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Research article

On some classical integral inequalities in the setting of new post quantum integrals

  • Received: 10 August 2022 Revised: 19 September 2022 Accepted: 23 September 2022 Published: 26 October 2022
  • MSC : 26D10, 26D15, 26A51, 05A33

  • In this article, we introduce the notion of aˉTp,q-integrals. Using the definition of aˉTp,q-integrals, we derive some new post quantum analogues of some classical results of Young's inequality, Hölder's inequality, Minkowski's inequality, Ostrowski's inequality and Hermite-Hadamard's inequality.

    Citation: Bandar Bin-Mohsin, Muhammad Uzair Awan, Muhammad Zakria Javed, Sadia Talib, Hüseyin Budak, Muhammad Aslam Noor, Khalida Inayat Noor. On some classical integral inequalities in the setting of new post quantum integrals[J]. AIMS Mathematics, 2023, 8(1): 1995-2017. doi: 10.3934/math.2023103

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  • In this article, we introduce the notion of aˉTp,q-integrals. Using the definition of aˉTp,q-integrals, we derive some new post quantum analogues of some classical results of Young's inequality, Hölder's inequality, Minkowski's inequality, Ostrowski's inequality and Hermite-Hadamard's inequality.



    Quantum calculus is one of the most fascinating subjects of mathematics. It serves as the bridge between mathematics and physics. The history of quantum calculus is old but in the past few decades it experienced rapid development. The concepts of quantum calculus have been used as tools in different areas of mathematics. For example Sudsutad et al. [13] and Tariboon and Ntouyas [14,15] have utilized the concepts of quantum calculus in obtaining the quantum analogues of inequalities involving convexity. This motivated many researchers and consequently number of new quantum analogues of classical inequalities have been obtained in the literature. For instance Noor et al. [12] have obtained new quantum analogues of the Hermite-Hadamard type of inequalities. Du et al. [7] and Zhang et al. [19] obtained new generalized quantum integral identities and obtained several new associated quantum analogues of integral inequalities.

    In 2017, Alp and Sarikaya [1] introduced another new definition of q-integral which they called as aˉTq -integral. They discussed several basic properties pertaining to this so called aˉTq-integral and also obtained new aˉTq-analogue of Hermite-Hadamard's inequality. In another article [2] Alp and Sarikaya obtained the aˉTq-analogues of Ostrowski's, Young's, Hölder's and Minkowski's inequalities. On the other hand in [10], Kara et al. introduced new quantum integral which is called bˉTq-integral. The authors proved also the corresponding Hermite-Hadamard inequalities for bˉTq-integrals.

    Chakarabarti and Jagannathan [5] considered a new generalization of quantum calculus (also known as q-calculus) which is called as post-quantum calculus (also known as (p,q)-calculus). In q-calculus, we deal with a q–number with one base q, however, in (p,q)-calculus we have p- and q-numbers with two independent variables p and q. Kunt et al. [11] derived (p,q)-analogues of Hermite–Hadamard's inequality. Since then several new post quantum analogues of classical inequalities have been obtained in the literature. For example, Chu et al. [6] obtained new (p,q)–analogues of Ostrowski type of inequalities using new definitions of left–right (p,q)–derivatives and definite integrals. Yu et al. [18] obtained some new refinements of inequalities via (p,q)–calculus and also discussed their applications. For some more details and basic properties, see [16]. For more details see [20,21,22,23,24].

    The aim of this paper is to introduce aˉTp,q-integrals. We derive new aˉTp,q-analogues of certain classical inequalities. For example, we obtain aˉTp,q-Young's inequality, aˉTp,q-Hölder's inequality, aˉTp,q-Minkowski's inequality, aˉTp,q-Ostrowski's inequality and aˉTp,q-Hermite-Hadamard's inequality respectively. To the best of our knowledge these results are new in the literature. We hope that the ideas and techniques of this paper will inspire interested readers working in this field.

    In this section, for the sake of completeness, we now recall some basic concepts from quantum and post quantum calculus.

    In this part, we give some of the necessary explanations and related inequalities regarding q-calculus. Also, here and further we use the following notation (see [9]):

    [˜ϖ]q=1q˜ϖ1q=1+q+q2++q˜ϖ1, q(0,1).

    In [8], Jackson gave the q-Jackson integral from 0 to b for 0<q<1 as follows:

    b0ˉΞ(x) dqx=(1q)b˜ϖ=0q˜ϖˉΞ(bq˜ϖ), (2.1)

    provided the sum converges absolutely.

    Jackson [8] gave the q-Jackson integral in a generic interval [a,b] as:

    baˉΞ(x) dqx=b0ˉΞ(x) dqxa0ˉΞ(x) dqx.

    Definition 2.1 ([14]). Let ˉΞ:[a,b]R be a continuous function. Then, the ˜q2ˉa1-definite integral on [a,b] is defined as

    baˉΞ(x) adqx=(1q)(ba)˜ϖ=0q˜ϖˉΞ(q˜ϖb+(1q˜ϖ)a)=(ba)10ˉΞ((1ˉϱ)a+ˉϱb) dqˉϱ. (2.2)

    In [3], Alp et al. proved the following ˜q2ˉa1 -Hermite-Hadamard inequalities for convex functions in the setting of quantum calculus:

    Theorem 2.1. Let ˉΞ:[a,b]R be a convex differentiable function on [a,b] and 0<q<1. Then q-Hermite-Hadamard inequalities are as follows:

    ˉΞ(qa+b1+q)1babaˉΞ(x) adqxqˉΞ(a)+ˉΞ(b)1+q. (2.3)

    Definition 2.2 ([14,15]). Let ˉΞ:J:=[a,b]RR be an arbitrary function. Then the q–derivative of ˉΞ on J at ˉϱ, is defined

    aDqˉΞ(ˉϱ)=ˉΞ(ˉϱ)ˉΞ(qˉϱ+(1q)a)(1q)(ˉϱa),ˉϱaandDqˉΞ(a)=limˉϱaDqˉΞ(ˉϱ),

    where 0<q<1 is a constant.

    On the other hand, Bermudo et al. gave the following new definition and related Hermite-Hadamard type inequalities:

    Definition 2.3 ([4]). Let ˉΞ:[a,b]R be a continuous function. Then, the qb-definite integral on [a,b] is defined as

    baˉΞ(x) bdqx=(1q)(ba)˜ϖ=0q˜ϖˉΞ(q˜ϖa+(1q˜ϖ)b)=(ba)10ˉΞ(ta+(1ˉϱ)b) dqˉϱ. (2.4)

    Theorem 2.2 ([4]). Let ˉΞ:[a,b]R be a convex function on [a,b] and 0<q<1. Then, q -Hermite-Hadamard inequalities are as follows:

    ˉΞ(a+qb1+q)1babaˉΞ(x) bdqxˉΞ(a)+qˉΞ(b)1+q. (2.5)

    In this part, we review some fundamental notions and notations of (p,q)-calculus.

    The [˜ϖ]p,q is said to be (p,q) integers and is expressed as:

    [˜ϖ]p,q=p˜ϖq˜ϖpq,

    with 0<q<p1.

    Definition 2.4 ([16]). The definite (p,q)a-integral of mapping ˉΞ:[a,b]R on [a,b] is stated as:

    xaˉΞ(ˉϱ)adp,qˉϱ=(pq)(xa)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a), (2.6)

    for x[a, qb+(1p)a] and 0<q<p1.

    Definition 2.5 ([16]). Let ˉΞ:JR be a continuous function and let xJ and 0<q<p1. Then the (p,q)-derivative on J of function ˉΞ at x is defined as

    Dp,qˉΞ(x)=ˉΞ(px+(1p)a))ˉΞ(qx+(1q)a)(pq)(xa),xa.

    Definition 2.6. From [17] , the definite (p,q)b-integral of mapping ˉΞ:[a,b]R on [a,b] is stated as:

    bxˉΞ(ˉϱ)bdp,qˉϱ=(pq)(bx)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)b), (2.7)

    for x[pa+(1p)b, b] and 0<q<p1.

    Remark 2.1. It is evident that if we pick p=1 in (2.6) and (2.7), then the equalities (2.6) and (2.7) change into (2.2) and (2.4), respectively.

    In [11], Kunt et al. proved the following Hermite-Hadamard type inequalities for convex functions via (p,q)a integral:

    Theorem 2.3. For a convex mapping ˉΞ:[a,b]R which is differentiable on [a,b], the following inequalities hold for (p,q)a integral:

    ˉΞ(qa+pb[2]p,q)1p(ba)qb+(1p)aaˉΞ(x)adp,qxqˉΞ(a)+pˉΞ(b)[2]p,q, (2.8)

    where 0<q<p1.

    In [17], Vivas-Cortez et al. proved the following Hermite-Hadamard type inequalities for convex functions via (p,q)b integral:

    Theorem 2.4. For a convex mapping ˉΞ:[a,b]R which is differentiable on [a,b], the following inequalities hold for (p,q)b integral:

    ˉΞ(pa+qb[2]p,q)1p(ba)bpa+(1p)bˉΞ(x)bdp,qxpˉΞ(a)+qˉΞ(b)[2]p,q, (2.9)

    where 0<q<p1.

    In this subsection, we present definitions and properties given by using trapezoids.

    Alp and Sarikaya, by using the area of trapezoids, introduced the following generalized quantum integral which is called aˉTq-integral.

    Definition 2.7 ([1,2,10]). Let ˉΞ:[a,b]R is continuous function. for x[a,b]

    baˉΞ(ˉϱ)adˉTqˉϱ=(1q)(ba)2q[(1+q)˜ϖ=0q˜ϖˉΞ(q˜ϖb+(1q˜ϖ)a)ˉΞ(b)].

    Theorem 2.5 (aˉTq-Hermite-Hadamard). [1,2] Let ˉΞ:[a,b]R be a convex continuous function on [a,b]. Then we have

    ˉΞ(a+b2)1babaˉΞ(ˉϱ)adˉTqˉϱˉΞ(a)+ˉΞ(b)2.

    In [10], Kara et al. introduced the following generalized quantum integral which is called bˉTq-integral.

    Definition 2.8 ([10]). Let ˉΞ:[a,b]R is continuous function. for x[a,b],

    baˉΞ(ˉϱ) bdˉTqˉϱ=(1q)(ba)2q[(1+q)˜ϖ=0q˜ϖˉΞ(q˜ϖa+(1q˜ϖ)b)ˉΞ(a)].

    Theorem 2.6 (bˉTq-Hermite-Hadamard). [10] Let ˉΞ:[a,b]R be a convex continuous function on [a,b]. Then we have

    ˉΞ(a+b2)1babaˉΞ(ˉϱ) bdˉTqˉϱˉΞ(a)+ˉΞ(b)2.

    First we define aˉTp,q-integral.

    Definition 3.1. Let ˉΞ:IR be continuous functions, then for 0<q<p1 we have

    xaˉΞ(ˉϱ) adˉTp,qˉϱ=(pq)(xa)2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a)ˉΞ(x+(p1)ap)],

    where x[a,qb+(1p)a].

    Remark 3.1. If we take p=1, then we obtain aˉTq integrals.

    We now discuss some basic properties of aˉTp,q-integral.

    Theorem 3.1. For αR{1}, then

    xa(ˉϱa)α adˉTp,qˉϱ=(pα+qα)(xa)α+12pα[α+1]p,q.

    Proof. From the definition of aˉTp,q-integrals, we have

    xa(ˉϱa)α adˉTp,qˉϱ=(pq)(xa)2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)aa)α(x+(p1)apa)α]=(pq)(xa)2q[(p+q)˜ϖ=0(q˜ϖp˜ϖ+1)α+1(xa)α(xap)α]=(pq)(xa)α+12q[(p+q)pα+1˜ϖ=0(q˜ϖp˜ϖ)α+1(1p)α]=(pq)(pα+qα)(xa)α+12pα(pα+1qα+1).

    This completes the proof.

    Note that if we take p=1, we have the following result.

    Corollary 3.1. For αR{1}, then

    xa(ˉϱa)α adˉΞqˉϱ=(1+qα)(xa)α+12[α+1]q.

    Lemma 3.1. Let p>1 and (aˉϱ)˜ϖp,q is (p,q)-binomial, then

    x0(aˉϱ)˜ϖp,q 0dˉTp,qˉϱ=a˜ϖ+1(p+q)2p[˜ϖ+1]p,q(q(apxq)˜ϖ+1+p(ax)˜ϖ+12p[˜ϖ+1]p,q).

    Proof. By using (p,q)-binomial formula and Gauss binomial formula

    x0(aˉϱ)˜ϖp,q 0dˉTp,qˉϱ=x0˜ϖk=0(1)k(˜ϖk)p,qp(˜ϖk)(˜ϖk1)2qk(k1)2ˉϱka˜ϖk 0dˉTp,qˉϱ=x0˜ϖk=0(1)k[˜ϖ]p,q![k]p,q![˜ϖk]p,q!p,qp(˜ϖk)(˜ϖk1)2qk(k1)2ˉϱka˜ϖk 0dˉTp,qˉϱ=˜ϖk=0(1)k[˜ϖ]p,q![k]p,q![˜ϖk]p,q!p,qp(˜ϖk)(˜ϖk1)2qk(k1)2a˜ϖkx0ˉϱk 0dˉTp,qˉϱ=˜ϖk=0(1)k[˜ϖ]p,q![k]p,q![˜ϖk]p,q!p(˜ϖk)(˜ϖk1)2qk(k1)2a˜ϖk(pk+qk)xk+12[k+1]p,q=˜ϖk=0(1)k[˜ϖ]p,q![k+1]p,q![˜ϖk]p,q!p(˜ϖk)(˜ϖk1)2qk(k1)2a˜ϖk(pk+qk)xk+12.

    Replacing k by k1

    =˜ϖ+1k=1(1)k1[˜ϖ+1]p,q![k]p,q![˜ϖ+1k]p,q!p(˜ϖk+1)(˜ϖk)2q(k1)(k2)2a˜ϖ+1k(pk+qk)xk2=12[˜ϖ+1]p,q[qp˜ϖ+1k=1(1)k1[˜ϖ]p,q![k]p,q![˜ϖ+1k]p,q!p(˜ϖk+1)(˜ϖk)2qk(k1)2a˜ϖ+1kxkqk2+˜ϖ+1k=1(1)k1[˜ϖ+1]p,q![k]p,q![˜ϖ+1k]p,q!p(˜ϖk+1)(˜ϖk)2qk(k1)2a˜ϖ+1kxk]=a˜ϖ+1(p+q)2p[˜ϖ+1]p,q(q(apxq)˜ϖ+1+p(ax)˜ϖ+12p[˜ϖ+1]p,q).

    Example 3.1. Let p>1 and (1ˉϱ)˜ϖp,q is (p,q)-binomial, then

    10(1ˉϱ)˜ϖp,q 0dˉTp,qˉϱ=(p+q)2p[˜ϖ+1]p,q.

    Theorem 3.2. Let ˉΞ1,ˉΞ2:IR be continuous functions, then

    (1) xa[ˉΞ1(ˉϱ)+ˉΞ2(ˉϱ)] adˉTp,qˉϱ=xaˉΞ1(ˉϱ) adˉTp,qˉϱ+xaˉΞ2(ˉϱ) adˉTp,qˉϱ;

    (2) xaαˉΞ(ˉϱ) adˉTp,qˉϱ=αxaˉΞ(ˉϱ) adˉTp,qˉϱ;

    (3) aDp,qxaˉΞ(ˉϱ) adˉTp,qˉϱ=ˉΞ(x)+ˉΞ((qx+(pq)a)p)2.

    Proof. We omit the details of the proof of (1) and (2). For (3), from the definition of aˉTp,q-integrals, we have

    xaˉΞ(ˉϱ) adˉTp,qˉϱ=(pq)(xa)2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a)ˉΞ(x+(p1)ap)].

    Taking (p,q)-derivative , we have

    aDp,qxaˉΞ(ˉϱ) adˉTp,qˉϱ=p2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ(q˜ϖp˜ϖ+1(px+(1p)a)+(1q˜ϖp˜ϖ+1)a)ˉΞ(x)]q2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ(q˜ϖp˜ϖ+1(qx+(1q)a)+(1q˜ϖp˜ϖ+1)a)ˉΞ(qx+(pq)ap)]=12q[(p+q)˜ϖ=0q˜ϖp˜ϖˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)pˉΞ(x)(p+q)˜ϖ=0q˜ϖ+1p˜ϖ+1ˉΞ(q˜ϖ+1p˜ϖ+1x+(1q˜ϖ+1p˜ϖ+1)a)+qˉΞ(qx+(pq)ap)]=12q[(p+q)˜ϖ=0q˜ϖp˜ϖˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)pˉΞ(x)(p+q)˜ϖ=1q˜ϖp˜ϖˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)+qˉΞ(qx+(pq)ap)]=12q[(p+q)˜ϖ=0q˜ϖp˜ϖˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)pˉΞ(x)(p+q)˜ϖ=0q˜ϖp˜ϖˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)+(p+q)ˉΞ(x)+qˉΞ(qx+(pq)ap)]=12q[(p+q)ˉΞ(x)pˉΞ(x)+qˉΞ(qx+(pq)ap)]=ˉΞ(x)+ˉΞ((qx+(pq)a)p)2.

    This completes the proof.

    Theorem 3.3 (Change of variable property). Let ˉΞ:IR be a function with 0<q<p1 , then

    p0ˉΞ(ˉϱb+(1ˉϱ)a) 0dˉTp,qˉϱ=1baqb+(1p)aaˉΞ(ˉϱ) adˉTp,qˉϱ.

    Proof. From the definition of aˉTp,q-integral, we have

    p0ˉΞ(ˉϱb+(1ˉϱ)a) 0dˉTp,qˉϱ=(pq)p2q[(p+q)˜ϖ=0ˉΞ(q˜ϖp˜ϖb+(1q˜ϖp˜ϖ)a)ˉΞ(b)]=(pq)p2q[(p+q)˜ϖ=0ˉΞ(q˜ϖp˜ϖ+1(qb+(1p)a)+(1q˜ϖp˜ϖ+1)a)ˉΞ(b)]=1baqb+(1p)aaˉΞ(ˉϱ) adˉTp,qˉϱ.

    This completes the proof.

    Theorem 3.4. Let ˉΞ:IR be a continuous function with c(a,x), then

    xcaDp,qˉΞ(ˉϱ) adˉTp,qˉϱ=qˉΞ(x)qˉΞ(c)+pˉΞ(qx+(pq)ap)pˉΞ(qc+(pq)ap)2q.

    Proof. Applying (p,q)-derivative, change of variable and aˉTp,q-integrals, we have

    xcaDp,qˉΞ(ˉϱ) adˉTp,qˉϱ=xaaDp,qˉΞ(ˉϱ) adˉTp,qˉϱcaaDpqˉΞ(ˉϱ) adˉTp,qˉϱ=xaˉΞ(pˉϱ+(1p)a)ˉΞ(qˉϱ+(1q)a)(pq)(ˉϱa) adˉTp,qˉϱcaˉΞ(pˉϱ+(1p)a)ˉΞ(qˉϱ+(1q)a)(pq)(ˉϱa)adˉp,ˉqˉϱ=xaˉΞ(pˉϱ+(1p)a)(pq)(ˉϱa) adˉTp,qˉϱxaˉΞ(qˉϱ+(1q)a)(pq)(ˉϱa) adˉTp,qˉϱcaˉΞ(pˉϱ+(1p)a)(pq)(ˉϱa) adˉTp,qˉϱ+caˉΞ(qˉϱ+(1q)a)(pq)(ˉϱa) adˉTp,qˉϱ=px+(1p)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱqx+(1q)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱpc+(1p)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱ+qc+(1q)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱ=I1I2I3+I4, (3.1)

    where

    I1=px+(1p)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱ=12q[(p+q)˜ϖ=0ˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)pˉΞ(x)],I2=qx+(1q)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱ=12q[(p+q)˜ϖ=0ˉΞ(q˜ϖp˜ϖx+(1q˜ϖp˜ϖ)a)(p+q)ˉΞ(x)pˉΞ(qx+(pq)ap)],I3=pc+(1p)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱ=12q[(p+q)˜ϖ=0ˉΞ(q˜ϖp˜ϖc+(1q˜ϖp˜ϖ)a)pˉΞ(c)],I4=qc+(1q)aaˉΞ(ˉϱ)(pq)(ˉϱa) adˉTp,qˉϱ=12q[(p+q)˜ϖ=0ˉΞ(q˜ϖp˜ϖc+(1q˜ϖp˜ϖ)a)(p+q)ˉΞ(c)pˉΞ(qc+(pq)ap)].

    Substituting the values of I1,I2,I3 and I4 in (3.1), we obtain the required result.

    Theorem 3.5. Let ˉΞ1,ˉΞ2:IR be continuous functions with c(a,x), then

    xcˉΞ1(pˉϱ+(1p)a)aDp,qˉϱˉΞ2(ˉϱ) adˉTp,qˉϱ=qˉΞ1(ˉϱ)ˉΞ2(ˉϱ)+pˉΞ1(qˉϱ+(pq)ap)ˉΞ2(qˉϱ+(pq)ap)2q|xcxcˉΞ2(qˉϱ+(1q)a)aDp,qˉϱˉΞ1(ˉϱ) adˉTp,qˉϱ. (3.2)

    Proof. Using the definition of (p,q)-derivative, we have

    aDp,qˉϱˉΞ1(ˉϱ)ˉΞ2(ˉϱ)=ˉΞ1(pˉϱ+(1p)a)ˉΞ2(pˉϱ+(1p)a)ˉΞ1(qˉϱ+(1q)a)ˉΞ1(qˉϱ+(1q)a)(pq)(ˉϱa)=ˉΞ1(pˉϱ+(1p)a)aDp,qˉϱˉΞ2(ˉϱ)+ˉΞ2(qˉϱ+(1q)a)aDp,qˉϱˉΞ1(ˉϱ).

    Now by taking aˉTp,q-integrals, we have

    xcaDp,qˉϱˉΞ1(ˉϱ)ˉΞ2(ˉϱ) adˉTp,qˉϱ=xcˉΞ1(pˉϱ+(1p)a)aDp,qˉϱˉΞ2(ˉϱ) adˉTp,qˉϱ+xcˉΞ2(qˉϱ+(1q)a)aDp,qˉϱˉΞ1(ˉϱ) adˉTp,qˉϱ.

    Now by making use of Theorem 3.4, we have

    xcˉΞ1(pˉϱ+(1p)a)aDp,qˉϱˉΞ2(ˉϱ) adˉTp,qˉϱ=qˉΞ1(ˉϱ)ˉΞ2(ˉϱ)+pˉΞ1(qˉϱ+(pq)ap)ˉΞ2(qˉϱ+(pq)ap)2q|xcxcˉΞ2(qˉϱ+(1q)a)aDp,qˉϱˉΞ1(ˉϱ) adˉTp,qˉϱ.

    This completes the proof.

    Theorem 3.6. Let ˉΞ1,ˉΞ2:IR be continuous functions such that ˉΞ1(ˉϱ)ˉΞ2(ˉϱ) for all ˉϱ[a,b], then we have

    xaˉΞ1(ˉϱ) adˉTp,qˉϱxaˉΞ2(ˉϱ) adˉTp,qˉϱ,

    for x[a,qb+(1p)a].

    Proof. By definition aˉTp,q integral, we have

    xaˉΞ1(ˉϱ) adˉTp,qˉϱ=(pq)(xa)2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ1(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a)ˉΞ1(x+(p1)ap)]=(pq)(xa)(p+q)2q˜ϖ=1q˜ϖp˜ϖ+1ˉΞ1(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a)(pq)(xa)(p+q)2q˜ϖ=1q˜ϖp˜ϖ+1ˉΞ2(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a)=(pq)(xa)2q[(p+q)˜ϖ=0q˜ϖp˜ϖ+1ˉΞ2(q˜ϖp˜ϖ+1x+(1q˜ϖp˜ϖ+1)a)ˉΞ2(x+(p1)ap)]=xaˉΞ2(ˉϱ) adˉTp,qˉϱ.

    This completes the proof.

    In this section, we derive some new aˉTp,q-analogues of certain classical inequalities.

    Theorem 4.1. Let a,b>0 and 1α+1β=1 with α>1, then

    a.b12[(pα1+qα1)[α]p,qaα+(pβ1+qβ1)[β]p,qbβ].

    Proof. Choose g=xα1 functions for α>1 with 1α+1β=1. We draw the graph of g=xα1.

    c1=a0xα1 0dˉTp,qx=(pα1+qα1)2[α]p,qaα.

    and

    c2=b0g1α1 0dˉTp,qg=(pβ1+qβ1)2[β]p,qbβ.

    According to Figure 1,

    a.bc1+c212[(pα1+qα1)[α]p,qaα+(pβ1+qβ1)[β]p,qbβ].
    Figure 1.  g=xα1.

    This completes the proof.

    Theorem 4.2. Let 1α+1β=1 with α>1, then we have

    baˉΞ1(ˉϱ)ˉΞ2(ˉϱ) adˉTp,qˉϱ=[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q]ˉΞ1αˉΞ2β,

    where ˉΞ1r=(ba|ˉΞ1(ˉϱ)|radˉp,ˉqˉϱ)1r.

    Proof. By taking a=|ˉΞ1(ˉϱ)|ˉΞ1α , b=|ˉΞ2(ˉϱ)|ˉΞ2β and using aˉTp,q-Young Inequality,

    |ˉΞ1(ˉϱ)|ˉΞ1α|ˉΞ2(ˉϱ)|ˉΞ2β(pα1+qα1)2[α]p,q|ˉΞ1(ˉϱ)|αˉΞ1αα+(pβ1+qβ1)2[β]p,q|ˉΞ2(ˉϱ)|βˉΞ2ββ.

    Now by taking aˉTp,q-integral of the above inequality,

    ba|ˉΞ1(ˉϱ)||ˉΞ2(ˉϱ)| adˉTp,qˉϱˉΞ1αˉΞ2βba(pα1+qα1)2[α]p,q|ˉΞ1(ˉϱ)|αftαα adˉTp,qˉϱ+(pβ1+qβ1)2[β]p,qba|ˉΞ2(ˉϱ)|βˉΞ2ββ adˉTp,qˉϱ(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q.

    This completes the proof.

    Theorem 4.3. Let 1α+1β=1 with α>1, then we have

    ˉΞ1+ˉΞ2α=[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q][ˉΞ1α+ˉΞ2α],

    where ˉΞ1r=(ba|ˉΞ1(ˉϱ)|r adˉTp,qˉϱ)1r.

    Proof. From aˉTp,q Hölder's inequality, we get

    ˉΞ1+ˉΞ2ααba|ˉΞ1(ˉϱ)||ˉΞ1(ˉϱ)+ˉΞ2(ˉϱ)|α1 adˉTp,qˉϱ+ba|ˉΞ2(ˉϱ)||ˉΞ1(ˉϱ)+ˉΞ2(ˉϱ)|α1 adˉTp,qˉϱ[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q](ba|ˉΞ1(ˉϱ)+ˉΞ2(ˉϱ)|β(α1) adˉTp,qˉϱ)1β×((ba|ˉΞ1(ˉϱ)|α adˉTp,qˉϱ)1α+(ba|ˉΞ1(ˉϱ)|α adˉTp,qˉϱ)1α)ˉΞ1+ˉΞ2α1α[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q][ˉΞ1α+ˉΞ2α].

    Dividing both sides by ˉΞ1+ˉΞ2α1α,we obtain our required result.

    Lemma 4.1. Let ˉΞ:[a,b]R be (p,q) -differentiable function (a,b), then for 0<q<p1, then

    12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ=1p(ba)[xap(ˉϱa)aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ+qb+(1p)axp(ˉϱb)aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ].

    Proof. Consider the right hand side and making use of Theorem 3.5, we have

    1ba[xap(ˉϱa)aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ+bx(pˉϱ+(1p)ab)aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ]=1ba[I3+I4], (4.1)

    where

    I3=xap(ˉϱa)aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ=q(ˉϱa)ˉΞ(ˉϱ)+(qˉϱ+(pq)apa)ˉΞ(qˉϱ+(pq)ap)2q|xaxaˉΞ(qˉϱ+(1q)a)aDp,q(ˉϱa) adˉTp,qˉϱ=q(xa)ˉΞ(x)+q(xa)ˉΞ(qx+(pq)ap)2qxaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ.

    Similarly

    I4=qb+(1p)axp(ˉϱb)aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ=qb+(1p)ax(pˉϱ+(1p)a(qb+(1p)a))aDp,qˉΞ(ˉϱ) adˉTp,qˉϱ=q(qb+(1p)ax)ˉΞ(x)p(pq)(ba)ˉΞ(qb+(1q)a)[q(xa)p2(ba)]ˉΞ(qx+(pq)ap)2qqb+(1p)axˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ.

    By substituting the values of I1 and I2 in (4.1), we obtain our required result.

    Lemma 4.2. Let p>1 and (aˉϱ)˜ϖp,q is (p,q)-binomial, then

    x0(aˉϱ)˜ϖp,q 0dˉTp,qˉϱ=a˜ϖ+1(p+q)2p[˜ϖ+1]p,q(q(apxq)˜ϖ+1+p(ax)˜ϖ+12p[˜ϖ+1]p,q).

    Proof. By using (p,q)-binomial formula and Gauss binomial formula

    x0(aˉϱ)˜ϖp,q 0dˉTp,qˉϱ=x0˜ϖk=0(1)k(˜ϖk)p,qp(˜ϖk)(˜ϖk1)2qk(k1)2ˉϱka˜ϖk 0dˉTp,qˉϱ=x0˜ϖk=0(1)k[˜ϖ]p,q![k]p,q![˜ϖk]p,q!p,qp(˜ϖk)(˜ϖk1)2qk(k1)2ˉϱka˜ϖk 0dˉTp,qˉϱ=˜ϖk=0(1)k[˜ϖ]p,q![k]p,q![˜ϖk]p,q!p,qp(˜ϖk)(˜ϖk1)2qk(k1)2a˜ϖkx0ˉϱk 0dˉTp,qˉϱ=˜ϖk=0(1)k[˜ϖ]p,q![k]p,q![˜ϖk]p,q!p(˜ϖk)(˜ϖk1)2qk(k1)2a˜ϖk(pk+qk)xk+12[k+1]p,q=˜ϖk=0(1)k[˜ϖ]p,q![k+1]p,q![˜ϖk]p,q!p(˜ϖk)(˜ϖk1)2qk(k1)2a˜ϖk(pk+qk)xk+12.

    Replacing k by k1

    =˜ϖ+1k=1(1)k1[˜ϖ+1]p,q![k]p,q![˜ϖ+1k]p,q!p(˜ϖk+1)(˜ϖk)2q(k1)(k2)2a˜ϖ+1k(pk+qk)xk2=12[˜ϖ+1]p,q[qp˜ϖ+1k=1(1)k1[˜ϖ]p,q![k]p,q![˜ϖ+1k]p,q!p(˜ϖk+1)(˜ϖk)2qk(k1)2a˜ϖ+1kxkqk2+˜ϖ+1k=1(1)k1[˜ϖ+1]p,q![k]p,q![˜ϖ+1k]p,q!p(˜ϖk+1)(˜ϖk)2qk(k1)2a˜ϖ+1kxk]=a˜ϖ+1(p+q)2p[˜ϖ+1]p,q(q(apxq)˜ϖ+1+p(ax)˜ϖ+12p[˜ϖ+1]p,q).

    Example 4.1. Let p>1 and (1ˉϱ)˜ϖp,q is (p,q)-binomial, then

    10(1ˉϱ)˜ϖp,q 0dˉTp,qˉϱ=(p+q)2p[˜ϖ+1]p,q.

    Theorem 4.4. Let ˉΞ:I=[a,b]R be a (p,q)-differentiable convex function on (a,b) and |aDp,qˉΞ(x)|M for all x[a,b], then we have

    |12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ|Mp(ba)((xa)2+p2(ba)22p(xa)(ba)).

    Proof. Using Lemma 4.1, modulus property and |aDp,qˉΞ(ˉϱ)|M

    |12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ|1p(ba)[xap|(ˉϱa)||aDp,qˉΞ(ˉϱ)| adˉTp,qˉϱ+qb+(1p)ax|p(ˉϱb)||aDp,qˉΞ(ˉϱ)| adˉTp,qˉϱ]Mp(ba)[xap(ˉϱa) adˉTp,qˉϱ+qb+(1p)axp(bˉϱ) adˉTp,qˉϱ]Mp(ba)((xa)22+p2(ba)22p(xa)(ba)+(xa)22)Mp(ba)((xa)2+p2(ba)22p(xa)(ba)).

    This completes the proof.

    Theorem 4.5. Let ˉΞ:I=[a,b]R be a (p,q)-differentiable on (a,b) and |aDp,qˉΞ(x)| is convex for all x[a,b], then we have

    |12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ|1(ba)2[(p2+q2p2[3]p,q(xa)3+p2q2[3]p,q(ba)3(ba)(xa)22p)|aDp,qˉΞ(b)|+(p2+q22[3]p,q(ba)3(ba)(xa)22pp2+q2p2[3]p,q(xa)3(ba)2(xa))|aDp,qˉΞ(a)|].

    Proof. Using Lemma 4.1, modulus property and convexity of |aDp,qˉΞ(ˉϱ)|, then we have

    |12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ|1p(ba)[xap|(ˉϱa)||aDp,qˉΞ(ˉϱ)| adˉTp,qˉϱ+qb+(1p)ax|p(ˉϱb)||aDp,qˉΞ(ˉϱ)| adˉTp,qˉϱ]1p(ba)[xap(ˉϱa)[ˉϱaba|aDp,qˉΞ(b)|+bˉϱba|aDp,qˉΞ(a)|] adˉTp,qˉϱqb+(1p)axp(bˉϱ)[ˉϱaba|aDp,qˉΞ(b)|+bˉϱba|aDp,qˉΞ(a)|] adˉTp,qˉϱ]=1(ba)2[|aDp,qˉΞ(b)|(xa(ˉϱa)2 adˉTp,qˉϱ+qb+(1p)ax(bˉϱ)(ˉϱa) adˉTp,qˉϱ)++|aDp,qˉΞ(a)|(xa(ˉϱa)(bˉϱ) adˉTp,qˉϱ+qb+(1p)ax(bˉϱ)2 adˉTp,qˉϱ)]=1(ba)2[|aDp,qˉΞ(b)|(p2+q2p2[3]p,q(xa)3+p2q2[3]p,q(ba)3(ba)(xa)22p)+|aDp,qˉΞ(a)|(p2+q22[3]p,q(ba)3(ba)(xa)22pp2+q2p2[3]p,q(xa)3(ba)2(xa))],

    which proves the required result.

    Theorem 4.6. Let ˉΞ:I=[a,b]R be a (p,q)-differentiable on (a,b) and |aDp,qˉΞ(x)|β is convex for all x[a,b] with 1α+1β=1 then we have

    |12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ|1(ba)2[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q]×[(xa)2((pα+qα)2pα[α+1]p,q)1α×((xa)|aDp,qˉΞ(b)|β+(2p(ba)(xa))|aDp,qˉΞ(a)|β2p)1β+(qb+(1p)ax(bˉϱ)α adˉTp,qˉϱ)1α×(((p2(ba)2(xa)2)|aDp,qˉΞ(b)|β+(p(ba)(xa))2|aDp,qˉΞ(a)|β2p))1β].

    Proof. Using Lemma 4.1, modulus property, Hölder's inequality and convexity of |aDp,qˉΞ(ˉϱ)|β, then

    |12q[qˉΞ(x)+pˉΞ(qx+(pq)ap)(pq)ˉΞ(qb+(pq)a)]1p(ba)qb+(1p)aaˉΞ(qˉϱ+(1q)a) adˉTp,qˉϱ|1p(ba)[xap|(ˉϱa)||aDp,qˉΞ(ˉϱ)| adˉTp,qˉϱ+qb+(1p)ax|p(ˉϱb)||aDp,qˉΞ(ˉϱ)| adˉTp,qˉϱ]1ba[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q]×[(xa(ˉϱa)α adˉTp,qˉϱ)1α(xa|aDp,qˉΞ(ˉϱ)|β adˉTp,qˉϱ)1β+(qb+(1p)ax(bˉϱ)α adˉTp,qˉϱ)1α(qb+(1p)ax|aDp,qˉΞ(ˉϱ)|β adˉTp,qˉϱ)1β]1ba[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q]×[((pα+qα)(xa)α+12pα[α+1]p,q)1α(xa[ˉϱaba|aDp,qˉΞ(b)|β+bˉϱba|aDp,qˉΞ(a)|β] adˉTp,qˉϱ)1β+(qb+(1p)ax(bˉϱ)α adˉTp,qˉϱ)1α(qb+(1p)ax[ˉϱaba|aDp,qˉΞ(b)|β+bˉϱba|aDp,qˉΞ(a)|β] adˉTp,qˉϱ)1β]=1(ba)2[(pα1+qα1)2[α]p,q+(pβ1+qβ1)2[β]p,q]×[((pα+qα)(xa)α+12pα[α+1]p,q)1α×((xa)22p|aDp,qˉΞ(b)|β+((ba)(xa)(xa)22p)|aDp,qˉΞ(a)|β)1β+(qb+(1p)ax(bˉϱ)α adˉTp,qˉϱ)1α×((p2(ba)2(xa)22p)|aDp,qˉΞ(b)ψβ+(p(ba)22(ba)(xa)+(xa)22p)|aDp,qˉΞ(a)|β)1β].

    This completes the proof.

    We now derive some new variants of classical Hermite-Hadamard's inequality essentially using aˉTp,q-integrals.

    Theorem 4.7. Let ˉΞ:I=[a,b]R be a continuous and convex function on (a,b) with 0<q<p1, we have

    ˉΞ(a+b2)1p(ba)pb+(1p)aaˉΞ(x) adˉTp,qxˉΞ(a)+ˉΞ(b)2. (4.2)

    Proof. Since ˉΞ is a differentiable function at [a,b] then there exist a tangent at a+b2(a,b), tangent line (see Figure 2) can be expressed as function ˉΞ1(x)=ˉΞ(a+b2)+ˉΞ(a+b2)(xa+b2).

    Figure 2.  ˉΞ1(x)=ˉΞ(a+b2)+ˉΞ(a+b2)(xa+b2).

    Since ˉΞ is convex function on [a,b], then

    ˉΞ1(x)=ˉΞ(a+b2)+ˉΞ(a+b2)(xa+b2)ˉΞ(x),

    for all x[a,b]. By Theorem 3.6, we have

    pb+(1p)aaˉΞ1(x) adˉTp,qxpb+(1p)aaˉΞ(x) adˉTp,qx.

    By definiton of aˉTp,q-integrals, we have

    pb+(1p)aaˉΞ1(x) adˉTp,qx=pb+(1p)aa[ˉΞ(a+b2)+ˉΞ(a+b2)(xa+b2)] adˉTp,qx=p(ba)ˉΞ(a+b2)+ˉΞ(a+b2)pb+(1p)aa(xaba2) adˉTp,qx=p(ba)ˉΞ(a+b2). (4.3)

    This completes the first ineqıality in (4.2).

    Also the line connecting the points (a,ˉΞ(a)) and (b,ˉΞ(b)) is expressed as

    ˉΞ(x)k(x)=ˉΞ(a)+ˉΞ(b)ˉΞ(a)ba(xa).

    This implies

    pb+(1p)aaˉΞ(x) adˉTp,qxpb+(1p)aa(ˉΞ(a)+ˉΞ(b)ˉΞ(a)ba(xa)) adˉTp,qxˉΞ(a)p(ba)+ˉΞ(b)ˉΞ(a)bap(ba)22p(ba)ˉΞ(a)+ˉΞ(b)2. (4.4)

    This compeletes the proof.

    Theorem 4.8. Let ˉΞ:I=[a,b]R be a convex continuous function on (a,b) with 0<q<p1, we have

    ˉΞ(qa+qbp+q)(ba)(pq)2(p+q)ˉΞ(qa+qbp+q)1p(ba)pb+(1p)aaˉΞ(x) adˉTp,qxˉΞ(a)+ˉΞ(b)2. (4.5)

    Proof. Since ˉΞ is a differentiable function at [a,b] then there exist a tangent at a+b2(a,b), tangent line can be expressed as function ˉΞ2(x)=ˉΞ(qa+qbp+q)+ˉΞ(qa+qbp+q)(xqa+qbp+q). since ˉΞ is convex function on [a,b], then

    ˉΞ2(x)=ˉΞ(qa+qbp+q)+ˉΞ(qa+qbp+q)(xqa+qbp+q)ˉΞ(x),x[a,b].

    Taking aˉTp,q-integrals of the above inequality over [a,b],

    pb+(1p)aaˉΞ2(x) adˉTp,qx=pb+(1p)aa(ˉΞ(qa+qbp+q)+ˉΞ(qa+qbp+q)(xqa+qbp+q)) adˉTp,qx=p(ba)ˉΞ(qa+qbp+q)+ˉΞ(qa+qbp+q)pb+(1p)aa(xapbap+q) adˉTp,qx=p(ba)ˉΞ(qa+qbp+q)ˉΞ(qa+qbp+q)p(ba)2(pq)2(p+q)pb+(1p)aaˉΞ(x) adˉTp,qx. (4.6)

    Comparing (4.6) and (4.4), we obtain our required result.

    Theorem 4.9. Let ˉΞ:I=[a,b]R be a convex continuous function on (a,b) with 0<q<p1, then

    ˉΞ(pa+qbp+q)+(ba)(pq)2(p+q)ˉΞ(pa+qbp+q)1p(ba)pb+(1p)aaˉΞ(x) adˉTp,qxˉΞ(a)+ˉΞ(b)2. (4.7)

    Proof. As ˉΞ is differentiable function on (a,b), so there exist a tangent at pa+qbp+q(a,b), tangent line can be expressed as function ˉΞ3(x)=ˉΞ(pa+qbp+q)+ˉΞ(pa+qbp+q)(xpa+qbp+q). since ˉΞ is convex function on [a,b], then

    ˉΞ3(x)=ˉΞ(pa+qbp+q)+ˉΞ(pa+qbp+q)(xpa+qbp+q)ˉΞ(x),x[a,b].

    Taking aˉTp,q-integrals of the above inequality over [a,b], we have

    pb+(1p)aaˉΞ3(x) adˉTp,qx=pb+(1p)aaˉΞ(pa+qbp+q)+ˉΞ(pa+qbp+q)(xpa+qbp+q) adˉTp,qx=p(ba)ˉΞ(pa+qbp+q)+ˉΞ(pa+qbp+q)pb+(1p)aa(xaqbap+q) adˉTp,qx=p(ba)ˉΞ(pa+qbp+q)+ˉΞ(pa+qbp+q)(p(ba)22qp(ba)2p+q)=p(ba)ˉΞ(pa+qbp+q)+p(ba)2(pq)2(p+q)ˉΞ(pa+qbp+q)pb+(1p)aaˉΞ(x) adˉTp,qx. (4.8)

    Comparing (4.8) and (4.4), we obtain our required result.

    Theorem 4.10. Let ˉΞ:I=[a,b]R be a convex continuous function on (a,b) with 0<q<p1, we have

    max{N1,N2,N3}1p(ba)pb+(1p)aaˉΞ(x) adˉTp,qxˉΞ(a)+ˉΞ(b)2,

    where

    N1=ˉΞ(a+b2),N2=ˉΞ(qa+qbp+q)(ba)(pq)2(p+q)ˉΞ(qa+qbp+q),N3=ˉΞ(pa+qbp+q)+(ba)(pq)2(p+q)ˉΞ(pa+qbp+q).

    Proof. Combining (4.2), (4.5) and (4.7), we can obtain our required result easily.

    We defined new post quantum integrals using trapezoidal strips and discussed their basic properties. Utilizing these integrals, we have established new (p,q) variants of Young's, Hölder's, Minkowski's, Ostrowski's and Hermite-Hadamard's inequalities. It is worth mentioning that by using these integrals, one can also develop new analogues of Chebychev and, Hermite-Hadamard-Mercer, Gruss-like inequalities. In future, we will extend the idea for q-fractional and interval analysis to obtain some new refinements of classical inequalities.

    The authors are thankful to the editor and the anonymous reviewers for their valuable comments and suggestions. Muhammad Uzair Awan is thankful to HEC Pakistan for 8081/Punjab/NRPU/R&D/HEC/2017.

    This research is supported by Researchers Supporting Project number (RSP-2021/158), King Saud University, Riyadh, Saudi Arabia.

    The authors declare that they have no conflicts of interests.



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