [10,Theorem 1] by using a parametric function and developed the result:

    Let α,β,ˆu,ˆvΓ,α<β,Ψ:[α,β]ΓR is differentiable, and ζ:[0,1][0,1], then

    |Gζ(η)2Ψ(ˆv)+ζ(η)Ψ(α)+HΨ(β)21βββαΨΔ(ˆu)Δˆu|Mβα(Ω2(α,α1)+Ω2(ˆv,α1)+Ω2(ˆv,α2)+Ω2(β,α2)) (1.5)

    for all η[0,1] such that α1 and α2 are in Γ, and t[α1,α2]Γ, where M=supα<ˆv<β|ΨΔ(ˆv)|<. Its sharpness is conditioned with

    ζ2(η)2ζ(η)2αζ2(η)2βαα1ˆuΔˆu.

    We refer [11,12,13] for detail textual-cum-mathematical description on the weighted Ostrowski type inequalities (wOTIs) via time scales. Such as Liu, Tuna and Jiang [11] developed few wOTIs by using weighted Montgomery identity on Γ. Liu and Tuna [13] characterized wOTIs on Γ by utilizing the concept of combined dynamic derivatives on Γ. Several scholars investigated multivariate OTIs on Γ in [14,15,16]. Few authors [17,18,19,20,21,22,23,24,25,26,27,28] utilized various methods to develop OTIs on Γ for functions of two variables. Motivated by the mentioned work, we extend OTIs for bivariate functions, which can be considered as generalizations of OTIs proved by Liu & Ngô, Xu & Fang and by Dragomir et al. [29].

    (1) Time scale: It is a closed subset of the real numbers. In present study, it is denoted by the symbol Γ. Examples of time scales include Cantor set, closed intervals, Z.

    (2) Forward jump operator: For tΓ, the Forward jump operator ρ:ΓΓ is defined as

    σ(ˆv):=inf{ˆuΓ:ˆu>ˆv}.

    (3) Γk notation:

    Γk={Γ(ρ(supΓ),supΓ],supΓ<,Γ,supΓ=.

    (4) Ψσ notation: If Ψ:ΓR is a function, then we define a function Ψσ:ΓR by

    Ψσ(ˆv)=Ψ(σ(ˆv)),ˆvΓ,

    i.e., Ψσ=Ψσ.

    (5) rd-continuous function: A function ζ:ΓR is stated as rd-continuous if it is continuous and its left-sided limits exist at right-dense points and left dense points respectively in Γ. The symbol Crd stands for the family of all such functions.

    (6) The delta derivative: Let Ψ:ΓR with ˆvΓk. The term ΨΔ(ˆv) is said to be a number (if exists) if for any ϵ>0, a neighbourhood U of ˆv (i.e., U=(ˆvδ,ˆv+δ)Γ for any δ>0) such that

    |[Ψ(σ(ˆv))Ψ(ˆu)]ΨΔ(ˆv)[σ(ˆv)ˆu]|ε|σ(ˆv)ˆu|forallˆuU.

    The value ΨΔ(ˆv) is known as the delta (or Hilger) derivative of Ψ at ˆv. Furthermore, the function Ψ is said to be delta differentiable on Γk if ΨΔ(ˆv) exists ˆvΓk.

    (7) Integration by parts: If α,βΓ and Ψ,ζCrd then

    βαΨ(σ(ˆv))ζΔ(ˆv)Δˆv=(Ψζ)(β)(Ψζ)(α)βαΨΔ(ˆv)ζ(ˆv)Δˆv, (1.6)

    or

    βαΨ(ˆv)ζΔ(ˆv)Δˆv=(Ψζ)(β)(Ψζ)(α)βαΨΔ(ˆv)ζ(σ(ˆv))Δˆv. (1.7)

    (8) Polynomials on time scales: The generalized polynomials are the functions ζk,Ωk:Γ2R,kN0, defined recursively as follow: the functions Ω0 and ζ0 are ζ0(ˆv,ˆu)=Ω0(ˆv,ˆu)=1 for all ˆu,ˆvΓ, and, for given ζk and Ωk with kN0, the functions ζk+1 and Ωk+1 are ζk+1(ˆv,ˆu)=ˆvˆuζk(σ(τ),ˆu)Δτ, for all ˆu,ˆvΓ and Ωk+1(ˆv,ˆu)=ˆvˆuΩk(τ,ˆu)Δτ for all ˆu,ˆvΓ. If ΩΔk(ˆv,ˆu) represents the derivative of Ωk(ˆv,ˆu) w.r.t. ˆv for any ˆu, then ΩΔk(ˆv,ˆu)=Ωk1(ˆv,ˆu),kN,ˆvΓk. Similarly ζΔk(ˆv,ˆu)=ζk1(σ(ˆv),ˆu)kN,ˆvΓk.

    For further study of time scale calculus, readers are referred to [30,31].

    Note: In this article, several abbreviations are used in order to lighten the notation and shorten the proofs; we refer to the table at the end just after the section of conclusions. Moreover throughout the paper we consider [a,b]Γ=[a,b]Γ.

    This section presents the characterization of novel wOTIs on Γ through the generalization of Montgomery identity with parameter functions.

    Lemma 2.1. Let α,β,γ,θ,ˆu,ˆvΓ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ:[0,1][0,1]. Then we have the following identity

    {Jθγ2}βαΨσ(ˆv,ˆy)Δˆv +{ζ(η)θγ2}βαΨσ(ˆv,γ)Δˆv+{Hθγ2}βαΨσ(ˆv,θ)Δˆv+βαk1(ˆx,ˆv){θγΨΔσ(ˆv,ˆu)Δˆu}Δˆv+βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv={J2.(θγ)(βα)4}Ψ(ˆx,ˆy)+{ζ(η)J(θγ)(βα)4}{Ψ(α,ˆy)+Ψ(ˆx,γ)}+{HJ.(θγ)(βα)4}{Ψ(β,ˆy)+Ψ(ˆx,θ)}+{ζ2(η).(θγ)(βα)4}Ψ(α,γ)+{ζ(η)H.(θγ)(βα)4}{Ψ(β,γ)+Ψ(α,θ)}+{H2.(θγ)(βα)4}Ψ(β,θ), (2.1)

    where

    k1(ˆx,ˆv)={ˆvα1,ˆv[α,ˆx);ˆvα2,ˆv[ˆx,β],&k2(ˆy,ˆu)={ˆuγ1,ˆu[γ,ˆy);ˆuγ2,ˆu[ˆy,θ].

    Proof. Since

    βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv=βαk1(ˆx,ˆv)[θγk2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)Δˆu]Δˆv. (2.2)

    Denote

    I1=θγk2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)Δˆu=ˆyγ[ˆuγ1]ΨΔΔ(ˆv,ˆu)Δˆu+θˆy[ˆuγ2]ΨΔΔ(ˆv,ˆu)ΔˆuI2+I3. (2.3)

    By using (1.6), we integrate I2 to get

    I2=[ˆyγ1]ΨΔ(ˆv,ˆy)+ζ(η)θγ2)ΨΔ(ˆv,γ)ˆyγΨΔσ(ˆv,ˆu)Δˆu.

    Similarly

    I3=[θγ2]ΨΔ(ˆv,θ)[ˆyγ2]ΨΔ(ˆv,ˆy)θˆyΨΔσ(ˆv,ˆu)Δˆu.

    By using I2 and I3 in (2.3), we have

    I1=[ˆyγζ(η)θγ2ˆy+γ+Gθγ2]ΨΔ(ˆv,ˆy)+ζ(η)θγ2)ΨΔ(ˆv,γ)+[{2H}θγ2]ΨΔ(ˆv,θ)θγΨΔσ(ˆv,ˆu)Δˆu=[Jθγ2]ΨΔ(ˆv,ˆy)+ζ(η)θγ2)ΨΔ(ˆv,γ)+[Hθγ2]ΨΔ(ˆv,θ)θγΨΔσ(ˆv,ˆu)Δˆu.

    Use I1 in (2.2) to find

    βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv=[Jθγ2]I4+{ζ(η)θγ2}I5+[Hθγ2]I6βαk1(ˆx,ˆv){θγΨΔσ(ˆv,ˆu)Δˆu}Δˆv, (2.4)

    where I4=βαk1(ˆx,ˆv)ΨΔ(ˆv,ˆy)Δˆv, I5=βαk1(ˆx,ˆv)ΨΔ(ˆv,γ)Δˆv and I6=βαk1(ˆx,ˆv)ΨΔ(ˆv,θ)Δˆv.

    Now,

    I4=ˆxα[ˆvα1]ΨΔ(ˆv,ˆy)Δˆv+βˆx[ˆvα2]ΨΔ(ˆv,ˆy)ΔˆvI7+I8. (2.5)

    Use (1.6) to find

    I7={ˆxα1}Ψ(ˆx,ˆy)+ζ(η)βα2)Ψ(α,ˆy)ˆxαΨσ(ˆv,ˆy)Δˆv.

    In similar fashion,

    I8={Hβα2}Ψ(β,ˆy){ˆxα2}Ψ(ˆx,ˆy)βˆxΨσ(ˆv,ˆy)Δˆv.

    By adding I7 and I8, we have

    I4=[{J}βα2]Ψ(ˆx,ˆy)+{ζ(η)βα2}Ψ(α,ˆy)+{Hβα2}Ψ(β,ˆy)βαΨσ(ˆv,ˆy)Δˆv.

    Similar calculations for I5 and I6 give

    I5=[{J}βα2]Ψ(ˆx,γ)+{ζ(η)βα2}Ψ(α,γ)+{Hβα2}Ψ(β,γ)βαΨσ(ˆv,γ)Δˆv

    and

    I6={Jβα2}Ψ(ˆx,θ)+{ζ(η)βα2}Ψ(α,θ)+{Hβα2}Ψ(β,θ)βαΨσ(ˆv,θ)Δˆv.

    Using I4, I5 and I6 in (2.4), we have

    βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv=[Jθγ2][{Jβα2}Ψ(ˆx,ˆy)+{ζ(η)βα2}Ψ(α,ˆy)+{Hβα2}Ψ(β,ˆy)βαΨσ(ˆv,ˆy)Δˆv}]+{ζ(η)θγ2}[{Jβα2}Ψ(ˆx,γ)+{ζ(η)βα2}Ψ(α,γ)+{Hβα2}Ψ(β,γ)βαΨσ(ˆv,γ)Δˆv]+[Hθγ2][{(1ζ(η)+ζ(1η))βα2}Ψ(ˆx,θ)+ζ(η)βα2Ψ(α,θ)+{Hβα2}Ψ(β,θ)βαΨσ(ˆv,θ)Δˆv]βαk1(ˆx,ˆv){θγΨΔσ(ˆv,ˆu)Δˆu}Δˆv.

    Simplification yields

    {Jθγ2}βαΨσ(ˆv,ˆy)Δˆv+{ζ(η)θγ2}βαΨσ(ˆv,γ)Δˆv+{Hθγ2}βαΨσ(ˆv,θ)Δˆv+βαk1(ˆx,ˆv){θγΨΔσ(ˆv,ˆu)Δˆu}Δˆv+βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv={J2.(θγ)(βα)4}Ψ(ˆx,ˆy)+{ζ(η)J(θγ)(βα)4}{Ψ(α,ˆy)+Ψ(ˆx,γ)}+{HJ.(θγ)(βα)4}{Ψ(β,ˆy)+Ψ(ˆx,θ)}+{ζ2(η).(θγ)(βα)4}Ψ(α,γ)+{ζ(η)H.(θγ)(βα)4}{Ψ(β,γ)+Ψ(α,θ)}+{H2.(θγ)(βα)4}Ψ(β,θ).

    Remark 2.1. If Ψ is single valued function then Eq (2.1) coincides with [10,Lemma 1].

    Corollary 2.1. Let α,β,γ,θ,ˆu,ˆvΓ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and η[0,1]. We then have the equation

    {(1η)2(θγ)(βα)}Ψ(ˆx,ˆy)+{η(1η)(θγ)βα2}[Ψ(α,ˆy)+Ψ(β,ˆy)+Ψ(ˆx,γ)+Ψ(ˆx,θ)]+{η2.(θγ)(βα)4}[Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)]=(θγ)βαΨσ(ˆv,θ)Δˆv+βαk1(ˆx,ˆv)[θγΨΔσ(ˆv,ˆu)Δˆu]Δˆv+βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv, (2.6)

    where

    k1(ˆx,ˆv)={ˆv(α+(η)βα2),ˆv[α,ˆx),ˆv(α+(2η)βα2),ˆv[ˆx,β],k2(ˆy,ˆu)={ˆu(γ+(η)θγ2),ˆu[γ,ˆy),ˆu(γ+(2η)θγ2),ˆu[ˆy,θ].

    Proof. If we choose ζ(η)=η in Lemma 2.1, we get the required estimate.

    Remark 2.2. If Ψ is single valued function then Eq (2.6) coincides with [9,Lemma 3.2].

    Theorem 2.1. Suppose that α,β,γ,θ,ˆu,ˆvΓ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ:[0,1][0,1]. We then have the inequality

    |J2Ψ(ˆx,ˆy)+ζ(η)J{Ψ(α,ˆy)+Ψ(ˆx,γ)}+HJ{Ψ(β,ˆy)+Ψ(ˆx,θ)}+ζ2(η)Ψ(α,γ)+ζ(η)H{Ψ(β,γ)+Ψ(α,θ)}+{H2Ψ(β,θ){J2βα}βαΨσ(ˆv,ˆy)Δˆv}{ζ(η)2βα}βαΨσ(ˆv,γ)Δˆv{H2βα}βαΨσ(ˆv,θ)Δˆv|4M(βα)(θγ)H2(α,β,ˆx,α1,α2)((θγ)+H2(γ,θ,ˆy,γ1,γ2)), (2.7)

    for all η[0,1] such that α1 and α2 are in Γ1, and ˆx[α,β]Γ1, γ1 and γ2 are in Γ2, ˆy[γ,θ]Γ2, where M1=Supα<ˆv<βγ<ˆu<θ|ΨΔΔ(ˆv,ˆu)|<, M2=Supα<ˆv<βγ<ˆu<θ|ΨΔσ(ˆv,ˆu)|< and M:=Max{M1,M2}.

    Proof. By taking absolute value on both sides of (2.1), one yields

    |J2Ψ(ˆx,ˆy)+ζ(η)J{Ψ(α,ˆy)+Ψ(ˆx,γ)}+HJ{Ψ(β,ˆy)+Ψ(ˆx,θ)}+ζ2(η)Ψ(α,γ)+ζ(η)H{Ψ(β,γ)+Ψ(α,θ)}+{H2Ψ(β,θ){J2βα}βαΨσ(ˆv,ˆy)Δˆv}{ζ(η)2βα}βαΨσ(ˆv,γ)Δˆv{H2βα}βαΨσ(ˆv,θ)Δˆv|=4(βα)(θγ)|βαk1(ˆx,ˆv){θγΨΔσ(ˆv,ˆu)Δˆu}Δˆv+βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv|.

    Further we have to use |Ψ+ζ||Ψ|+|ζ|, |βαΨdˆx|βα|Ψ|dˆx & |ΨΔσ(ˆv,ˆu)|M;|ΨΔΔ(ˆv,ˆu)|M;|Ψ.ζ|=|Ψ||ζ| toestimate as following:

    |J2Ψ(ˆx,ˆy)+ζ(η)J{Ψ(α,ˆy)+Ψ(ˆx,γ)}+HJ{Ψ(β,ˆy)+Ψ(ˆx,θ)}+ζ2(η)Ψ(α,γ)+ζ(η)H{Ψ(β,γ)+Ψ(α,θ)}+{H2Ψ(β,θ){J2βα}βαΨσ(ˆv,ˆy)Δˆv}{ζ(η)2βα}βαΨσ(ˆv,γ)Δˆv{H2βα}βαΨσ(ˆv,θ)Δˆv|4M(βα)(θγ)((θγ){ˆxα|k1(ˆx,ˆv)|Δˆv+βˆx|k1(ˆx,ˆv)|Δˆv}+βα|k1(ˆx,ˆv)|{ˆyγ|k2(ˆy,ˆu)|Δˆu+θˆy|k2(ˆy,ˆu)|Δˆu}Δˆv)4M(βα)(θγ)((θγ)H2(α,β,x,α1,α2)+βα|k1(ˆx,ˆv)|H2(γ,θ,y,γ1,γ2)Δˆv).

    Simplifications give the required result.

    Corollary 2.2. If all the assumptions of Corollary 2.1 hold, then we find the following Ostrowski type inequality for parameter η

    |4(1η)2Ψ(ˆx,ˆy)+2η(1η)[Ψ(α,ˆy)+Ψ(β,ˆy)+Ψ(ˆx,γ)+Ψ(ˆx,θ)]+η2[Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)]4βαβαΨσ(ˆv,θ)Δˆv|4M(βα)(θγ)(Ω2(α,α+ηβα2)+Ω2(ˆx,α+ηβα2)+Ω2(ˆx,α+(2η)βα2)+Ω2(β,α+(2η)βα2))((θγ)+Ω2(γ,γ+ηθγ2)+Ω2(ˆy,γ+ηθγ2)+Ω2(ˆy,γ+(2η)θγ2)+Ω2(θ,(γ+(2η)θγ2)). (2.8)

    Proof. The proof is similar to proof of Theorem 2.1.

    Remark 2.3. The inequality (2.8) can be considered as extension of [9,Theorem 3.1]. Since if Ψ is single valued in (2.8), we get [9,Theorem 3.1].

    Corollary 2.3. Under the assumptions of Theorem 2.1, we have the following Ostrowski type inequality:

    |4(1η)2Ψ(ˆx,ˆy)+2η2(1η){Ψ(α,ˆy)+Ψ(ˆx,γ)}+2η(2η)(1η){Ψ(β,ˆy)+Ψ(ˆx,θ)}+η4Ψ(α,γ)+η3(2η){Ψ(β,γ)+Ψ(α,θ)}+η2(2η)2Ψ(β,θ)4βαβα{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|4M(βα)(θγ)(Ω2(α,α+η2βα2)+Ω2(ˆx,α+η2βα2)+Ω2(ˆx,α+(1+(1η)2)βα2)+Ω2(β,α+(1+(1η)2)βα2))((θγ)+Ω2(γ,γ+η2θγ2)+Ω2(ˆy,γ+η2θγ2)+Ω2(ˆy,γ+(1+(1η)2)θγ2)+Ω2(θ,(γ+(1+(1η)2)θγ2)). (2.9)

    Proof. Take ζ(η)=η2 in Theorem 2.1 to meet the requirement.

    Remark 2.4. The following Ostrowski type inequalities are obtained by choosing η=0,η=12 and η=1 in Corollary 2.3 respectively:

    (a) |Ψ(ˆx,ˆy)1βαβα{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv| M(βα)(θγ){Ω2(ˆx,α)+Ω2(ˆx,β)}[(θγ)+Ω2(ˆy,γ)+Ω2(ˆy,θ)].

    (b) |Ψ(ˆx,ˆy)+14{Ψ(α,ˆy)+Ψ(ˆx,γ)}+34{Ψ(β,ˆy)+Ψ(ˆx,θ)}+116Ψ(α,γ)+316{Ψ(β,γ)+Ψ(α,θ)}+916Ψ(β,θ)4βαβα{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|4M(βα)(θγ)

    (Ω2(α,7α+β8)+Ω2(ˆx,7α+β8)+Ω2(ˆx,3α+5β8)+Ω2(β,3α+5β8))((θγ)+Ω2(γ,7γ+θ8)+Ω2(ˆy,7γ+θ8)+Ω2(ˆy,3γ+5θ8)+Ω2(θ,3γ+5θ8)).

    (c) |Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)4βαβα{Ψσ(ˆv,ˆy)+Ψσ(ˆv,γ)+Ψσ(ˆv,θ)}Δˆv|4M(βα)(θγ)

    (Ω2(α,α+β2)+Ω2(ˆx,α+β2)+Ω2(ˆx,α+β2)+Ω2(β,α+β2))((θγ)+Ω2(γ,γ+θ2)+Ω2(ˆy,γ+θ2)+Ω2(ˆy,γ+θ2)+Ω2(θ,γ+θ2)).

    Lemma 2.2. Let α,β,γ,θ,ˆu,ˆvΓ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ1,ζ2:[0,1][0,1]. We then have the equation

    (J1.J2)Ψ(ˆx,ˆy)+(ζ1(η)J2)Ψ(α,ˆy)+(J2.H1)Ψ(β,ˆy)+(ζ2(μ).J1)Ψ(ˆx,γ)+(ζ1(η).ζ2(μ))Ψ(α,γ)+(ζ2(μ)H1)Ψ(β,γ)+(H2J1)Ψ(ˆx,θ)+(ζ1(η)H2)Ψ(α,θ)+(H1H2)Ψ(β,θ)(J22βα).βαΨσ(ˆv,ˆy)Δˆv(ζ2(μ)2βα).βαΨσ(ˆv,γ)Δˆv(H22βα)βαΨσ(ˆv,θ)Δˆv=4(βα)(θγ)[βαk1(ˆx,ˆv)(θγΨΔσ(ˆv,ˆu)Δˆu)Δˆv+βαθγk1(ˆx,ˆv)k2(ˆy,ˆu)ΨΔΔ(ˆv,ˆu)ΔˆuΔˆv], (2.10)

    where

    k1(ˆx,ˆv)={ˆve1,ˆv[α,ˆx);ˆve2,ˆv[ˆx,β].k2(ˆy,ˆu)={ˆue3,ˆu[γ,ˆy);ˆue4,ˆu[ˆy,θ].

    Proof. It can easily be proved by following the steps of Lemma 2.1.

    Remark 2.5. If ζ1(η) = ζ2(μ) in Lemma 2.2, it becomes Lemma 2.1.

    Theorem 2.2. Suppose that α,β,γ,θ,ˆu,ˆvΓ with α<β,γ<θ,Ψ:I=[α,β]Γ1×[γ,θ]Γ2R is differentiable. Assume that the delta derivatives ΨΔΔ(ˆv,ˆu) exist on I and ζ1,ζ2:[0,1][0,1]. We then have the inequality

    |{J1.J2}Ψ(ˆx,ˆy)+{ζ1(η)J2}Ψ(α,ˆy)+{J2.H1}Ψ(β,ˆy)+{ζ2(μ).J1}Ψ(ˆx,γ)+{ζ1(η).ζ2(μ)}Ψ(α,γ)+{ζ2(μ)H1}Ψ(β,γ)+{H2J1}Ψ(ˆx,θ)+{ζ1(η)H2}Ψ(α,θ)+{H1H2}Ψ(β,θ){(1ζ2(μ)+ζ2(1μ))2βα}βαΨσ(ˆv,ˆy)Δˆv}{ζ2(μ)2βα}βαΨσ(ˆv,γ)Δˆv{H22βα}βαΨσ(ˆv,θ)Δˆv|4M(βα)(θγ)H2(α,β,x,e1,e2)((θγ)+H2(γ,θ,y,e3,e4)). (2.11)

    Proof. By using Lemma 2.2 and adopting the technique of proof of Theorem 2.1, we get the desired result.

    Remark 2.6. In similar fashion, remaining results of Section 2.2 can be extended for (2.11).

    Some important examples of time scales include continuous time scale Γ=R (set of all real numbers, which gives rise to differential equations), discrete time scale Z (set of integers, which gives rise to difference equations) and quantum time Scale qN0,q>1. In this section we have discussed Ostrowski type Inequality (2.8) for these special time scales.

    Example 3.1. If we take Γ1=Γ2=R, then the delta integral is the usual Riemann integral i.e. βαΨσ(ˆv)Δˆv=βαΨ(ˆv)dˆv as σ(ˆv)=ˆv. In this case, the generalized polynomial Ω2 is

    Ω2(ˆv,ˆu)=(ˆvˆu)22forallˆu,ˆvR,

    which implies the following relations:

    Ω2(α,α+ηβα2)=(ααηβα2)22=η2(βα)28,
    Ω2(ˆx,α+ηβα2)=4(ˆxα)2+η2(βα)24η(ˆxα)(βα)8,
    Ω2(ˆx,α+(2η)βα2)=4(ˆxα)2+(2η)2(βα)24(ˆxα)(2η)(βα)8,
    Ω2(β,α+(2η)βα2)=(βα)2η28,
    Ω2(γ,γ+ηθγ2)=η2(θγ)28,
    Ω2(ˆy,γ+ηθγ2)=4(ˆyγ)2+η2(θγ)24η(ˆyγ)(θγ)8,
    Ω2(ˆy,γ+(2η)θγ2)=4(ˆyγ)2+(2η)2(θγ)24(ˆyγ)(2η)(θγ)8,
    Ω2(θ,γ+(2η)θγ2)=(θγ)2η28.

    The Eq (2.8) takes the following form

    |4(1η)2Ψ(ˆx,ˆy)+2η(1η)[Ψ(α,ˆy)+Ψ(β,ˆy)+Ψ(ˆx,γ)+Ψ(ˆx,θ)]+η2[Ψ(α,γ)+Ψ(β,γ)+Ψ(α,θ)+Ψ(β,θ)]4βαβαΨ(ˆv,θ)dˆv|M(βα)(θγ)((βα)2(η2η)+(αˆx)2+(βˆx)2)(2(θγ)+(θγ)2(η2η)+(γˆy)2+(θˆy)2). (3.1)

    Remark 3.1. If Ψ is single valued function then assumptions and calculations made in Example 3.1 coincide with [29,Theorem 2].

    Example 3.2. Using Γ1=Γ2=Z,α=0=γ,β=n,θ=m,s=j,t=i,Ψ(p,q)=ˆxpˆyq,p=k,q=l and Ψσ(p,q)=σ(ˆxpˆyq)=ˆxp+1ˆyq+1; βαΨσ(ˆv,θ)Δˆv=4nn1t=0ˆxt+1ˆym in Eq (2.8), with the known result

    Ω2(ˆv,ˆu)=(ˆvˆu)!2!(ˆvs2)!=(ˆvˆu)(ˆvs1)2=(ˆvˆu2),forallˆu,ˆvZ,

    we have

    |4(1η)2ˆxkˆyl+2η(1η)[ˆx0ˆyl+ˆxnˆyl+ˆxkˆy0+ˆxkˆym]+η2[ˆx0ˆy0+ˆxnˆy0+ˆx0ˆym+ˆxnˆym]4nn1i=0ˆxi+1ˆym|M4mn(n2(η2η+1)+2k(kn1)+n)(5m+4m2(η2η+1)+8l(lm1)).

    Example 3.3. If we take Γ1=Γ2=qN0,q>1,α=γ=qm,β=θ=qn,m<n,η=1 in Eq (2.8), we have

    |Ψ(qm,qm)+Ψ(qn,qm)+Ψ(qm,qn)+Ψ(qn,qn)4qnqmqnqmΨσ(ˆv,qn)Δˆv|M(qnqm)2.14(1+q)2((qmqn)(qm(2q)qn+1)+2(2ˆxqnqm)(2ˆxqn+1qm+1)+(qnqm)(qn(2q)qm+1))(4(qnqm)(1+q)+(qmqn)(qm(2q)qn+1)+2(2ˆyqnqm)(2ˆyqn+1qm+1)+(qnqm)(qn(2q)qm+1)).

    Remark 3.2. If Ψ is single valued then Example 3.1 to Example 3.3 coincide with [9,Corollaries 3.6–3.8]. Furthermore, it is also possible to reset Eq (2.7) instead of Eq (2.8) for these particular time scales, which will be extensions of [10,Corollary 1,2] in case of continuous and discrete time scales.

    In this study, a novel approach is employed for the establishment of Ostrowski type integral inequalities for double integrals via Montgomery identity under the setting of time scales calculus. In addition, certain generalizations are made for some weighted and parameterized functions. Moreover, some particular cases, applications and examples are discussed for some specific time scales. It is also worth mentioning that the results of the paper extend the results of [9,10,29]. Further extensions can be sought by the expansion of this proposed study for multiple integrals.

    Following notations have been used in the paper for vivid understanding of the concept:

    Notations Used For
    G 1+ζ(1η)
    H 1ζ(1η)
    J 1ζ(η)+ζ(1η)
    α1 α+ζ(η)βα2
    α2 α+Gβα2
    γ1 γ+ζ(η)θγ2
    γ2 γ+Gθγ2
    H2(α,β,x,α1,α2) Ω2(α,α1)+Ω2(ˆx,α1)+Ω2(ˆx,α2)+Ω2(β,α2)
    H2(γ,θ,y,γ1,γ2) Ω2(γ,γ1)+Ω2(ˆy,γ1)+Ω2(ˆy,γ2)+Ω2(θ,γ2)
    e1 α+ζ1(η)βα2
    e2 α+G1βα2
    e3 γ+ζ2(η)θγ2
    e4 γ+G2θγ2
    H2(α,β,x,e1,e2) Ω2(α,e1)+Ω2(ˆx,e1)+Ω2(ˆx,e2)+Ω2(β,e2)
    H2(γ,θ,y,e3,e4) Ω2(γ,e3)+Ω2(ˆy,e3)+Ω2(ˆy,e4)+Ω2(θ,e4)
    G1 1+ζ1(1η)
    G2 1+ζ2(1μ)
    H1 1ζ1(1η)
    H2 1ζ2(1μ)
    J1 1ζ1(η)+ζ1(1η)
    J2 1ζ2(μ)+ζ2(1μ).

     | Show Table
    DownLoad: CSV

    This research received funding support from the NSRF via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation (grant number B05F640088).

    The authors declare that there is no conflict of interests regarding the publication of this paper.



    [1] S. S. Dragomir, T. M. Rassias, Ostrowski type inequalities and applications in numerical integration, Springer Dordrecht, 2002. http://dx.doi.org/10.1007/978-94-017-2519-4
    [2] S. Hilger, Ein Mabkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten [thesis], Universitat Wurzburg, 1988.
    [3] A. Ekinci, Inequalities for convex functions on time scales, TWMS J. Appl. Eng. Math., 9 (2019), 64–72.
    [4] B. Benaissa, M. Z. Sarikaya, A. Senouci, On some new Hardy-type inequalities, Math. Method. Appl. Sci., 43 (2020), 8488–8495, http://dx.doi.org/10.1002/mma.6503 doi: 10.1002/mma.6503
    [5] M. Gürbüz, M. E. Özdemír, On some inequalities for product of different kinds of convex functions, Turk. J. Sci., 5 (2020), 23–27.
    [6] M. Hu, L. Wang, Dynamic inequalities on time scales with applications in permanence of predator-prey system, Discrete Dyn. Nat. Soc., 2012 (2012), 281052. http://dx.doi.org/10.1155/2012/281052 doi: 10.1155/2012/281052
    [7] S. H. Saker, Applications of Opial inequalities on time scales on dynamic equations with damping terms, Math. Comput. Model., 58 (2013), 1777–1790, http://dx.doi.org/10.1016/j.mcm.2013.04.006 doi: 10.1016/j.mcm.2013.04.006
    [8] M. Bohner, T. Matthews, Ostrowski inequalities on time scales, J. Inequal. Pure Appl. Math., 9 (2008), 8.
    [9] W. J. Liu, Q. A. Ngô, W. Chen, A new generalization of Ostrowski type inequality on time scales, An. St. Univ. Ovidius Constanta, 17 (2009), 101–114, http://dx.doi.org/10.48550/arXiv.0804.4310 doi: 10.48550/arXiv.0804.4310
    [10] G. Xu, Z. B. Fang, A new Ostrowski type inequality on time scales, J. Math. Inequal., 10 (2016), 751–760, http://dx.doi.org/10.7153/jmi-10-61 doi: 10.7153/jmi-10-61
    [11] W. Liu, A. Tuna, Y. Jiang, On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski-Grüss like inequalities on time scales, Appl. Anal., 93 (2014), 551–571, https://doi.org/10.1080/00036811.2013.786045 doi: 10.1080/00036811.2013.786045
    [12] W. Liu, A. Tuna, Y. Jiang, New weighted Ostrowski and Ostrowski-Grüss type inequalities on time scales, Annals of the Alexandru Ioan Cuza University-Mathematics, 60 (2014), 57–76, http://dx.doi.org/10.2478/aicu-2013-0002 doi: 10.2478/aicu-2013-0002
    [13] W. Liu, A. Tuna, Diamond-α weighted Ostrowski type and Grüss type inequalities on time scales, Appl. Math. Comput., 270 (2015), 251–260, http://dx.doi.org/10.1016/j.amc.2015.06.132 doi: 10.1016/j.amc.2015.06.132
    [14] G. A. Anastassiou, Representations and Ostrowski type inequalities on time scales, Comput. Math. Appl., 62 (2011), 3933–3958, http://dx.doi.org/10.1016/j.camwa.2011.09.046 doi: 10.1016/j.camwa.2011.09.046
    [15] Q. Feng, F. Meng, Generalized Ostrowski type inequalities for multiple points on time scales involving functions of two independent variables, J. Inequal. Appl., 2012 (2012), 74, http://dx.doi.org/10.1186/1029-242x-2012-74 doi: 10.1186/1029-242x-2012-74
    [16] B. Zheng, Q. Feng, Generalized dimensional Ostrowski type and Grüss type inequalities on time scales, J. Appl. Math., 2014 (2014), 434958. http://dx.doi.org/10.1155/2014/434958 doi: 10.1155/2014/434958
    [17] B. Meftah, B. Khaled, Some New Ostrowski type inequalities on time scales for functions of two independent variables, J. Interdiscip. Math., 20 (2017), 397–415, http://dx.doi.org/10.1080/09720502.2015.1026463 doi: 10.1080/09720502.2015.1026463
    [18] A. A. El-Deeb, H. A. Elsennary, E. R. Nwaeze, Generalized weighted Ostrowski, trapezoid and Grüss type inequalities on time scales, Fasciculi Math., 60 (2018), 123–144, http://dx.doi.org/10.1515/fascmath-2018-0008 doi: 10.1515/fascmath-2018-0008
    [19] G. Xu, Z. B. Fang, A Generalization of Ostrowski type inequality on time scales with k points, J. Math. Inequal., 11 (2017), 41–48, http://dx.doi.org/10.7153/jmi-11-04 doi: 10.7153/jmi-11-04
    [20] S. Kermausuor, E. R. Nwaeze, New Generalized 2D Ostrowski type inequalities on time scales with k2 points using a parameter, Filomat, 32 (2018), 3155–3169, http://dx.doi.org/10.2298/fil1809155k doi: 10.2298/fil1809155k
    [21] S. Kermausuor, E. R. Nwaeze, Ostrowski-Grüss type inequalities and a 2D Ostrowski type inequality on time scales involving a combination of Δ-integral means, Kragujev. J. Math., 44 (2020), 127–143, http://dx.doi.org/10.46793/kgjmat2001.127k doi: 10.46793/kgjmat2001.127k
    [22] H. Budak, E. Pehlivan, Weighted Ostrowski, trapezoid and midpoint type inequalities for RiemannLiouville fractional integrals, AIMS Math., 5 (2020), 1960–1984, http://dx.doi.org/10.3934/math.2020131 doi: 10.3934/math.2020131
    [23] A. Tuna, E. R. Nwaeze, Ostrowski and generalized Trapezoid type inequalities on time scales, J. King Saud Univ. Sci., 32 (2020), 496–500, http://dx.doi.org/10.1016/j.jksus.2018.04.011 doi: 10.1016/j.jksus.2018.04.011
    [24] A. Tuna, A new generalization of Ostrowski type inequalities on arbitrary time scale, TWMS J. App. Eng. Math., 9 (2019), 172–185.
    [25] S. Fatima Tahir, M. Mushtaq, M. Muddassar, A note on integral inequalities on time scales associated with Ostrowski's type, J. Funct. Space., 2019 (2019), 4748373. http://dx.doi.org/10.1155/2019/4748373 doi: 10.1155/2019/4748373
    [26] A. A. El-Deeb, S. Rashid, On some new double dynamic inequalities associated with Leibniz integral rule on time scales, Adv. Differ. Equ., 2021 (2021), 125. http://dx.doi.org/10.1186/s13662-021-03282-3 doi: 10.1186/s13662-021-03282-3
    [27] R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl., 4 (2001), 535–558, http://dx.doi.org/10.7153/mia-04-48 doi: 10.7153/mia-04-48
    [28] R. Hilscher, A time scales version of a Wirtinger-type inequality and applications, J. Comput. Appl. Math., 141 (2002), 219–226, http://dx.doi.org/10.1016/S0377-0427(01)00447-2 doi: 10.1016/S0377-0427(01)00447-2
    [29] S. S. Dragomir, P. Cerone, J. Roumelitis, A new generalization of Ostrowski integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett., 13 (2000), 19–25. https://doi.org/10.1016/S0893-9659(99)00139-1 doi: 10.1016/S0893-9659(99)00139-1
    [30] M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Boston: Birkhäuser, 2001, http://dx.doi.org/10.1007/978-1-4612-0201-1
    [31] M. Bohner, G. Guseinov, A. Peterson, Introduction to the time scales calculus, In: Advances in dynamic equations on time scales, Boston: Birkhäuser, 2003, http://dx.doi.org/10.1007/978-0-8176-8230-9_1
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