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Research article Special Issues

Weighted generalized Quasi Lindley distribution: Different methods of estimation, applications for Covid-19 and engineering data

  • Received: 16 June 2021 Accepted: 25 July 2021 Published: 16 August 2021
  • MSC : 62E15, 60E05, 62F10

  • Recently, a new lifetime distribution known as a generalized Quasi Lindley distribution (GQLD) is suggested. In this paper, we modified the GQLD and suggested a two parameters lifetime distribution called as a weighted generalized Quasi Lindley distribution (WGQLD). The main mathematical properties of the WGQLD including the moments, coefficient of variation, coefficient of skewness, coefficient of kurtosis, stochastic ordering, median deviation, harmonic mean, and reliability functions are derived. The model parameters are estimated by using the ordinary least squares, weighted least squares, maximum likelihood, maximum product of spacing's, Anderson-Darling and Cramer-von-Mises methods. The performances of the proposed estimators are compared based on numerical calculations for various values of the distribution parameters and sample sizes in terms of the mean squared error (MSE) and estimated values (Es). To demonstrate the applicability of the new model, four applications of various real data sets consist of the infected cases in Covid-19 in Algeria and Saudi Arabia, carbon fibers and rain fall are analyzed for illustration. It turns out that the WGQLD is empirically better than the other competing distributions considered in this study.

    Citation: SidAhmed Benchiha, Amer Ibrahim Al-Omari, Naif Alotaibi, Mansour Shrahili. Weighted generalized Quasi Lindley distribution: Different methods of estimation, applications for Covid-19 and engineering data[J]. AIMS Mathematics, 2021, 6(11): 11850-11878. doi: 10.3934/math.2021688

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  • Recently, a new lifetime distribution known as a generalized Quasi Lindley distribution (GQLD) is suggested. In this paper, we modified the GQLD and suggested a two parameters lifetime distribution called as a weighted generalized Quasi Lindley distribution (WGQLD). The main mathematical properties of the WGQLD including the moments, coefficient of variation, coefficient of skewness, coefficient of kurtosis, stochastic ordering, median deviation, harmonic mean, and reliability functions are derived. The model parameters are estimated by using the ordinary least squares, weighted least squares, maximum likelihood, maximum product of spacing's, Anderson-Darling and Cramer-von-Mises methods. The performances of the proposed estimators are compared based on numerical calculations for various values of the distribution parameters and sample sizes in terms of the mean squared error (MSE) and estimated values (Es). To demonstrate the applicability of the new model, four applications of various real data sets consist of the infected cases in Covid-19 in Algeria and Saudi Arabia, carbon fibers and rain fall are analyzed for illustration. It turns out that the WGQLD is empirically better than the other competing distributions considered in this study.



    In recent years, a useful extension has been proposed from the classical calculus by permitting derivatives and integrals of arbitrary orders is known as fractional calculus. It emerged from a celebrated logical conversation between Leibniz and L'Hopital in 1695 and was enhanced by different scientists like Laplace, Abel, Euler, Riemann, and Liouville [1]. Fractional calculus has gained popularity on the account of diverse applications in various areas of science and technology [2,3,4]. The concept of this new calculus was applied in several distinguished areas previously with excellent developments in the frame of novel approaches and posted scholarly papers, see [5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Various notable generalized fractional integral operators such as the Riemann-Liouville, Hadamard, Caputo, Marichev-Saigo-Maeda, Riez, the Gaussian hypergeometric operators and so on, their attempts helpful for researchers to recognize the real world phenomena. Therefore, the Caputo and Riemann-Liouville was the most used fractional operators having singular kernels. It is remarkable that all the above mentioned operators are the particular cases of the operators investigated by Jarad et al. [19]. The utilities to weighted generalized fractional operators are undertaking now.

    Adopting the excellency of the above work, we introduce a new weighted framework of generalized proportional fractional integral operator with respect to monotone function Ψ. Also, some new characteristics of the aforesaid operator are apprehended to explore new ideas to amplify the fractional operators and acquire fractional integral inequalities via generalized fractional operators (see Remark 2 and 3 below).

    Recently, by employing the fractional integral operators, several researchers have established a bulk of fractional integral inequalities and their variant forms with fertile applications. These sorts of speculations have noteworthy applications in fractional differential/difference equations and fractional Schrödinger equations [20,21]. By the use of Riemann-Liouville fractional integral operator, Belarbi and Dahmani [22] contemplated the subsequent integral inequalities as follows:

    If f1 and g1 are two synchronous functions on [0,), then

    Ωα(f1g1)(ϰ)Γ(α+1)ϰαΩα(f1)(ϰ)Ωα(g1)(ϰ) (1.1)

    and

    ϰαΓ(α+1)Ωβ(f1g1)(ϰ)+ϰβΓ(β+1)Ωα(f1g1)(ϰ)Ωα(f1)(ϰ)Ωβ(g1)(ϰ)+Ωβ(f1)(ϰ)Ωα(g1)(ϰ), (1.2)

    for all ϰ>0,α,β>0. Butt et al. [23], Rashid et al. [24] and Set et al. [25] established the fractional integral inequalities via generalized fractional integral operator having Raina's function, generalized K-fractional integral and Katugampola fractional integral inequalities similar to the variants (1.1) and (1.2), respectively. Here we should emphasize that, inequalities (1.1) and (1.2) are a remarkable instrument for reconnoitering plentiful scientific regions of investigation encompassing probability theory, statistical analysis, physics, meteorology, chaos and henceforth.

    More general version of inequalities (1.1) and (1.2) proposed by Dahmani [26] by employing Riemann-Liouville fractional integral operator.

    Let f1 and g1 be two synchronous functions on [0,) and let r,s:[0,)[0,). Then

    ΩαP(ϰ)Ωα(Qf1g1)(ϰ)+ΩαQ(ϰ)Ωα(Pf1g1)(ϰ)Ωα(Qf1)(ϰ)Ωα(Pg1)(ϰ)+Ωα(Pf1)(ϰ)Ωα(Qg1)(ϰ) (1.3)

    and

    ΩαP(ϰ)Ωβ(Qf1g1)(ϰ)+ΩβQ(ϰ)Ωα(Pf1g1)(ϰ)Ωα(Qf1)(ϰ)Ωβ(Pg1)(ϰ)+Ωβ(Pf1)(ϰ)Ωα(Qg1)(ϰ) (1.4)

    for all ϰ>0,α,β>0. Chinchane and Pachpatte [27], Brahim and Taf [28] and Shen et al. [29] explored the Hadamard fractional integral inequalities, the fractional version of integral inequalities in two variable quantum deformation and the Riemann-Liouville fractional integral operator on time scale analysis coincide to variants (1.3) and (1.4), respectively.

    Let us define the most distinguished Chebyshev functional [30]:

    T(f1,g1)=1b1a1b1a1f1(ϰ)g1(ϰ)dϰ1b1a1b1a1f1(ϰ)dϰ1b1a1b1a1g1(ϰ)dϰ, (1.5)

    where f1 and g1 are two integrable functions on [a1,b1]. In [31], Grüss proposed the well-known generalization:

    |T(f1,g1)|14(Φϕ)(Υγ), (1.6)

    where f1 and g1 are two integrable functions on [a1,b1] satisfying the assumptions

    ϕf1(ϰ)Φ,γg1(ϰ)Υ,ϕ,Φ,γ,ΥR,ϰ[a1,b1]. (1.7)

    The inequality (1.6) is known to be Grüss inequality. In recent years, the Grüss type integral inequality has been the subject of very active research. Mathematicians and scientists can see them in research papers, monographs, and textbooks devoted to the theory of inequalities [32,33,34,35] such as, Dragomir [36] demonstrated certain variants with the supposition of vectors and continuous mappings of selfadjoint operators in Hilbert space similar to (1.6). In this context, f1 and g1 are holding the assumptions (1.7), Dragomir [37] derived several functionals in two and three variable sense as follows:

    |S(f1,g1,P)|14(Φϕ)(Υγ)(b1a1P1(ϰ)dϰ)2, (1.8)

    where

    S(f1,g1,P)=12T(f1,g1,P)=b1a1P(ϰ)dϰb1a1P(ϰ)f1(ϰ)g1(ϰ)dϰb1a1P(ϰ)f1(ϰ)dϰb1a1P(ϰ)g1(ϰ)dϰ (1.9)

    and

    T(f1,g1,P,Q)=b1a1Q(ϰ)dϰb1a1P(ϰ)f1(ϰ)g1(ϰ)dϰ+b1a1P(ϰ)dϰb1a1Q(ϰ)f1(ϰ)g1(ϰ)dϰb1a1Q(ϰ)f1(ϰ)dϰb1a1P(ϰ)g1(ϰ)dϰb1a1P(ϰ)f1(ϰ)dϰb1a1Q(ϰ)g1(ϰ)dϰ. (1.10)

    In [37], Dragomir established the inequality:

    If f1,g1L(a1,b1), then

    |S(f1,g1,P)|f1g1(b1a1P(ϰ)dϰb1a1ϰ2P(ϰ)dϰ(b1a1ϰP(ϰ)dϰ)2). (1.11)

    Moreover, author [37] proved numerous variants for Lipschitzian functions as follows:

    If f1 is L-g1-Lipschitzian on [a1,b1], that is

    |f1(μ)fν|L|g1(μ)g1(ν)|,L>0,μ,ν[a1,b1]. (1.12)

    and

    |S(f1,g1,P)|L(b1a1P(ϰ)dϰb1a1g21(ϰ)P(ϰ)dϰ(b1a1g1(ϰ)P(ϰ)dϰ)2). (1.13)

    Furthermore, if f1 and g1 are L1 and L2-Lipschitzian functions on [a1,b1], then

    |S(f1,g1,P)|L1L2(b1a1P(ϰ)dϰb1a1ϰ2P(ϰ)dϰ(b1a1ϰP(ϰ)dϰ)2). (1.14)

    Owing to the above tendency, Dhamani et al. [38] proposed the fractional integral inequalities in the Riemann-Liouville parallel to variant (1.6) with the suppositions (1.7). Additionally, Dahamani and Benzidane [39] introduced weighted Grüss type inequality via (α,β)-fractional q-integral inequality resemble to (1.8) under the hypothesis of (1.5). Author [40,41] derived the extended functional of (1.10) by employing Riemann-Liouville integral corresponds to variants (1.11), (1.13) and (1.14), respectively. In this flow, Set et al. [42] contemplated the Grüss type inequalities considering the generalized K-fractional integral. Chen et al. [43] obtained the novel refinements of Hermite-Hadamard type inequalities for n-polynomial p-convex functions within the generalized fractional integral operators. Abdeljawad et al. [44] derived the Simpson's type inequalities for generalized p-convex functions involving fractal set. Jarad et al. [45] investigated the properties of the more general form of generalized proportional fractional operators in Laplace transforms.

    The motivation of this paper is twofold. First, we propose a novel framework named weighted generalized proportional fractional integral operator based on characteristics, as well as considering the boundedness and semi-group property and able to be widely applied to many scientific results. Second, the current operator employed to the extended weighted Chebyshev and Grüss type inequalities for exploring the analogous versions of (1.5) and (1.6). Some special cases are pictured with new fractional operators which are not computed yet. Interestingly, particular cases are designed for Riemann-Liouville fractional integral, generalized Riemann-Liouville fractional integral and generalized proportional fractional integral inequalities. It is worth mentioning that these operators have the ability to recapture several generalizations in the literature by considering suitable assumptions of Ψ,ω and ρ.

    In this section, we demonstrate the space where the weighted fractional integrals are bounded and also, provide certain specific features of these operators.

    Definition 2.1 ([19])Let ω0 be a mapping defined on [a1,b1], g1 is a differentiable strictly increasing function on [a1,b1]. The space χpω(a1,b1),1p< is the space of all Lebesgue measurable functions f1 defined on [a1,b1] for which f1χpω, where

    f1χpω=(b1a1|ω(ϰ)f1(ϰ)|pg1(ϰ)dϰ)1p,1<p< (2.1)

    and

    f1χpω=esssupa1ϰb1|ω(ϰ)f1(ϰ)|<. (2.2)

    Remark 1. Clearly we see that f1χpω(a1,b1) ω(ϰ)f1(ϰ)(g11(ϰ))1/pLp(a1,b1) for 1p< and f1χpω(a1,b1) ω(ϰ)f1(ϰ)L(a1,b1).

    Now, we show a novel fractional integral operator which is known as the weighted generalized proportional fractional integral operator with respect to monotone function Ψ.

    Definition 2.2. Let f1χpω(a1,b1) and ω0 be a function on [a1,b1]. Also, assume that Ψ is a continuously differentiable function on [a1,b1] with ψ>0 on [a1,b1]. Then the left and right-sided weighted generalized proportional fractional integral operator with respect to another function Ψ of order α>0 are described as:

    ΨωΩρ;αa1f1(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αf1(μ)ω(μ)Ψ(μ)dμ,a1<ϰ (2.3)

    and

    ΨωΩρ;αb1f1(ϰ)=ω1(ϰ)ραΓ(α)b1ϰexp[ρ1ρ(Ψ(μ)Ψ(ϰ))](Ψ(μ)Ψ(ϰ))1αf1(μ)ω(μ)Ψ(μ)dμ,ϰ<b1, (2.4)

    where ρ(0,1] is the proportionality index, αC,(α)>0 and Γ(ϰ)=0μϰ1eμdμ is the Gamma function.

    Remark 2. Some particular fractional operators are the special cases of (2.3) and (2.4).

    (1) Setting Ψ(ϰ)=ϰ, in Definition (2.2), then we get the weighted generalized proportional fractional operators stated as follows:

    ωΩρ;αa1f1(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(ϰμ)](ϰμ)1αf1(μ)ω(μ)dμ,a1<ϰ (2.5)

    and

    ωΩρ;αb1f1(ϰ)=ω1(ϰ)ραΓ(α)b1ϰexp[ρ1ρ(μϰ)](μϰ)1αf1(μ)ω(μ)dμ,ϰ<b1. (2.6)

    (2) Setting Ψ(ϰ)=ϰ and ρ=1 in Definition (2.2), then we get the weighted Riemann-Liouville fractional operators stated as follows:

    ωΩαa1f1(ϰ)=ω1(ϰ)Γ(α)ϰa1f1(μ)ω(μ)dμ(ϰμ)1α,a1<ϰ (2.7)

    and

    ωΩαb1f1(ϰ)=ω1(ϰ)Γ(α)b1ϰf1(μ)ω(μ)dμ(μϰ)1α,ϰ<b1. (2.8)

    (3) Setting Ψ(ϰ)=lnϰ and a1>0 in Definition (2.2), we get the weighted generalized proportional Hadamard fractional operators stated as follows:

    ωΩρ;αa1f1(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(lnϰμ)](lnϰμ)1αf1(μ)ω(μ)μdμ,a1<ϰ (2.9)

    and

    ωΩρ;αb1f1(ϰ)=ω1(ϰ)ραΓ(α)b1ϰexp[ρ1ρ(lnμϰ)](lnμϰ)1αf1(μ)ω(μ)μdμ,ϰ<b1. (2.10)

    (4) Setting Ψ(ϰ)=lnϰ and a1>0 along with ρ=1 in Definition (2.2), then we get the weighted Hadamard fractional operators stated as follows:

    ωΩαa1f1(ϰ)=ω1(ϰ)Γ(α)ϰa1f1(μ)ω(μ)dμμ(lnϰμ)1α,a1<ϰ (2.11)

    and

    ωΩαb1f1(ϰ)=ω1(ϰ)Γ(α)b1ϰf1(μ)ω(μ)dμμ(lnμϰ)1α,ϰ<b1. (2.12)

    (5) Setting Ψ(ϰ)=ϰττ(τ>0) in Definition (2.2), then we get the weighted generalized fractional operators in terms of Katugampola stated as follows:

    ωΩαa1f1(ϰ)=ω1(ϰ)Γ(α)ϰa1(ϰτμττ)α1f1(μ)ω(μ)dμμ1τ,a1<ϰ (2.13)

    and

    ωΩαb1f1(ϰ)=ω1(ϰ)Γ(α)b1ϰ(μτϰττ)α1f1(μ)ω(μ)dμμ1τ,ϰ<b1. (2.14)

    Remark 3. Several existing integral operators can be derived from Definition 2.2 as follows:

    (1) Letting ω(ϰ)=1, then we get the Definition 4 proposed by Rashid et al. [46] and Definition 3.2 introduced by Jarad et al. [47], independently.

    (2) Letting ω(ϰ)=1,Ψ(ϰ)=ϰ, then we get the Definition 3.4 defined by Jarad et al. [48].

    (3) Letting ω(ϰ)=1 and Ψ(ϰ)=lnϰ along with a1>0, then we get the Definition 2.1 defined by Rahman et al. [49].

    (4) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=lnϰ along with a1>0, then we get the operator defined by Kilbas et al. [3] and Smako et al. [5], respectively.

    (5) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=ϰ, then we get the operator defined by Kilbas et al [3].

    (6) Letting ω(ϰ)=1 and Ψ(ϰ)=ϰττ,(τ>0), then we get the operator defined by Katugampola et al. [7].

    (7) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=ϰτ+sτ+s,τ(0,1],sR, then we get the Definition 2 defined by Khan and Khan et al [50].

    (8) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=(ϰa1)ττ, and Ψ(ϰ)=(b1ϰ)ττ,(τ>0), then we get the operator defined by Jarad et al. [51].

    Theorem 2.3. For α>0,ρ(0,1],1p and f1χpω(a1,b1). Then ΨωΩρ;αa1 is bounded in χpω(a1,b1) and

    ΨωΩρ;αa1f1χpω(Ψ(b1)Ψ(a1))αf1χpωραΓ(α+1).

    Proof. For 1p, we have

    ΨωΩρ;αa1f1χpω=1ραΓ(α)(b1a1|ϰa1exp[ρ1ρΨ(ϰ)Ψ(μ)](Ψ(ϰ)Ψ(μ))1αω(μ)f1(μ)Ψ(μ)dμ|pΨ(ϰ)dϰ)1/p=1ραΓ(α)(Ψ(b1)Ψ(a1)|t2Ψ(a1)exp[ρ1ρ(t2t1)](t2t1)1αω(Ψ1(t1))f1(Ψ1(t1))|pdt2)1/p.

    Using the fact that |exp[ρ1ρ(t2t1)]|<1. Taking into account the generalized Minkowski inequality [5], we can write

    ΨωΩρ;αa1f1χpω1ραΓ(α)Ψ(b1)Ψ(a1)(|ω(Ψ1(t1))f1(Ψ1(t1))|pΨ(b1)t1(t2t1)p(α1)dt2)1/pdt1=1ραΓ(α)Ψ(b1)Ψ(a1)(|ω(Ψ1(t1))f1(Ψ1(t1))|((Ψ(b1)t1)p(α1)+1p(α1)+1)1/pdt1.

    By employing the well-known Hölder inequality satisfying p1+q1=1, we obtain

    ΨωΩρ;αa1f1χpω1ραΓ(α)(Ψ(b1)Ψ(a1)|ω(Ψ1(t1))f1(Ψ1(t1))|pdt1)1/p(Ψ(b1)Ψ(a1)((Ψ(b1)t1)p(α1)+1p(α1)+1)q/pdt1)1/q1ραΓ(α)(b1a1|ω(ϰ)f1(ϰ)|pΨ(ϰ)dϰ)1/p(Ψ(b1)Ψ(a1)((Ψ(b1)t1)p(α1)+1p(α1)+1)q/pdt1)1/q(Ψ(b1)Ψ(a1))αf1χpωραΓ(α+1).

    Now, for p=, we have

    |ω(ϰ)ΨωΩρ;αa1f1(ϰ)|=1ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αf1(μ)ω(μ)Ψ(μ)dμ1ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1α|f1(μ)ω(μ)|Ψ(μ)dμ,Since(|exp[ρ1ρ(t2t1)]|<1)f1χωραΓ(α)ϰa1(Ψ(ϰ)Ψ(μ))α1dμ(Ψ(ϰ)Ψ(a1))αf1χωραΓ(α+1)=(Ψ(b1)Ψ(a1))αf1χωραΓ(α+1).

    This ends the proof.

    Our next result is the semi group property for weighted generalized proportional fractional integral operator with respect to monotone function.

    Theorem 2.4. For α,β>0,ρ(0,1] with 1p and let f1χpω(a1,b1). Then

    (ΨωΩρ;αa1ΨωΩρ;βa1)f1=(ΨωΩρ;α+βa1)f1. (2.15)

    Proof.

    (ΨωΩρ;αa1ΨωΩρ;βa1f1)(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αω(μ)(ΨωΩρ;βa1f1)(μ)Ψ(μ)dμ=ω1(ϰ)ρα+βΓ(α)Γ(β)ϰa1μa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αexp[ρ1ρ(Ψ(μ)Ψ(ν))](Ψ(μ)Ψ(ν))1β×ω(ν)f1(ν)Ψ(ν)Ψ(μ)dμdν.

    By making change of variable technique θ=Ψ(μ)Ψ(a1)Ψ(ϰ)Ψ(a1), we can write

    (ΨωΩρ;αa1ΨωΩρ;βa1f1)(ϰ)=ω1(ϰ)ρα+βΓ(α)Γ(β)10θβ1(1θ)α1dθϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(ν))](Ψ(ϰ)Ψ(ν))1αβω(ν)f1(ν)Ψ(ν)dν=ω1(ϰ)ρα+βΓ(α)Γ(β)Γ(α)Γ(β)Γ(α+β)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(ν))](Ψ(ϰ)Ψ(ν))1αβω(ν)f1(ν)Ψ(ν)dν=(ΨωΩρ;α+βa1f1)(ϰ),

    where B(α,β)=Γ(α)Γ(β)Γ(α+β)=10θβ1(1θ)α1dθ is known to be Euler Beta function.

    This section contains some significant generalizations for weighted integral inequalities by employing weighted generalized proportional fractional integral operator, for the consequences relating to (1.1) and (1.2), it is suppose that all mappings are integrable in the Riemann sense.

    Throughout this investigation, we use the following assumptions:

    I. Let f1 and g1 be two synchronous functions on [0,).

    II. Let Ψ:[0,)(0,) is an increasing function with continuous derivative Ψ on the interval (0,).

    Lemma 3.1. If the supposition \boldsymbol{I} and \boldsymbol{II} are satisfied and let \mathcal{Q} and \mathcal{P} be two non-negative continuous mappings on [0, \infty). Then the inequality

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray} (3.1)

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Since f_{1} and g_{1} are two synchronous functions on [0, \infty) , then for all \mu > 0 and \nu > 0, we have

    \begin{eqnarray} \big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big)\geq0. \end{eqnarray} (3.2)

    By (3.2), we write

    \begin{eqnarray} f_{1}(\mu)g_{1}(\mu)+ f_{1}(\nu)g_{1}(\nu)\geq g_{1}(\mu)f_{1}(\nu)+g_{1}(\nu)f_{1}(\mu). \end{eqnarray} (3.3)

    If we multiply both sides of (3.3) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{Q}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}} and integrating the resulting inequality with respect to \mu from 0 to \varkappa, we get

    \begin{eqnarray} &&\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{Q}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}(\mu)g_{1}(\mu)d\mu\\&&\quad+ \frac{f_{1}(\nu)g_{1}(\nu)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{Q}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}d\mu\\&&\geq \frac{f_{1}(\nu)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{Q}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}g_{1}(\nu)d\nu\\&&\quad+\frac{g_{1}(\nu)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{Q}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}(\mu)d\mu. \end{eqnarray} (3.4)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we have

    \begin{eqnarray} \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+f_{1}(\nu)g_{1}(\nu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\geq g_{1}(\nu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+f_{1}(\nu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa). \end{eqnarray} (3.5)

    Further, if we multiply both sides of (3.5) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]\mathcal{P}(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} and integrating the resulting inequality with respect to \nu from 0 to \varkappa. Then, multiplying by \omega^{-1}(\varkappa) and in view of Definition 2.2, we obtain

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray} (3.6)

    which implies (3.1).

    Theorem 3.2. Under the assumption of \boldsymbol{I}, \boldsymbol{II} and let r, s and t be three non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray} &&2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big)\\&&\quad+2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\\&&\geq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big) \\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big)\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray} (3.7)

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. By means of Lemma 3.1 and setting \mathcal{P} = r, \, \mathcal{Q} = s, we can write

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa). \end{eqnarray} (3.8)

    Conducting product both sides of (3.8) by \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho; \alpha}r(\varkappa), we obtain

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big). \end{eqnarray} (3.9)

    By means of Lemma 3.1 and setting \mathcal{P} = r, \, \mathcal{Q} = t, we can write

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa). \end{eqnarray} (3.10)

    Conducting product of (3.10) by \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho; \alpha}s(\varkappa), we obtain

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big). \end{eqnarray} (3.11)

    By similar argument as we did before, yields

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\Big). \end{eqnarray} (3.12)

    Adding (3.9), (3.11) and (3.12), we get the desired inequality (3.8).

    Lemma 3.3. Under the assumption of \boldsymbol{I}, \boldsymbol{II} and let \mathcal{Q} and \mathcal{P} be two non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray*} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\mathcal{P}f_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\mathcal{Q}(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\mathcal{P}(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1}g_{1})(\varkappa)\nonumber\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\mathcal{P}f_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\mathcal{P}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1})(\varkappa), \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. If we multiply both sides of (3.2) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]\mathcal{Q}(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\beta}\Gamma(\beta)(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}} and integrating the resulting inequality with respect to \nu from 0 to \varkappa, we have

    \begin{eqnarray} &&\frac{f_{1}(\mu)g_{1}(\mu)}{\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]\mathcal{Q}(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}d\nu\\&&\quad+ \frac{f_{1}(\nu)g_{1}(\nu)}{\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]\mathcal{Q}(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}d\nu\\&&\geq \frac{g_{1}(\mu)}{\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]\mathcal{Q}(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}f_{1}(\nu)d\nu\\&&\quad+\frac{f_{1}(\mu)}{\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]\mathcal{Q}(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}g_{1}(\nu)d\nu.\\ \end{eqnarray} (3.13)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we have

    \begin{eqnarray} f_{1}(\mu)g_{1}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\mathcal{Q}(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1}g_{1})(\varkappa)\geq f_{1}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}g_{1})(\varkappa)+g_{1}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1})(\varkappa). \end{eqnarray} (3.14)

    Again, multiplying both sides of (3.14) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{P}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}} and integrating the resulting inequality with respect to \nu from 0 to \varkappa, we have

    \begin{eqnarray} &&\frac{\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\mathcal{Q}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{P}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}(\mu)g_{1}(\mu)d\mu\\&&\quad+\frac{\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1}g_{1})(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{P}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}d\mu\\&&\geq \frac{\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}g_{1})(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{P}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}(\mu)d\mu\\&&\quad+\frac{\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1})(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]\mathcal{P}(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}g_{1}(\mu)d\mu. \end{eqnarray} (3.15)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray*} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\mathcal{P}f_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\mathcal{Q}(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\mathcal{P}(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1}g_{1})(\varkappa)\nonumber\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\mathcal{P}f_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\mathcal{P}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\mathcal{Q}f_{1})(\varkappa), \end{eqnarray*}

    which implies (3.13).

    Theorem 3.4. Under the assumptions \boldsymbol{I}, \boldsymbol{II} and let r, s and t be three non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)+2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1}g_{1})(\varkappa)\Big)\\&&\quad+\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\Big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1})(\varkappa)\Big)\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1})(\varkappa)\Big)\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\Big) \end{eqnarray} (3.16)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. By means of Lemma 3.3 and setting \mathcal{P} = s, \mathcal{Q} = t, we can write

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1}g_{1})(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1})(\varkappa). \end{eqnarray} (3.17)

    Conducting product both sides of (3.17) by \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho; \alpha}r(\varkappa), we obtain

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1}g_{1})(\varkappa)\Big)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1})(\varkappa)\Big). \end{eqnarray} (3.18)

    Again, by means of Lemma 3.3 and setting \mathcal{P} = r, \mathcal{Q} = t, we can write

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1}g_{1})(\varkappa)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1})(\varkappa). \end{eqnarray} (3.19)

    Conducting product both sides of (3.19) by \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho; \alpha}s(\varkappa), we obtain

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}t(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1}g_{1})(\varkappa)\Big)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(tf_{1})(\varkappa)\Big). \end{eqnarray} (3.20)

    By similar arguments as we did before, yields

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(rf_{1}g_{1})(\varkappa)\Big)\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\Big). \end{eqnarray} (3.21)

    Adding (3.18), (3.20) and (3.21), we get the desired inequality (3.16).

    Remark 4. Theorem 3.2 and Theorem 3.4 lead to the following conclusions:

    (1) Let f_{1} and g_{1} are the asynchronous functions on [0, \infty), then (3.8) and (3.16) are reversed.

    (2) Let r, \, s and t are negative on [0, \infty), then (3.8) and (3.16) are reversed.

    (3) Let r, \, s are positive t is negative on [0, \infty), then (3.8) and (3.16) are reversed.

    In the next, we derive certain novel Grüss-type integral inequalities via weighted generalized proportional fractional integral operators.

    Lemma 3.5. Suppose an integrable function f_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let a continuous function r defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}x(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\Big) \end{eqnarray} (3.22)

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. By the given hypothesis and utilizing (1.7). For any \mu, \nu\in[0, \infty), we have

    \begin{eqnarray} &&\big(\Phi-f_{1}(\nu)\big)\big(f_{1}(\mu)-\phi\big)+\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\nu)-\phi\big)-\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)-\big(\Phi-f_{1}(\nu)\big)\big(f_{1}(\nu)-\phi\big)\\&&\leq f_{1}^{2}(\mu)+f_{1}^{2}(\nu)-2f_{1}(\mu)f_{1}(\nu). \end{eqnarray} (3.23)

    Multiplying both sides of (3.23) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} and integrating the resulting inequality with respect to \nu from 0 to \varkappa, we have

    \begin{eqnarray} &&\frac{\big(f_{1}(\mu)-\phi\big)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}\big(\Phi-f_{1}(\nu)\big)d\nu\\&&\quad+\frac{\big(\Phi-f_{1}(\mu)\big)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}\big(f_{1}(\nu)-\phi\big)d\nu\\&&\quad-\frac{\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}d\nu\\&&\quad-\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}\big(\Phi-f_{1}(\nu)\big)\big(f_{1}(\nu)-\phi\big)d\nu\\&&\leq\frac{f_{1}^{2}(\mu)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}d\nu\\&&\quad+\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}f_{1}^{2}(\nu)d\nu\\&&\quad-2\frac{f_{1}(\mu)}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}f_{1}(\nu)d\nu. \end{eqnarray} (3.24)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(f_{1}(\mu)-\phi\big)+\big(\Phi-f_{1}(\mu)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r(\varkappa)(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\big)\\&&\leq f_{1}^{2}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-2f_{1}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa). \end{eqnarray} (3.25)

    Multiplying both sides of (3.25) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}} and integrating the resulting inequality with respect to \mu from 0 to \varkappa, we have

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\nu)\big)\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\big(f_{1}(\mu)-\phi\big)d\mu\\&&\quad+\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\big(\Phi-f_{1}(\mu)\big)d\mu\\&&\quad-\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r(\varkappa)(\Phi-f_{1}(\nu)\big)\big(f_{1}(\nu)-\phi\big)\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}d\nu\\&&\leq\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}^{2}(\mu)d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad+\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\\&&\quad-2\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}(\mu)d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa). \end{eqnarray} (3.26)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad+\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r(\varkappa)(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)(f_{1}(\varkappa)-\phi)\Big)\\&&\leq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa), \end{eqnarray} (3.27)

    which gives (3.22) and proves the lemma.

    Theorem 3.6. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let a continuous function r defined on [0, \infty) . Then the inequality

    \begin{eqnarray} \Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\leq\frac{(\Phi-\phi)(\Upsilon-\gamma)}{4}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)^{2} \end{eqnarray} (3.28)

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. By the given hypothesis stated in Theorem 3.6. Also, assume that \mathfrak{\mu, \nu} be defined by

    \begin{eqnarray} \mathfrak{T}(\mu, \nu) = \big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big), \quad\mu, \nu\in[0, \varkappa], \quad\varkappa > 0. \end{eqnarray} (3.29)

    Multiplying both sides of (3.30) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} and integrating the resulting inequality with respect to \mu and \nu from 0 to \varkappa, we can state that

    \begin{eqnarray} &&\frac{1}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = \frac{1}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}\\&&\quad\times\big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big)d\mu d\nu. \end{eqnarray} (3.30)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray} &&\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = 2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa). \end{eqnarray} (3.31)

    Thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we can write that

    \begin{eqnarray} &&\Bigg(\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \mathfrak{T}(\mu, \nu)d\mu d\nu\Bigg)^{2}\\&&\leq \bigg(\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \big(f_{1}(\mu)-f_{1}(\nu)\big)d\mu d\nu\bigg)\\&&\quad\bigg(\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \big(g_{1}(\mu)-g_{1}(\nu)\big)d\mu d\nu\bigg)\\&& = 4\bigg(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)^{2}\bigg)\\&&\quad\times\bigg(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\bigg). \end{eqnarray} (3.32)

    Since \big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)\geq0 and \big(\Upsilon-g_{1}(\mu)\big)\big(g_{1}(\mu)-\gamma\big)\geq0, we have

    \begin{eqnarray} \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)\Big)\geq0, \end{eqnarray} (3.33)

    and

    \begin{eqnarray} \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Upsilon-g_{1}(\mu)\big)\big(g_{1}(\mu)-\gamma\big)\Big)\geq0. \end{eqnarray} (3.34)

    Therefore, from (3.33), (3.34) and Lemma 3.5, we get

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big) \end{eqnarray} (3.35)

    and

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big). \end{eqnarray} (3.36)

    Combining (3.30), (3.31), (3.35) and (3.36), we deduce that

    \begin{eqnarray} &&\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(xf_{1}g_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf)(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\\&&\quad\times\Big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big). \end{eqnarray} (3.37)

    Taking into consideration the elementary inequality 4a_{1}a_{2}\leq(a_{1}+a_{2})^{2}, \, a_{1}, a_{2}\in\mathbb{R}, we can state that

    \begin{eqnarray} 4\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\leq\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)(\Phi-\phi)\Big)^{2} \end{eqnarray} (3.38)

    and

    \begin{eqnarray} 4\Big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\leq\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)(\Upsilon-\gamma)\Big)^{2}. \end{eqnarray} (3.39)

    From (3.37)-(3.39), we obtain (3.28). This completes the proof of Theorem 3.6.

    Lemma 3.7. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} && \Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\Big)\\&&\quad\times\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\Big)\\ \end{eqnarray} (3.40)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Taking product (3.30) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\beta}\Gamma(\beta)(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}} and integrating the resulting inequality with respect to \mu and \nu from 0 to \varkappa, we can state that

    \begin{eqnarray} &&\frac{1}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = \frac{1}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\\&&\quad\times\big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big)d\mu d\nu. \end{eqnarray} (3.41)

    Taking product both sides of the above equation by \omega^{-2}(\varkappa) and utilizing Definition (2.2), we have

    \begin{eqnarray} &&\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa). \end{eqnarray} (3.42)

    Then, thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we conclude (3.40).

    Lemma 3.8. Suppose an integrable function f_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I} and \boldsymbol{II} on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} && \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\\&&\leq\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\Big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Phi-f_{1}(\varkappa))\big(f_{1}(\varkappa)-\phi\big)\Big) \end{eqnarray} (3.43)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Multiplying both sides of (3.25) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\beta}\Gamma(\beta)(\Psi(\varkappa)-\Psi(\mu))^{1-\beta}} and integrating the resulting inequality with respect to \mu from 0 to \varkappa. Then, by multiplying with \omega^{-1}(\varkappa) and in view of Definition 2.2, concludes

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\Big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Phi-f_{1}(\varkappa))\big(f_{1}(\varkappa)-\phi\big)\Big)\\&&\leq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa), \end{eqnarray} (3.44)

    which gives (3.43) and proves the lemma.

    Theorem 3.9. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} && \Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big\{\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\big)\Big\}\\&&\quad\times\Big\{\big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\big)\Big\} \end{eqnarray} (3.45)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Since (\Phi-f_{1}(\mu))(f_{1}(\mu)-\phi)\geq0 and (\Upsilon-g_{1}(\mu))(g_{1}(\mu)-\gamma)\geq0, we have

    \begin{eqnarray} -\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)(\Phi-f_{1}(\varkappa))(f_{1}(\varkappa)-\phi)\Big)- \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Phi-f_{1}(\varkappa))(f_{1}(\varkappa)-\phi)\Big)\leq0 \end{eqnarray} (3.46)

    and

    \begin{eqnarray} -\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)(\Upsilon-g_{1}(\varkappa))(g_{1}(\varkappa)-\gamma)\Big)- \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Upsilon-g_{1}(\varkappa))(g_{1}(\varkappa)-\gamma)\Big)\leq0. \end{eqnarray} (3.47)

    Utilizing Lemma 3.8 to f_{1} and g_{1}, and utilizing Lemma 3.7 and the inequalities (3.46) and (3.47), yields (3.45).

    Theorem 3.10. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)(\Phi-\phi)(\Upsilon-\gamma) \end{eqnarray} (3.48)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Taking into consideration the assumption (1.7), we have

    \begin{eqnarray} \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\leq \Phi-\phi, \quad\quad \Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq \Upsilon-\gamma, \quad\mu, \nu\in[0, \infty), \end{eqnarray} (3.49)

    which implies that

    \begin{eqnarray} \big\vert\mathfrak{T}(\mu, \nu)\big\vert = \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq (\Phi-\phi)(\Upsilon-\gamma). \end{eqnarray} (3.50)

    From (3.42) and (3.50), we obtain that

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&&\leq\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\Big((\Phi-\phi)(\Upsilon-\gamma)\Big)d\mu d\nu\\&& = \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)(\Phi-\phi)(\Upsilon-\gamma). \end{eqnarray} (3.51)

    This ends the proof.

    Theorem 3.11. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq L\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\Big)\\ \end{eqnarray} (3.52)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Taking into consideration the assumption (1.12), we have

    \begin{eqnarray} \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\leq L\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\quad\mu, \nu\in[0, \infty), \end{eqnarray} (3.53)

    which implies that

    \begin{eqnarray} \big\vert\mathfrak{T}(\mu, \nu)\big\vert = \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq L\big( g_{1}(\mu)-g_{1}(\nu)\big)^{2}. \end{eqnarray} (3.54)

    From (3.42) and (3.54), we obtain that

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&&\leq L\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\big( g_{1}(\mu)-g_{1}(\nu)\big)^{2}d\mu d\nu\\&& = L\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\Big). \end{eqnarray} (3.55)

    This ends the proof.

    Theorem 3.12. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and the lipschitzian condition with the constants \mathcal{M}_{1} and \mathcal{M}_{2} and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \mathcal{M}_{1}\mathcal{M}_{2}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa^{2}s(\varkappa))+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa^{2} r(\varkappa))\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa r(\varkappa))\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa s(\varkappa))\Big)\\ \end{eqnarray} (3.56)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. By the given hypothesis, we have

    \begin{eqnarray} \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\leq \mathcal{M}_{1}\big\vert \mu-\nu \big\vert\quad \Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq \mathcal{M}_{2}\big\vert \mu-\nu \big\vert\quad\mu, \nu\in[0, \infty), \end{eqnarray} (3.57)

    which implies that

    \begin{eqnarray} \big\vert\mathfrak{T}(\mu, \nu)\big\vert = \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq \mathcal{M}_{1}\mathcal{M}_{2}\big( \mu-\nu\big)^{2}. \end{eqnarray} (3.58)

    From (3.42) and (3.58), we obtain that

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&&\leq L\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}(\mu-\nu)^{2}d\mu d\nu\\&& = \mathcal{M}_{1}\mathcal{M}_{2}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa^{2}s(\varkappa))+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa^{2} r(\varkappa))\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa r(\varkappa))\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa s(\varkappa))\Big). \end{eqnarray} (3.59)

    This ends the proof.

    Corollary 1. Let f_{1} and g_{1} be two differentiable functions on [0, \infty) and let r and s be two non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \|f_{1}^{\prime}\|_{\infty}\|g_{1}^{\prime}\|_{\infty}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa^{2}s(\varkappa))+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa^{2} r(\varkappa))\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa r(\varkappa))\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa s(\varkappa))\Big)\\ \end{eqnarray} (3.60)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. We have f_{1}(\mu)-f_{1}(\nu) = \int\limits_{\nu}^{\mu}f_{1}^{\prime}(\varkappa)d\varkappa and g_{1}(\mu)-g_{1}(\nu) = \int\limits_{\nu}^{\mu}g_{1}^{\prime}(\varkappa)d\varkappa. That is, \big\vert f_{1}(\mu)-f_{1}(\nu)\big\vert\leq\|f_{1}^{\prime}\|_{\infty}\big\vert \mu-\nu \big\vert, \big\vert g_{1}(\mu)-g_{1}(\nu)\big\vert\leq\|g_{1}^{\prime}\|_{\infty}\big\vert \mu-\nu \big\vert, \mu, \nu\in[0, \infty), and the immediate consequence follows from Theorem 3.12. This completes the proof.

    Example 3.13. Let \rho, \, \alpha > 0, \, \, q_{1}, q_{2} > 1 with q_{1}^{-1}+q_{2}^{-1} = 1, and \omega\neq0 be a function on [0, \infty). Let f_{1} be an integrable function defined on [0, \infty) and \, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho; \alpha}f_{1} be the weighted generalized proportional fractional integral operator satisfying assumption \bf{II}. Then we have

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert\leq\Theta\|(f_{1}\circ\omega)(\mu)\|_{L_{1}(a_{1}, \varkappa)}, \end{eqnarray*}

    where

    \begin{eqnarray*} \Theta = \frac{\omega^{-1}(\varkappa)(-1)^{\alpha-1}}{\Gamma(\alpha)}\Big\{\Big(\frac{\rho}{q_{1}(\rho-1)}\Big)^{\alpha-1+{1/q_{1}}}\Big\}^{1/q_{1}}\Phi^{1/q_{1}}\Big(q_{1}(\alpha-1)+1, \frac{q_{1}(\rho-1)}{\rho}\big(\Psi(\varkappa)-\Psi(a_{1})\big)\Big) \end{eqnarray*}

    and

    \Phi(\alpha, \varkappa) = \int\limits_{0}^{\varkappa}e^{-v}v^{\alpha-1}dv

    is the incomplete gamma function [52,53].

    Proof. It follows from Definition 2.2 and the modulus property that

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert\leq\frac{\omega^{-1}(\varkappa)}{\rho^{\alpha}\Gamma(\rho)}\int\limits_{a_{1}}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]}{\big(\Psi(\varkappa)-\Psi(\mu)\big)^{1-\alpha}}\Psi^{\prime}(\mu)\big\vert f_{1}(\mu)\omega(\mu)\big\vert d\mu \end{eqnarray*}

    for \varkappa > a_{1}.

    Making use of the well-known Hölder inequality, we obtain

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert\leq\frac{\omega^{-1}(\varkappa)}{\rho^{\alpha}\Gamma(\rho)}\Bigg(\int\limits_{a_{1}}^{\varkappa}\frac{q_{1}\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]}{\big(\Psi(\varkappa)-\Psi(\mu)\big)^{q_{1}(1-\alpha)}}\Psi^{\prime}(\mu)d\mu\Bigg)^{1/q_{1}}\| f_{1}\circ\omega(\mu)\|_{L_{1}(a_{1}, \varkappa)}. \end{eqnarray*}

    Let \theta = \Psi(\varkappa)-\Psi(\mu). Then elaborated computations lead to

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert&&\leq\frac{(-1)^{\alpha-1}\omega^{-1}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)}\Big\{\Big(\frac{\rho}{q_{1}(\rho-1)}\Big)^{\alpha-1+{1/q_{1}}}\Big\}^{1/q_{1}}\nonumber\\&&\quad\times\Phi^{1/q_{1}}\Big(q_{1}(\alpha-1)+1, \frac{q_{1}(\rho-1)}{\rho}\big(\Psi(\varkappa)-\Psi(a_{1})\big)\Big)\| f_{1}\circ\omega(\mu)\|_{L_{1}(a_{1}, \varkappa)}. \end{eqnarray*}

    Here, we aim at present some new generalizations via weighted generalized proportional fractional, weighted generalized Riemann-Liouville and weighted Riemann-Liouville fractional integral operators, which are the new estimates of the main consequences.

    Lemma 4.1. Let f_{1} and g_{1} be two synchronous functions on [0, \infty). Assume that \mathcal{Q} and \mathcal{P} be two non-negative continuous mappings on [0, \infty). Then the inequality

    \begin{eqnarray*} &&\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \Psi(\varkappa) = \varkappa and Lemma 3.1 yields the proof of Lemma 4.1.

    Lemma 4.2. Let f_{1} and g_{1} be two synchronous functions on [0, \infty). Assume that \mathcal{Q} and \mathcal{P} be two non-negative continuous mappings on [0, \infty). Then the inequality

    \begin{eqnarray*} &&\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \Psi(\varkappa) = \varkappa and Lemma 3.1 yields the proof of Lemma 4.2.

    Lemma 4.3. Under the assumption of Lemma 3.1, then the inequality

    \begin{eqnarray*} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1 and Lemma 3.1 yields the proof of Lemma 4.3.

    Lemma 4.4. Under the assumption of Lemma 4.2, then the inequality

    \begin{eqnarray*} &&\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1, \, \Psi(\varkappa) = \varkappa and Lemma 3.1 yields the proof of Lemma 4.4.

    Theorem 4.5. Let f_{1} and g_{1} be two synchronous functions on [0, \infty). Assume that r, s and t be three non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray*} &&2\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big)\nonumber\\&&\quad+2\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\nonumber\\&&\geq\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big) \nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big)\nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \Psi(\varkappa) = \varkappa and Theorem 3.2 yields the proof of Theorem 4.5.

    Theorem 4.6. Under the assumption of \boldsymbol{I}, \boldsymbol{II} and let r, s and t be three non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray*} &&2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}t(\varkappa)\Big)\nonumber\\&&\quad+2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}t(\varkappa)\nonumber\\&&\geq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big) \nonumber\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big)\nonumber\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1 and Theorem 3.2 yields the proof of Theorem 4.6.

    Theorem 4.7. Under the assumption of Theorem 4.5, then the inequality

    \begin{eqnarray*} &&2\, _{\omega}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}t(\varkappa)\Big)\nonumber\\&&\quad+2\, _{\omega}\Omega_{0^{+}}^{\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}t(\varkappa)\nonumber\\&&\geq\, _{\omega}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big) \nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big)\nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1, \, \, \Psi(\varkappa) = \varkappa and Theorem 3.2 yields the proof of Theorem 4.7.

    Remark 5. The computed results lead to the following conclusion:

    (1) Setting \rho = 1, \Psi(\varkappa) = \varkappa and r(\varkappa) = s(\varkappa) = 1, and using the relation (2.7), (2.8) and the assumption \omega(\varkappa) = 1 , then Theorem 3.6 and Theorem 3.9 reduces to the known results due to Dahmani et al. [38].

    (2) Setting \rho = 1, \Psi(\varkappa) = \varkappa and using the relation (2.7), (2.8) and the assumption \omega(\varkappa) = 1 , then Theorem 3.10–3.12, and Corollary 1 reduces to the known results due to Dahmani et al. [38] and Dahmani [40], respectively.

    A new generalized fractional integral operator is proposed in this paper. The novel investigation is used to generate novel weighted fractional operators in the Riemann-Liouville, generalized Riemann-Liouville, Hadamard, Katugampola, Generalized proportional fractional, generalized Hadamard proportional fractional and henceforth, which effectively alleviates the adverse effect of another function \Psi and proportionality index \rho. Utilizing the weighted generalized proportional fractional operator technique, we derived the analogous versions of the extended Chebyshev and Grüss type inequalities that improve the accuracy and efficiency of the proposed technique. Contemplating the Remark 2 and 3, several existing results can be identified in the literature. Some innovative particular cases constructed by this method are tested and analyzed for statistical theory, fractional Schrödinger equation [20,21]. The results show that the method proposed in this paper can stably and efficiently generate integral inequalities for convexity with better operators performance, thus providing a reliable guarantee for its application in control theory [54].

    The authors declare that they have no competing interests.

    The authors would like to express their sincere thanks to referees for improving the article and also thanks to Natural Science Foundation of China (Grant Nos. 61673169) for providing financial assistance to support this research. The authors would like to express their sincere thanks to the support of Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia.



    [1] H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control, 19 (1974), 716–723. doi: 10.1109/TAC.1974.1100705
    [2] A. K. Al-Khadim, A. N. Hussein, New proposed length biased weighted exponential and Rayleigh distribution with application, Math. Theo. Mod., 4 (2014), 2224–2235.
    [3] A. I. Al-Omari, Estimation of mean based on modified robust extreme ranked set sampling, J. Stat. Comput. Sim., 81 (2011), 1055–1066. doi: 10.1080/00949651003649161
    [4] A. I. Al-Omari, Ratio estimation of population mean using auxiliary information in simple random sampling and median ranked set sampling, Stat. Probab. Lett., 82 (2012), 1883–1990. doi: 10.1016/j.spl.2012.07.001
    [5] A. I. Al-Omari, I. K. Alsmairan, Length-biased Suja distribution: Properties and application, J. Appl. Prob. Stat., 14 (2019), 95–116.
    [6] A. I. Al-Omari, A. Al-Nasser, E. Ciavolino, A size-biased Ishita distribution: Application to real data, Qual. Quant., 53 (2019), 493–512. doi: 10.1007/s11135-018-0765-y
    [7] A. I. Al-Omari, M. Gharaibeh, Topp-Leone Mukherjee-Islam distribution: Properties and applications, Pakistan J. Stat., 34 (2018), 479–494.
    [8] A. I. Al-Omari, A. Al-khazaleh, M. Al-khazaleh, Exponentiated new Weibull-Pareto distribution, Rev. Invest. Operacional, 40 (2019), 165–175.
    [9] S. Benchiha, A. I. Al-Omari, Generalized Quasi Lindley distribution: Theoretical properties, estimation methods, and applications, Electron. J. Appl. Stat. Anal., 14 (2021), 167–196.
    [10] H. Bozdogan, Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions, Psychometrika, 52 (1987), 345–370. doi: 10.1007/BF02294361
    [11] R. C. H. Cheng, N. A. K. Amin, Maximum product-of-spacings estimation with applications to the log-normal distribution, Tech. rep., Department of Mathematics, University of Wales. 1979.
    [12] R. C. H. Cheng, N. A. K. Amin, Estimating parameters in continuous univariate distributions with a shifted origin, J. Royal Stat. Soc., 45 (1983), 394–403.
    [13] R. B. D'Agostino, M. A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker: New York, NY, USA, 1986.
    [14] H. David, H. Nagaraja, Order Statistics, John Wiley and Sons, New York, 2003.
    [15] R. A. Fisher, The efects of methods of ascertainment upon the estimation of frequencies, Ann. Eugen., 6 (1934), 13–25. doi: 10.1111/j.1469-1809.1934.tb02105.x
    [16] R. C. Gupta, J. P. Keating, Relations for reliability measures under length biased sampling, Scan. J. Statist, 13 (1985), 49–56.
    [17] R. C. Gupta, S. N. U. A. Kirmani, The role of weighted distributions in stochastic modeling, Commun. Stat., 19 (1990), 3147–3162. doi: 10.1080/03610929008830371
    [18] M. Haq, R. Usman, S. Hashmi, A. I. Al-Omari, The Marshall-Olkin length-biased exponential distribution and its applications, J. King Saud Univ. Sci., 31 (2019), 246–251. doi: 10.1016/j.jksus.2017.09.006
    [19] A. Haq, J. Brown, E. Moltchanova, A. I. Al-Omari, Ordered double ranked set samples and applications to inference, Am. J. Math. Manage. Sci., 33 (2014), 239–260.
    [20] A. Haq, J. Brown, E. Moltchanova, A. I. Al-Omari, Varied L ranked set sampling scheme, J. Stat. Theory Pract., 9 (2015), 741–767. doi: 10.1080/15598608.2015.1008606
    [21] E. J. Hannan, B. G. Quinn, The determination of the order of an autoregression, J. Royal Stat. Soc., 41 (1979), 190–195.
    [22] R. W. Katz, M. B. Parlange, P. Naveau, Statistics of extremes in hydrology, Adv. Water Res, 25 (2002), 1287–1304. doi: 10.1016/S0309-1708(02)00056-8
    [23] M. Kilany, Weighted Lomax Distribution, Springer Plus, 5 (2016).
    [24] A. Luceno, Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators, Comput. Stat. Data Anal., 51 (2006), 904–917. doi: 10.1016/j.csda.2005.09.011
    [25] Y. Benoist, P. Foulon, F. Labourie, A bootstrap control chart for Weibull percentiles, Qual. Reliab. Eng. Int., 22 (2006), 141–151 doi: 10.1002/qre.691
    [26] B. O. Oluyede, On inequalities and selection of experiments for length-biased distributions, Probab. Eng. Inform. Sci., 13 (1999), 169–185. doi: 10.1017/S0269964899132030
    [27] G. P. Patil, G. R. Rao, Weighted distributions and size biased sampling with applications to wildlife populations and human families, Biometrics, 34 (1978), 179–189. doi: 10.2307/2530008
    [28] J. Swain, S. Venkatraman, J. Wilson, Least squares estimation of distribution function in Johnson's translation system, J. Stat. Comput. Simul, 29 (1988), 271–297. doi: 10.1080/00949658808811068
    [29] M. Shaked, J. Shanthikumar, Stochastic orders and their applications, distribution and associated inference, Academic Press New York, (1994).
    [30] G. Schwarz, Estimating the dimension of a model, Ann. Stat., 6 (1978), 461–464.
    [31] C. R. Rao, On discrete distributions arising out of methods of ascertainment, In: classical and contagious discrete distribution, Pergamon Press and Statistical Publishing Society, Calcutta, (1965), 320–332.
    [32] E. Zamanzade, A. I. Al-Omari, New ranked set sampling for estimating the population mean and variance, Hacet. J. Math. Stat., 45 (2016), 1891–1905.
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