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The following inequality is named the Ostrowski type inequality [30].
Theorem 1. [10] Let f:[a,b]→R be a differentiable mapping on (a,b) and f′∈L[a,b] (i.e. f′ be an integrable function on [a,b]). If |f′(x)|<M on [a,b], then the following inequality holds:
|f(x)−1b−a∫baf(t)dt|≤Mb−a[(x−a)2+(b−x)22] | (1.1) |
for all x∈[a,b].
To prove the Ostrowski type inequality in (1.1), the following identity is used, (see [26]):
f(x)=1b−a∫baf(t)dt+∫xat−ab−af′(t)dt+∫bxt−bb−af′(t)dt, | (1.2) |
where f(x) is a continuous function on [a,b] with a continuous first derivative in (a,b). The identity (1.2) is known as Montgomery identity.
By changing variable, the Montgomery identity (1.2) can be expressed as:
f(x)−1b−a∫baf(t)dt=(b−a)∫10K(t)f′(tb+(1−t)a)dt, | (1.3) |
where
K(t)={t,t∈[0,x−ab−a],t−1,t∈(x−ab−a,1]. |
A number of different identities of the Montgomery and many inequalities of Ostrowski type were obtained by using these identities. Through the framework of Montgomery's identity, Cerone and Dragomir [9] developed a systematic study which produced some novel inequalities. By introducing some parameters, Budak and Sarıkaya [8] as well as Özdemir et al. [31] established the generalized Montgomery-type identities for differentiable mappings and certain generalized Ostrowski-type inequalities, respectively. Aljinović in [1], presented another simpler generalization of the Montgomery identity for fractional integrals by utilizing the weighted Montgomery identity. Furthermore, the generalized Montgomery identity involving the Ostrowski type inequalities in question with applications to local fractional integrals can be found in [32]. For more related results considering the different Montgomery identities, [2,4,7,11,12,13,15,16,21,22,23,24,25,33,34,36] and the references therein can be seen.
In the related literature of Montgomery type identity, it was not considered via quantum integral operators. The aim of this work is to set up a quantum Montgomery identity with respect to quantum integral operators. With the help of this new version of Montgomery identity, some new quantum integral inequalities such as Ostrowski type, midpoint type, etc are established. The absolute values of the derivatives of considered mappings are quantum differentiable convex mappings.
Throughout this paper, let 0<q<1 be a constant. It is known that quantum calculus constructs in a quantum geometric set. That is, if qx∈A for all x∈A, then the set A is called quantum geometric.
Suppose that f(t) is an arbitrary function defined on the interval [0,b]. Clearly, for b>0, the interval [0,b] is a quantum geometric set. The quantum derivative of f(t) is defined with the following expression:
Dqf(t):=f(t)−f(qt)(1−q)t,t≠0, | (1.4) |
Dqf(0):=limt→0Dqf(t). |
Note that
limq→1−Dqf(t)=df(t)dt, | (1.5) |
if f(t) is differentiable.
The quantum integral of f(t) is defined as:
∫b0f(t) dqt=(1−q)b∞∑n=0qnf(qnb) | (1.6) |
and
∫bcf(t) dqt=∫b0f(t) dqt−∫c0f(t) dqt, | (1.7) |
Note that if the series in right-hand side of (1.6) is convergence, then ∫b0f(t) dqt is exist, i.e., f(t) is quantum integrable on [0,b]. Also, provided that if ∫b0f(t) dt converges, then one has
limq→1−∫b0f(t) dqt=∫b0f(t) dt. (see [3, page 6]). | (1.8) |
These definitions are not sufficient in establishing integral inequalities for a function defined on an arbitrary closed interval [a,b]⊂R. Due to this fact, Tariboon and Ntouyas in [37,38] improved these definitions as follows:
Definition 1. [37,38]. Let f:[a,b]→R be a continuous function. The q-derivative of f at t∈[a,b] is characterized by the expression:
aDqf(t)=f(t)−f(qt+(1−q)a)(1−q)(t−a), t≠a, | (1.9) |
aDqf(a)=limt→aaDqf(t). |
The function f is said to be q-differentiable on [a,b], if aDqf(t) exists for all t∈[a,b].
Clearly, if a=0 in (1.9), then0Dqf(t)=Dqf(t), where Dqf(t) is familiar quantum derivatives given in (1.4).
Definition 2. [37,38]. Let f:[a,b]→R be a continuous function. Then the quantum definite integral on [a,b] is defined as
∫baf(t)adqt=(1−q)(b−a)∞∑n=0qnf(qnb+(1−qn)a) | (1.10) |
and
∫bcf(t)adqt=∫baf(t)adqt−∫caf(t)adqt, | (1.11) |
where a<c<b.
Clearly, if a=0 in (1.10), then
∫b0f(t)0dqt=∫b0f(t)dqt, |
where ∫b0f(t)dqt is familiar definite quantum integrals on [0,b] given in (1.6).
Definition 1 and Definition 2 have actually developed previous definitions and have been widely used for quantum integral inequalities. There is a lot of remarkable papers about quantum integral inequalities based on these definitions, including Kunt et al. [19] in the study of the quantum Hermite–Hadamard inequalities for mappings of two variables considering convexity and quasi-convexity on the co-ordinates, Noor et al. [27,28,29] in quantum Ostrowski-type inequalities for quantum differentiable convex mappings, quantum estimates for Hermite–Hadamard inequalities via convexity and quasi-convexity, quantum analogues of Iyengar type inequalities for some classes of preinvex mappings, as well as Tunç et al. [39] in the Simpson-type inequalities for convex mappings via quantum integral operators. For more results related to the quantum integral operators, the interested reader is directed to [5,18,20,35,41] and the references cited therein.
In [6], Alp et al. proved the following inequality named quantum Hermite–Hadamard type inequality. Also in [40], Zhang et al. proved the same inequality with the fewer assumptions and shorter method.
Theorem 2. Let f:[a,b]→R be a convex function with 0<q<1. Then we have
f(qa+b1+q)≤1b−a∫baf(t)adqt≤qf(a)+f(b)1+q. | (1.12) |
Firstly, we discuss the assumptions of the continuity of the function f(t) in Definition 1 and Definition 2. Also, under these conditions, we want to discuss that similar cases with (1.5) and (1.8) can exist.
By considering the Definition 1, it is not necessary that the function f(t) must be continuous on [a,b]. Indeed, for all t∈[a,b], qt+(1−q)a∈[a,b] and f(t)−f(qt+(1−q)a)∈R. It means that f(t)−f(qt+(1−q)a)(1−q)(t−a)∈R exists for all t∈(a,b], so the Definition 1 should be as follows:
Definition 3. (Quantum derivative on [a,b]) Let f:[a,b]→R be an arbitrary function. Then f is called quantum differentiable on (a,b] with the following expression:
aDqf(t)=f(t)−f(qt+(1−q)a)(1−q)(t−a)∈R, t≠a | (2.1) |
and f is called quantum differentiable on t=a, if the following limit exists:
aDqf(a)=limt→aaDqf(t). |
Lemma 1. (Similar case with (1.5)) Let f:[a,b]→R be a differentiable function. Then we have
limq→1−aDqf(t)=df(t)dt. | (2.2) |
Proof. Since f is differentiable on [a,b], clearly we have
limh→0−f(t+h)−f(t)h=df(t)dt | (2.3) |
for all t∈(a,b]. Since 0<q<1, for all a<t≤b, we have (1−q)(a−t)<0. Changing variable in (2.2) as (1−q)(a−t)=h, then q→1− we have h→0− and qt+(1−q)a=t+h. Using (2.3), we have
limq→1−aDqf(t)=limq→1−f(t)−f(qt+(1−q)a)(1−q)(t−a)=limq→1−f(qt+(1−q)a)−f(t)(1−q)(a−t)=limh→0−f(t+h)−f(t)h=limh→0f(t+h)−f(t)h=df(t)dt |
for all t∈(a,b]. On the other hand, for t=a we have
limq→1−aDqf(a)=limq→1−limt→aaDqf(t)=limt→alimq→1−aDqf(t)=limt→adf(t)dt=limt→alimh→0f(t+h)−f(t)h=limt→alimh→0+f(t+h)−f(t)h=limh→0+limt→af(t+h)−f(t)h=limh→0+f(a+h)−f(a)h=df(a)dt, |
which completes the proof.
In Definition 2, the condition of the continuity of the function f(t) on [a,b] is not required. For this purpose, it is enough to construct an example in which a function is discontinuous on [a,b], but quantum integrable on it.
Example 1. Let 0<q<1 be a constant, and the set A is defined as
A:={qn2+(1−qn)(−1):n=0,1,2,...,}⊂[−1,2]. |
Then the function f:[−1,2]→R defined as
f(t):={1,t∈A,0,t∈[−1,2]∖A. |
Clearly, it is not continuous on [−1,2]. On the other hand
∫2−1f(t)−1dqt=(1−q)(2−(−1))∞∑n=0qnf(qn2+(1−qn)(−1))=3(1−q)∞∑n=0qn=3(1−q)11−q=3, |
i.e., the function f(t) is quantum integrable on [−1,2].
Hence the Definition 2 should be described in the following way.
Definition 4. (Quantum definite integral on [a,b]) Let f:[a,b]→R be an arbitrary function. Then the quantum integral of f on [a,b] is defined as
∫baf(t)adqt=(1−q)(b−a)∞∑n=0qnf(qnb+(1−qn)a). | (2.4) |
If the series in right-hand side of (2.4) is convergent, then ∫baf(t)adqt is exist, i.e., f(t) is quantum integrable on [a,b].
Lemma 2. (Similar case with (1.8)) Let f:[a,b]→R, be an arbitrary function. It provided that if ∫baf(t) converges, then we have
limq→1−∫baf(t)adqt=∫baf(t)dt. | (2.5) |
Proof. If ∫baf(t) dt converges, then ∫10f(tb+(1−t)a) dt also converges. Using (1.8), we have that
limq→1−∫baf(t)adqt=limq→1−[(1−q)(b−a)∞∑n=0qnf(qnb+(1−qn)a)]=(b−a)limq→1−∫10f(tb+(1−t)a)0dqt=(b−a)∫10f(tb+(1−t)a)dt=∫baf(t) dt. |
Next we present an important quantum Montgomery identity, which is similar with the identity in (1.3).
Lemma 3. (Quantum Montgomery identity) Let f:[a,b]→R, be an arbitrary function with aDqf is quantum integrable on [a,b], then the following quantum identity holds:
f(x)−1b−a∫baf(t)adqt=(b−a)∫10Kq(t)aDqf(tb+(1−t)a)0dqt, | (2.6) |
where
Kq(t)={qt,t∈[0,x−ab−a],qt−1,t∈(x−ab−a,1]. |
Proof. By the Definition 3, f(t) is quantum differentiable on (a,b) and aDqf is exist. Since aDqf is quantum integrable on [a,b], by the Definition 4, the quantum integral for the right-side of (2.6) is exist. The integral of the right-side of (2.6), with the help of (2.1) and (2.4), is equal to
(b−a)∫10Kq(t)aDqf(tb+(1−t)a)0dqt=(b−a)[∫x−ab−a0qtaDqf(tb+(1−t)a)0dqt+∫1x−ab−a(qt−1)aDqf(tb+(1−t)a)0dqt]=(b−a)[∫x−ab−a0qtaDqf(tb+(1−t)a)0dqt+∫10(qt−1)aDqf(tb+(1−t)a)0dqt−∫x−ab−a0(qt−1)aDqf(tb+(1−t)a)0dqt]=(b−a)[∫10(qt−1)aDqf(tb+(1−t)a)0dqt+∫x−ab−a0aDqf(tb+(1−t)a)0dqt]=(b−a)[∫10qtaDqf(tb+(1−t)a)0dqt−∫10aDqf(tb+(1−t)a)0dqt+∫x−ab−a0aDqf(tb+(1−t)a)0dqt]=(b−a)[∫10qtf(tb+(1−t)a)−f(qtb+(1−qt)a)(1−q)t(b−a)0dqt−∫10f(tb+(1−t)a)−f(qtb+(1−qt)a)(1−q)t(b−a)0dqt+∫x−ab−a0f(tb+(1−t)a)−f(qtb+(1−qt)a)(1−q)t(b−a)0dqt]=11−q[q[∫10f(tb+(1−t)a)0dqt−∫10f(qtb+(1−qt)a)0dqt]−[∫10f(tb+(1−t)a)t0dqt−∫10f(qtb+(1−qt)a)t0dqt]+[∫x−ab−a0f(tb+(1−t)a)t0dqt−∫x−ab−a0f(qtb+(1−qt)a)t0dqt]]=11−q[q[(1−q)∞∑n=0qnf(qnb+(1−qn)a)−(1−q)∞∑n=0qnf(qn+1b+(1−qn+1)a)]−[(1−q)∞∑n=0qnf(qnb+(1−qn)a)qn−(1−q)∞∑n=0qnf(qn+1b+(1−qn+1)a)qn]+[(1−q)x−ab−a∞∑n=0qnf(qnx−ab−ab+(1−qnx−ab−a)a)qnx−ab−a−(1−q)x−ab−a∞∑n=0qnf(qn+1x−ab−ab+(1−qn+1x−ab−a)a)qnx−ab−a]]=[q[∞∑n=0qnf(qnb+(1−qn)a)−∞∑n=0qnf(qn+1b+(1−qn+1)a)]−[∞∑n=0f(qnb+(1−qn)a)−∞∑n=0f(qn+1b+(1−qn+1)a)]+[∞∑n=0f(qnx−ab−qb+(1−qnx−ab−q)a)−∞∑n=0f(qn+1x−ab−qb+(1−qn+1x−ab−q)a)]]=[q[∞∑n=0qnf(qnb+(1−qn)a)−1q∞∑n=1qnf(qnb+(1−qn)a)]−[∞∑n=0f(qnb+(1−qn)a)−∞∑n=1f(qnb+(1−qn)a)]+[∞∑n=0f(qn(x−ab−a)b+(1−qn(x−ab−a))a)−∞∑n=1f(qn(x−ab−a)b+(1−qn(x−ab−a))a)]]=q[(1−1q)∞∑n=0qnf(qnb+(1−qn)a)+f(b)q]−f(b)+f((x−ab−a)b+(1−(x−ab−a))a)=f(x)−(1−q)∞∑n=0qnf(qnb+(1−qn)a)=f(x)−1b−a∫baf(t)adqt, |
which completes the proof.
Remark 1. If one takes limit q→1− on the Quantum Montgomery identity in (2.6), one has the Montgomery identity in (1.3).
The following calculations of quantum definite integrals are used in next result:
∫x−ab−a0qt 0dqt=q(1−q)x−ab−a∞∑n=0qn(x−ab−aqn)=q(1−q)(x−ab−a)211−q2=q1+q(x−ab−a)2, | (2.7) |
∫x−ab−a0qt2 0dqt=q(1−q)x−ab−a∞∑n=0qn(x−ab−aqn)2=q(1−q)(x−ab−a)311−q3=q1+q+q2(x−ab−a)3, | (2.8) |
∫1x−ab−a(1−qt) 0dqt=∫10(1−qt) 0dqt−∫x−ab−a0(1−qt) 0dqt=(1−q)∞∑n=0qn(1−qqn)−(1−q)x−ab−a∞∑n=0qn(1−qqnx−ab−a)=(1−q)(11−q−q1−q2)−(1−q)x−ab−a(11−q−q1−q2x−ab−a)=11+q−x−ab−a(1−q1+qx−ab−a)=11+q−(1−b−xb−a)(11+q+q1+q−q1+q(1−b−xb−a))=11+q−(1−b−xb−a)(11+q+q1+q(b−xb−a))=[11+q−11+q−q1+q(b−xb−a)+q1+q(b−xb−a)+q1+q(b−xb−a)2]=q1+q(b−xb−a)2, | (2.9) |
and
∫1x−ab−a(t−qt2) 0dqt=∫10(t−qt2) 0dqt−∫x−ab−a0(t−qt2) 0dqt=(1−q)∞∑n=0qn(qn−qq2n)−(1−q)x−ab−a∞∑n=0qn(qnx−ab−a−qq2n(x−ab−a)2)=(1−q)(11−q2−q1−q3)−(1−q)x−ab−a(11−q2x−ab−a−q1−q3(x−ab−a)2) |
=(11+q−q1+q+q2)−x−ab−a(11+qx−ab−a−q1+q+q2(x−ab−a)2)=1(1+q)(1+q+q2)−11+q(x−ab−a)2+q1+q+q2(x−ab−a)3. | (2.10) |
Let us introduce some new quantum integral inequalities by the help of quantum power mean inequality and Lemma 3.
Theorem 3. Let f:[a,b]→R be an arbitrary function with aDqf is quantum integrable on [a,b]. If |aDqf|r, r≥1 is a convex function, then the following quantum integral inequality holds:
|f(x)−1b−a∫baf(t) adqt|≤(b−a)[K1−1r1(a,b,x,q)[|aDqf(a)|rK2(a,b,x,q)+|aDqf(b)|rK3(a,b,x,q)]1r+K1−1r4(a,b,x,q)[|aDqf(a)|rK5(a,b,x,q)+|aDqf(b)|rK6(a,b,x,q)]1r] | (2.11) |
for all x∈[a,b], where
K1(a,b,x,q)=∫x−ab−a0qt0dqt=q1+q(x−ab−a)2, |
K2(a,b,x,q)=∫x−ab−a0qt20dqt=q1+q+q2(x−ab−a)3, |
K3(a,b,x,q)=∫x−ab−a0qt−qt20dqt=K1(a,b,x,q)−K2(a,b,x,q), |
K4(a,b,x,q)=∫1x−ab−a(1−qt)0dqt=q1+q(b−xb−a)2, |
K5(a,b,x,q)=∫1x−ab−a(t−qt2)0dqt=1(1+q)(1+q+q2)−11+q(x−ab−a)2+q1+q+q2(x−ab−a)3, |
and
K6(a,b,x,q)=∫1x−ab−a(1−qt−t+qt2)0dqt=K4(a,b,x,q)−K5(a,b,x,q). |
Proof. Using convexity of |aDqf|r, we have that
|aDqf(tb+(1−t)a)|r≤t|aDqf(a)|r+(1−t)|aDqf(b)|r. | (2.12) |
By using Lemma 3, quantum power mean inequality and (2.12), we have that
|f(x)−1b−a∫baf(t) adqt|≤(b−a)∫10|Kq(t)||aDqf(tb+(1−t)a)| 0dqt≤(b−a)[∫x−ab−a0qt|aDqf(tb+(1−t)a)| 0dqt+∫1x−ab−a(1−qt)|aDqf(tb+(1−t)a)| 0dqt]≤(b−a)[(∫x−ab−a0qt 0dqt)1−1r(∫x−ab−a0qt|aDqf(tb+(1−t)a)|r 0dqt)1r+(∫1x−ab−a(1−qt)0dqt)1−1r(∫1x−ab−a(1−qt)|aDqf(tb+(1−t)a)|r0dqt)1r]≤(b−a)[(∫x−ab−a0qt0dqt)1−1r(∫x−ab−a0qt[t|aDqf(a)|r+(1−t)|aDqf(b)|r]0dqt)1r+(∫1x−ab−a(1−qt)0dqt)1−1r(∫1x−ab−a(1−qt)[t|aDqf(a)|r+(1−t)|aDqf(b)|r]0dqt)1r]≤(b−a)[(∫x−ab−a0qt0dqt)1−1r(|aDqf(a)|r∫x−ab−a0qt20dqt+|aDqf(b)|r∫x−ab−a0qt−qt20dqt)1r+(∫1x−ab−a(1−qt)0dqt)1−1r(|aDqf(a)|r∫1x−ab−a(t−qt2)0dqt+|aDqf(b)|r∫1x−ab−a(1−qt−t+qt2)0dqt)1r]. | (2.13) |
Using (2.7)–(2.10) in (2.13), we obtain the desired result in (2.11). This ends the proof.
Corollary 1. In Theorem 3, the following inequalities are held by the following assumptions:
1. If one takes r=1, one has
|f(x)−1b−a∫baf(t)adqt|≤(b−a)[|aDqf(a)|K2(a,b,x,q)+|aDqf(b)|K3(a,b,x,q)+|aDqf(a)|K5(a,b,x,q)+|aDqf(b)|K6(a,b,x,q)]. |
2. If one takes r=1 and |aDqf(x)|<M for all x∈[a,b], then one has (a quantum Ostrowski type inequality, see [27,Theorem 3.1])
|f(x)−1b−a∫baf(t)adqt|≤M(b−a)[K2(a,b,x,q)+K3(a,b,x,q)+K5(a,b,x,q)+K6(a,b,x,q)]≤M(b−a)[K1(a,b,x,q)+K4(a,b,x,q)]≤M(b−a)[q1+q(x−ab−a)2+q1+q(b−xb−a)2]≤qMb−a[(x−a)2+(b−x)21+q]. |
3. If one takes r=1, |aDqf(x)|<M for all x∈[a,b] and q→1−, then one has (Ostrowski inequality (1.1)).
4. If one takes r=1 and x=qa+b1+q, then one has (a new quantum midpoint type inequality)
|f(qa+b1+q)−1b−a∫baf(t)adqt|≤(b−a)[[|aDqf(a)|K2(a,b,qa+b1+q,q)+|aDqf(b)|K3(a,b,qa+b1+q,q)]+|aDqf(a)|K5(a,b,qa+b1+q,q)+|aDqf(b)|K6(a,b,qa+b1+q,q)]≤(b−a)[|aDqf(a)|q(1+q)3(1+q+q2)+|aDqf(b)|q2+q3(1+q)3(1+q+q2)+|aDqf(a)|2q(1+q)3(1+q+q2)+|aDqf(b)|−2q+q3+q4+q5(1+q)3(1+q+q2)]≤(b−a)[|aDqf(a)|3q(1+q)3(1+q+q2)+|aDqf(b)|−2q+q2+2q3+q4+q5(1+q)3(1+q+q2)]. |
5. If one takes r=1, x=qa+b1+q and q→1−, then one has (a midpoint type inequality, see [17,Theorem 4])
|f(a+b2)−1b−a∫baf(t)dt|≤(b−a)[|f′(a)|+|f′(b)|]8. |
6. If one takes r=1 and x=a+b2, then one has (a new quantum midpoint type inequality)
|f(a+b2)−1b−a∫baf(t)adqt|≤(b−a)[|aDqf(a)|K2(a,b,a+b2,q)+|aDqf(b)|K3(a,b,a+b2,q)+|aDqf(a)|K5(a,b,a+b2,q)+|aDqf(b)|K6(a,b,a+b2,q)]≤(b−a)[|aDqf(a)|q8(1+q+q2)+|aDqf(b)|q+q2+2q38(1+q)(1+q+q2)+|aDqf(a)|6−q−q28(1+q)(1+q+q2)+|aDqf(b)|3q+3q2+2q3−68(1+q)(1+q+q2)]≤(b−a)[|aDqf(a)|68(1+q)(1+q+q2)+|aDqf(b)|4q+4q2+4q3−68(1+q)(1+q+q2)]. |
7. If one takes |aDqf(x)|<M for all x∈[a,b], then one has (a quantum Ostrowski type inequality, see [27,Theorem 3.1])
|f(x)−1b−a∫baf(t)adqt|≤(b−a)M[K1−1r1(a,b,x,q)[K2(a,b,x,q)+K3(a,b,x,q)]1r+K1−1r4(a,b,x,q)[K5(a,b,x,q)+K6(a,b,x,q)]1r]≤(b−a)M[K1−1r1(a,b,x,q)K1r1(a,b,x,q)+K1−1r4(a,b,x,q)K1r4(a,b,x,q)]≤(b−a)M[K1(a,b,x,q)+K4(a,b,x,q)]≤M(b−a)[q1+q(x−ab−a)2+q1+q(b−xb−a)2]≤qMb−a[(x−a)2+(b−x)21+q]. |
8. If one takes x=qa+b1+q, then one has (a new quantum midpoint type inequality)
|f(qa+b1+q)−1b−a∫baf(t)adqt|≤(b−a)[K1−1r1(a,b,qa+b1+q,q)[|aDqf(a)|rK2(a,b,qa+b1+q,q)+|aDqf(b)|rK3(a,b,qa+b1+q,q)]1r+K1−1r4(a,b,qa+b1+q,q)[|aDqf(a)|rK5(a,b,qa+b1+q,q)+|aDqf(b)|rK6(a,b,qa+b1+q,q)]1r]≤(b−a)[[q(1+q)3]1−1r[|aDqf(a)|rq(1+q)3(1+q+q2)+|aDqf(b)|rq2+q3(1+q)3(1+q+q2)]1r+[q3(1+q)3]1−1r[|aDqf(a)|r2q(1+q)3(1+q+q2)+|aDqf(b)|r−2q+q3+q4+q5(1+q)3(1+q+q2)]1r]. |
9. If one takes x=qa+b1+q and q→1−, then one has (a midpoint type inequality, see [6,Corollary 17])
|f(a+b2)−1b−a∫baf(t)dt|≤(b−a)123−3r[(|f′(a)|r124+|f′(b)|r112)1r+(|f′(a)|r112+|f′(b)|r124)1r]. |
10. If one takes x=a+b2, then one has (a new quantum midpoint type inequality)
|f(a+b2)−1b−a∫baf(t)adqt|≤(b−a)[K1−1r1(a,b,a+b2,q)[|aDqf(a)|rK2(a,b,a+b2,q)+|aDqf(b)|rK3(a,b,a+b2,q)]1r+K1−1r4(a,b,a+b2,q)[|aDqf(a)|rK5(a,b,a+b2,q)+|aDqf(b)|rK6(a,b,a+b2,q)]1r]≤(b−a)(q4(1+q))1−1r[[|aDqf(a)|rq8(1+q+q2)+|aDqf(b)|rq+q2+2q38(1+q)(1+q+q2)]1r+[|aDqf(a)|r6−q−q28(1+q)(1+q+q2)+|aDqf(b)|r3q+3q2+2q3−68(1+q)(1+q+q2)]1r]. |
11. If one takes x=a+qb1+q, then one has (a new quantum midpoint type inequality)
|f(a+qb1+q)−1b−a∫baf(t)adqt|≤(b−a)[K1−1r1(a,b,a+qb1+q,q)[|aDqf(a)|rK2(a,b,a+qb1+q,q)+|aDqf(b)|rK3(a,b,a+qb1+q,q)]1r+K1−1r4(a,b,a+qb1+q,q)[|aDqf(a)|rK5(a,b,a+qb1+q,q)+|aDqf(b)|rK6(a,b,a+qb1+q,q)]1r]≤(b−a)[[q3(1+q)3]1−1r[|aDqf(a)|rq4(1+q)3(1+q+q2)+|aDqf(b)|rq3+q5(1+q)3(1+q+q2)]1r+[q(1+q)3]1−1r[|aDqf(a)|r1+2q−q3(1+q)3(1+q+q2)+|aDqf(b)|r−1−q+q2+2q3(1+q)3(1+q+q2)]1r]. |
Finally, we give the following calculated quantum definite integrals used as the next Theorem 4.
∫x−ab−a0t0dqt=(1−q)x−ab−a∞∑n=0qn(qnx−ab−a)=(1−q)(x−ab−a)211−q2=11+q(x−ab−a)2, | (2.14) |
∫x−ab−a0(1−t)0dqt=(1−q)x−ab−a∞∑n=0qn(1−qnx−ab−a)=(1−q)x−ab−a(11−q−(x−ab−a)11−q2)=x−ab−a(1−11+q(x−ab−a))=x−ab−a−11+q(x−ab−a)2, | (2.15) |
∫1x−ab−at0dqt=∫10t0dqt−∫x−ab−a0t0dqt=11+q−11+q(x−ab−a)2=11+q(1−(x−ab−a)2), | (2.16) |
and
∫1x−ab−a(1−t)0dqt=∫10(1−t)0dqt−∫x−ab−a0(1−t)0dqt=q1+q−x−ab−a+11+q(x−ab−a)2. | (2.17) |
Theorem 4. Let f:[a,b]→R be an arbitrary function with aDqf is quantum integrable on [a,b]. If |aDqf|r, r>1 and 1r+1p=1 is a convex function, then the following quantum integral inequality holds:
|f(x)−1b−a∫baf(t)adqt|≤(b−a)[(∫x−ab−a0qt0dqt)1p(|aDqf(a)|r[11+q(x−ab−a)2]+|aDqf(b)|r[x−ab−a−11+q(x−ab−a)2])1r+(∫1x−ab−a(1−qt)p0dqt)1p(|aDqf(a)|r[11+q(1−(x−ab−a)2)]+|aDqf(b)|r[q1+q−x−ab−a+11+q(x−ab−a)2])1r] | (2.18) |
for all x∈[a,b].
Proof. By using Lemma 3, quantum Hölder inequality and (2.8), we have that
|f(x)−1b−a∫baf(t)adqt|≤(b−a)∫10|Kq(t)||aDqf(tb+(1−t)a)|0dqt≤(b−a)[∫x−ab−a0qt|aDqf(tb+(1−t)a)|0dqt+∫1x−ab−a(1−qt)|aDqf(tb+(1−t)a)|0dqt]≤(b−a)[(∫x−ab−a0(qt)p0dqt)1p(∫x−ab−a0|aDqf(tb+(1−t)a)|r0dqt)1r+(∫1x−ab−a(1−qt)p0dqt)1p(∫1x−ab−a|aDqf(tb+(1−t)a)|r0dqt)1r] |
≤(b−a)[(∫x−ab−a0(qt)p0dqt)1p(∫x−ab−a0[t|aDqf(a)|r+(1−t)|aDqf(b)|r]0dqt)1r+(∫1x−ab−a(1−qt)p0dqt)1p(∫1x−ab−a[t|aDqf(a)|r+(1−t)|aDqf(b)|r]0dqt)1r]≤(b−a)[(∫x−ab−a0qt0dqt)1p(|aDqf(a)|r∫x−ab−a0t0dqt+|aDqf(b)|r∫x−ab−a0(1−t)0dqt)1r+(∫1x−ab−a(1−qt)p0dqt)1p(|aDqf(a)|r∫1x−ab−at0dqt+|aDqf(b)|r∫1x−ab−a(1−t)0dqt)1r]. | (2.19) |
Using (2.14)–(2.17) in (2.19), we obtain the desired result in (2.18). This ends the proof.
Remark 2. In Theorem 4, many different inequalities could be derived similarly to Corollary 1.
In the terms of quantum Montgomery identity, some quantum integral inequalities of Ostrowski type are established. The establishment of the inequalities is based on the mappings whose first derivatives absolute values are quantum differentiable convex. Furthermore, the important relevant connection obtained in this work with those which were introduced in previously published papers is investigated. By considering the special value for x∈[a,b], some fixed value for r, and as well as q→1−, many sub-results can be derived from the main results of this work. It is worthwhile to mention that certain quantum inequalities presented in this work are generalized forms of the very recent results given by Alp et al. (2018) and Noor et al. (2016). With the contribution of this work, the interested researchers will be motivated to explore this fascinating field of the quantum integral inequality based on the techniques and ideas developed in this article.
The first author would like to thank Ondokuz Mayıs University for being a visiting professor and providing excellent research facilities.
The authors declare that they have no competing interests.
[1] |
Hu C, Ashok D, Nisbet DR, et al. (2019) Bioinspired surface modification of orthopedic implants for bone tissue engineering. Biomaterials 119366. doi: 10.1016/j.biomaterials.2019.119366
![]() |
[2] |
Karsenty G, Olson EN (2016) Bone and muscle endocrine functions: unexpected paradigms of inter-organ communication. Cell 164: 1248-1256. doi: 10.1016/j.cell.2016.02.043
![]() |
[3] |
Rossi M, Battafarano G, Pepe J, et al. (2019) The endocrine function of osteocalcin regulated by bone resorption: A lesson from reduced and increased bone mass diseases. Int J Mol Sci 20: 4502. doi: 10.3390/ijms20184502
![]() |
[4] |
Loebel C, Burdick JA (2018) Engineering stem and stromal cell therapies for musculoskeletal tissue repair. Cell Stem Cell 22: 325-339. doi: 10.1016/j.stem.2018.01.014
![]() |
[5] |
Dimitriou R, Jones E, McGonagle D, et al. (2011) Bone regeneration: current concepts and future directions. BMC Med 9: 66. doi: 10.1186/1741-7015-9-66
![]() |
[6] | Nordin M, Frankel VH (2001) Basic Biomechanics of the Musculoskeletal System, 3 Eds USA: Lippincott Williams & Wilkins. |
[7] |
Kobayashi S, Takahashi HE, Ito A, et al. (2003) Trabecular minimodeling in human iliac bone. Bone 32: 163-169. doi: 10.1016/S8756-3282(02)00947-X
![]() |
[8] |
Bartl R, Bartl C (2019) Control and regulation of bone remodelling. The Osteoporosis Manual Cham: Springer, 31-39. doi: 10.1007/978-3-030-00731-7_4
![]() |
[9] |
Kenkre JS, Bassett JHD (2018) The bone remodelling cycle. Ann Clin Biochem 55: 308-327. doi: 10.1177/0004563218759371
![]() |
[10] |
Prendergast PJ, Huiskes R (1995) The biomechanics of Wolff's law: recent advances. Irish J Med Sci 164: 152-154. doi: 10.1007/BF02973285
![]() |
[11] |
Wegst UGK, Bai H, Saiz E, et al. (2015) Bioinspired structural materials. Nat Mater 14: 23-36. doi: 10.1038/nmat4089
![]() |
[12] |
Reznikov N, Shahar R, Weiner S (2014) Bone hierarchical structure in three dimensions. Acta Biomater 10: 3815-3826. doi: 10.1016/j.actbio.2014.05.024
![]() |
[13] |
Weiner S, Wagner HD (1998) The material bone: structure-mechanical function relations. Annu Rev Mater Sci 28: 271-298. doi: 10.1146/annurev.matsci.28.1.271
![]() |
[14] |
Recker RR, Kimmel DB, Dempster D, et al. (2011) Issues in modern bone histomorphometry. Bone 49: 955-964. doi: 10.1016/j.bone.2011.07.017
![]() |
[15] |
Eriksen EF, Vesterby A, Kassem M, et al. (1993) Bone remodeling and bone structure. Physiology and Pharmacology of Bone Heidelberg: Springer, 67-109. doi: 10.1007/978-3-642-77991-6_2
![]() |
[16] |
Augat P, Schorlemmer S (2006) The role of cortical bone and its microstructure in bone strength. Age Ageing 35: ii27-ii31. doi: 10.1093/ageing/afl081
![]() |
[17] |
Kozielski M, Buchwald T, Szybowicz M, et al. (2011) Determination of composition and structure of spongy bone tissue in human head of femur by Raman spectral mapping. J Mater Sci: Mater Med 22: 1653-1661. doi: 10.1007/s10856-011-4353-0
![]() |
[18] |
Cross LM, Thakur A, Jalili NA, et al. (2016) Nanoengineered biomaterials for repair and regeneration of orthopedic tissue interfaces. Acta Biomater 42: 2-17. doi: 10.1016/j.actbio.2016.06.023
![]() |
[19] |
Zebaze R, Seeman E (2015) Cortical bone: a challenging geography. J Bone Miner Res 30: 24-29. doi: 10.1002/jbmr.2419
![]() |
[20] |
Liu Y, Luo D, Wang T (2016) Hierarchical structures of bone and bioinspired bone tissue engineering. Small 12: 4611-4632. doi: 10.1002/smll.201600626
![]() |
[21] |
Brodsky B, Persikov AV (2005) Molecular structure of the collagen triple helix. Adv Protein Chem 70: 301-339. doi: 10.1016/S0065-3233(05)70009-7
![]() |
[22] |
Cui FZ, Li Y, Ge J (2007) Self-assembly of mineralized collagen composites. Mater Sci Eng R Rep 57: 1-27. doi: 10.1016/j.mser.2007.04.001
![]() |
[23] |
Wang Y, Azaïs T, Robin M, et al. (2012) The predominant role of collagen in the nucleation, growth, structure and orientation of bone apatite. Nat Mater 11: 724-733. doi: 10.1038/nmat3362
![]() |
[24] | Bentmann A, Kawelke N, Moss D, et al. (2010) Circulating fibronectin affects bone matrix, whereas osteoblast fibronectin modulates osteoblast function. J Bone Miner Res 25: 706-715. |
[25] |
Szweras M, Liu D, Partridge EA, et al. (2002) α2-HS glycoprotein/fetuin, a transforming growth factor-β/bone morphogenetic protein antagonist, regulates postnatal bone growth and remodeling. J Biol Chem 277: 19991-19997. doi: 10.1074/jbc.M112234200
![]() |
[26] |
Boskey AL, Robey PG (2013) The regulatory role of matrix proteins in mineralization of bone. Osteoporosis, 4 Eds Academic Press, 235-255. doi: 10.1016/B978-0-12-415853-5.00011-X
![]() |
[27] |
Boskey AL (2013) Bone composition: relationship to bone fragility and antiosteoporotic drug effects. Bonekey Rep 2: 447. doi: 10.1038/bonekey.2013.181
![]() |
[28] |
Stock SR (2015) The mineral–collagen interface in bone. Calcified Tissue Int 97: 262-280. doi: 10.1007/s00223-015-9984-6
![]() |
[29] |
Nikel O, Laurencin D, McCallum SA, et al. (2013) NMR investigation of the role of osteocalcin and osteopontin at the organic–inorganic interface in bone. Langmuir 29: 13873-13882. doi: 10.1021/la403203w
![]() |
[30] |
He G, Dahl T, Veis A, et al. (2003) Nucleation of apatite crystals in vitro by self-assembled dentin matrix protein 1. Nat Mater 2: 552-558. doi: 10.1038/nmat945
![]() |
[31] |
Clarke B (2008) Normal bone anatomy and physiology. Clin J Am Soc Nephro 3: S131-S139. doi: 10.2215/CJN.04151206
![]() |
[32] |
Olszta MJ, Cheng X, Jee SS, et al. (2007) Bone structure and formation: A new perspective. Mater Sci Eng R Rep 58: 77-116. doi: 10.1016/j.mser.2007.05.001
![]() |
[33] |
Nair AK, Gautieri A, Chang SW, et al. (2013) Molecular mechanics of mineralized collagen fibrils in bone. Nature Commun 4: 1724. doi: 10.1038/ncomms2720
![]() |
[34] |
Landis WJ (1995) The strength of a calcified tissue depends in part on the molecular structure and organization of its constituent mineral crystals in their organic matrix. Bone 16: 533-544. doi: 10.1016/8756-3282(95)00076-P
![]() |
[35] |
Hunter GK, Hauschka PV, POOLE RA, et al. (1996) Nucleation and inhibition of hydroxyapatite formation by mineralized tissue proteins. Biochem J 317: 59-64. doi: 10.1042/bj3170059
![]() |
[36] |
Oikeh I, Sakkas P, Blake D P, et al. (2019) Interactions between dietary calcium and phosphorus level, and vitamin D source on bone mineralization, performance, and intestinal morphology of coccidia-infected broilers. Poult Sci 11: 5679-5690. doi: 10.3382/ps/pez350
![]() |
[37] |
Boyce BF, Rosenberg E, de Papp AE, et al. (2012) The osteoclast, bone remodelling and treatment of metabolic bone disease. Eur J Clin Invest 42: 1332-1341. doi: 10.1111/j.1365-2362.2012.02717.x
![]() |
[38] |
Teitelbaum SL (2000) Bone resorption by osteoclasts. Science 289: 1504-1508. doi: 10.1126/science.289.5484.1504
![]() |
[39] |
Yoshida H, Hayashi SI, Kunisada T, et al. (1990) The murine mutation osteopetrosis is in the coding region of the macrophage colony stimulating factor gene. Nature 345: 442-444. doi: 10.1038/345442a0
![]() |
[40] |
Roodman GD (2006) Regulation of osteoclast differentiation. Ann NY Acad Sci 1068: 100-109. doi: 10.1196/annals.1346.013
![]() |
[41] |
Martin TJ (2013) Historically significant events in the discovery of RANK/RANKL/OPG. World J Orthop 4: 186-197. doi: 10.5312/wjo.v4.i4.186
![]() |
[42] |
Coetzee M, Haag M, Kruger MC (2007) Effects of arachidonic acid, docosahexaenoic acid, prostaglandin E2 and parathyroid hormone on osteoprotegerin and RANKL secretion by MC3T3-E1 osteoblast-like cells. J Nutr Biochem 18: 54-63. doi: 10.1016/j.jnutbio.2006.03.002
![]() |
[43] |
Steeve KT, Marc P, Sandrine T, et al. (2004) IL-6, RANKL, TNF-alpha/IL-1: interrelations in bone resorption pathophysiology. Cytokine Growth F R 15: 49-60. doi: 10.1016/j.cytogfr.2003.10.005
![]() |
[44] |
Mellis DJ, Itzstein C, Helfrich M, et al. (2011) The skeleton: a multi-functional complex organ. The role of key signalling pathways in osteoclast differentiation and in bone resorption. J Endocrinol 211: 131-143. doi: 10.1530/JOE-11-0212
![]() |
[45] | Silva I, Branco J (2011) Rank/Rankl/opg: literature review. Acta Reumatol Port 36: 209-218. |
[46] |
Martin TJ, Sims NA (2015) RANKL/OPG; Critical role in bone physiology. Rev Endocr Metab Dis 16: 131-139. doi: 10.1007/s11154-014-9308-6
![]() |
[47] |
Wang Y, Qin QH (2012) A theoretical study of bone remodelling under PEMF at cellular level. Comput Method Biomec 15: 885-897. doi: 10.1080/10255842.2011.565752
![]() |
[48] |
Weitzmann MN, Pacifici R (2007) T cells: unexpected players in the bone loss induced by estrogen deficiency and in basal bone homeostasis. Ann NY Acad Sci 1116: 360-375. doi: 10.1196/annals.1402.068
![]() |
[49] |
Duong LT, Lakkakorpi P, Nakamura I, et al. (2000) Integrins and signaling in osteoclast function. Matrix Biol 19: 97-105. doi: 10.1016/S0945-053X(00)00051-2
![]() |
[50] |
Stenbeck G (2002) Formation and function of the ruffled border in osteoclasts. Semin Cell Dev Biol 13: 285-292. doi: 10.1016/S1084952102000587
![]() |
[51] |
Jurdic P, Saltel F, Chabadel A, et al. (2006) Podosome and sealing zone: specificity of the osteoclast model. Eur J Cell Biol 85: 195-202. doi: 10.1016/j.ejcb.2005.09.008
![]() |
[52] |
Väänänen HK, Laitala-Leinonen T (2008) Osteoclast lineage and function. Arch Biochem Biophys 473: 132-138. doi: 10.1016/j.abb.2008.03.037
![]() |
[53] | Vaananen HK, Zhao H, Mulari M, et al. (2000) The cell biology of osteoclast function. J cell Sci 113: 377-381. |
[54] |
Sabolová V, Brinek A, Sládek V (2018) The effect of hydrochloric acid on microstructure of porcine (Sus scrofa domesticus) cortical bone tissue. Forensic Sci Int 291: 260-271. doi: 10.1016/j.forsciint.2018.08.030
![]() |
[55] |
Delaissé JM, Engsig MT, Everts V, et al. (2000) Proteinases in bone resorption: obvious and less obvious roles. Clin Chim Acta 291: 223-234. doi: 10.1016/S0009-8981(99)00230-2
![]() |
[56] |
Logar DB, Komadina R, Preželj J, et al. (2007) Expression of bone resorption genes in osteoarthritis and in osteoporosis. J Bone Miner Metab 25: 219-225. doi: 10.1007/s00774-007-0753-0
![]() |
[57] |
Lorget F, Kamel S, Mentaverri R, et al. (2000) High extracellular calcium concentrations directly stimulate osteoclast apoptosis. Biochem Bioph Res Co 268: 899-903. doi: 10.1006/bbrc.2000.2229
![]() |
[58] |
Nesbitt SA, Horton MA (1997) Trafficking of matrix collagens through bone-resorbing osteoclasts. Science 276: 266-269. doi: 10.1126/science.276.5310.266
![]() |
[59] |
Xing L, Boyce BF (2005) Regulation of apoptosis in osteoclasts and osteoblastic cells. Biochem Bioph Res Co 328: 709-720. doi: 10.1016/j.bbrc.2004.11.072
![]() |
[60] |
Hughes DE, Wright KR, Uy HL, et al. (1995) Bisphosphonates promote apoptosis in murine osteoclasts in vitro and in vivo. J Bone Miner Res 10: 1478-1487. doi: 10.1002/jbmr.5650101008
![]() |
[61] |
Choi Y, Arron JR, Townsend MJ (2009) Promising bone-related therapeutic targets for rheumatoid arthritis. Nat Rev Rheumatol 5: 543-548. doi: 10.1038/nrrheum.2009.175
![]() |
[62] |
Harvey NC, McCloskey E, Kanis JA, et al. (2017) Bisphosphonates in osteoporosis: NICE and easy? Lancet 390: 2243-2244. doi: 10.1016/S0140-6736(17)32850-7
![]() |
[63] |
Ducy P, Schinke T, Karsenty G (2000) The osteoblast: a sophisticated fibroblast under central surveillance. Science 289: 1501-1504. doi: 10.1126/science.289.5484.1501
![]() |
[64] |
Katagiri T, Takahashi N (2002) Regulatory mechanisms of osteoblast and osteoclast differentiation. Oral dis 8: 147-159. doi: 10.1034/j.1601-0825.2002.01829.x
![]() |
[65] |
Kretzschmar M, Liu F, Hata A, et al. (1997) The TGF-beta family mediator Smad1 is phosphorylated directly and activated functionally by the BMP receptor kinase. Gene Dev 11: 984-995. doi: 10.1101/gad.11.8.984
![]() |
[66] |
Bennett CN, Longo KA, Wright WS, et al. (2005) Regulation of osteoblastogenesis and bone mass by Wnt10b. P Natl A Sci 102: 3324-3329. doi: 10.1073/pnas.0408742102
![]() |
[67] | Wang Y, Qin QH, Kalyanasundaram S (2009) A theoretical model for simulating effect of parathyroid hormone on bone metabolism at cellular level. Mol Cell Biomech 6: 101-112. |
[68] |
Elefteriou F, Ahn JD, Takeda S, et al. (2005) Leptin regulation of bone resorption by the sympathetic nervous system and CART. Nature 434: 514-520. doi: 10.1038/nature03398
![]() |
[69] |
Proff P, Römer P (2009) The molecular mechanism behind bone remodelling: a review. Clin Oral Invest 13: 355-362. doi: 10.1007/s00784-009-0268-2
![]() |
[70] |
Katsimbri P (2017) The biology of normal bone remodelling. Eur J Cancer Care 26: e12740. doi: 10.1111/ecc.12740
![]() |
[71] |
Fratzl P, Weinkamer R (2007) Nature's hierarchical materials. Prog Mater Sci 52: 1263-1334. doi: 10.1016/j.pmatsci.2007.06.001
![]() |
[72] |
Athanasiou KA, Zhu CF, Lanctot DR, et al. (2000) Fundamentals of biomechanics in tissue engineering of bone. Tissue Eng 6: 361-381. doi: 10.1089/107632700418083
![]() |
[73] |
Takahashi N, Udagawa N, Suda T (1999) A new member of tumor necrosis factor ligand family, ODF/OPGL/TRANCE/RANKL, regulates osteoclast differentiation and function. Biocheml Bioph Res Co 256: 449-455. doi: 10.1006/bbrc.1999.0252
![]() |
[74] |
Nakashima T, Hayashi M, Fukunaga T, et al. (2011) Evidence for osteocyte regulation of bone homeostasis through RANKL expression. Nat Med 17: 1231-1234. doi: 10.1038/nm.2452
![]() |
[75] |
Prideaux M, Findlay DM, Atkins GJ (2016) Osteocytes: the master cells in bone remodelling. Curr Opin Pharmacol 28: 24-30. doi: 10.1016/j.coph.2016.02.003
![]() |
[76] |
Dallas SL, Prideaux M, Bonewald LF (2013) The osteocyte: an endocrine cell… and more. Endocr Rev 34: 658-690. doi: 10.1210/er.2012-1026
![]() |
[77] |
Rochefort GY, Pallu S, Benhamou CL (2010) Osteocyte: the unrecognized side of bone tissue. Osteoporosis Int 21: 1457-1469. doi: 10.1007/s00198-010-1194-5
![]() |
[78] |
Rowe PSN (2012) Regulation of bone–renal mineral and energy metabolism: The PHEX, FGF23, DMP1, MEPE ASARM pathway. Crit Rev Eukaryot Gene Expr 22: 61-86. doi: 10.1615/CritRevEukarGeneExpr.v22.i1.50
![]() |
[79] |
Pajevic PD, Krause DS (2019) Osteocyte regulation of bone and blood. Bone 119: 13-18. doi: 10.1016/j.bone.2018.02.012
![]() |
[80] | Frost HM (1987) The mechanostat: a proposed pathogenic mechanism of osteoporoses and the bone mass effects of mechanical and nonmechanical agents. Bone Miner 2: 73-85. |
[81] |
Tate MLK, Adamson JR, Tami AE, et al. (2004) The osteocyte. Int J Biochem Cell Biol 36: 1-8. doi: 10.1016/S1357-2725(03)00241-3
![]() |
[82] |
Bonewald LF, Johnson ML (2008) Osteocytes, mechanosensing and Wnt signaling. Bone 42: 606-615. doi: 10.1016/j.bone.2007.12.224
![]() |
[83] |
Manolagas SC, Parfitt AM (2010) What old means to bone. Trends Endocrinol Metab 21: 369-374. doi: 10.1016/j.tem.2010.01.010
![]() |
[84] |
Wang Y, Qin QH (2010) Parametric study of control mechanism of cortical bone remodeling under mechanical stimulus. Acta Mech Sinica 26: 37-44. doi: 10.1007/s10409-009-0313-z
![]() |
[85] |
Qu C, Qin QH, Kang Y (2006) A hypothetical mechanism of bone remodeling and modeling under electromagnetic loads. Biomaterials 27: 4050-4057. doi: 10.1016/j.biomaterials.2006.03.015
![]() |
[86] |
Parfitt AM (2002) Targeted and nontargeted bone remodeling: relationship to basic multicellular unit origination and progression. Bone 1: 5-7. doi: 10.1016/S8756-3282(01)00642-1
![]() |
[87] |
Hadjidakis DJ, Androulakis II (2006) Bone remodeling. Ann NYAcad Sci 1092: 385-396. doi: 10.1196/annals.1365.035
![]() |
[88] | Vaananen HK, Zhao H, Mulari M, et al. (2000) The cell biology of osteoclast function. J cell Sci 113: 377-381. |
[89] |
Goldring SR (2015) The osteocyte: key player in regulating bone turnover. RMD Open 1: e000049. doi: 10.1136/rmdopen-2015-000049
![]() |
[90] |
Silver IA, Murrills RJ, Etherington DJ (1988) Microelectrode studies on the acid microenvironment beneath adherent macrophages and osteoclasts. Exp Cell Res 175: 266-276. doi: 10.1016/0014-4827(88)90191-7
![]() |
[91] |
Delaissé JM, Andersen TL, Engsig MT, et al. (2003) Matrix metalloproteinases (MMP) and cathepsin K contribute differently to osteoclastic activities. Microsc Res Techniq 61: 504-513. doi: 10.1002/jemt.10374
![]() |
[92] |
Delaisse JM (2014) The reversal phase of the bone-remodeling cycle: cellular prerequisites for coupling resorption and formation. Bonekey Rep 3: 561. doi: 10.1038/bonekey.2014.56
![]() |
[93] | Bonewald LF, Mundy GR (1990) Role of transforming growth factor-beta in bone remodeling. Clin Orthop Relat R 250: 261-276. |
[94] |
Locklin RM, Oreffo ROC, Triffitt JT (1999) Effects of TGFβ and bFGF on the differentiation of human bone marrow stromal fibroblasts. Cell Biol Int 23: 185-194. doi: 10.1006/cbir.1998.0338
![]() |
[95] |
Lee B, Oh Y, Jo S, et al. (2019) A dual role of TGF-β in human osteoclast differentiation mediated by Smad1 versus Smad3 signaling. Immunol Lett 206: 33-40. doi: 10.1016/j.imlet.2018.12.003
![]() |
[96] |
Koseki T, Gao Y, Okahashi N, et al. (2002) Role of TGF-β family in osteoclastogenesis induced by RANKL. Cell Signal 14: 31-36. doi: 10.1016/S0898-6568(01)00221-2
![]() |
[97] |
Anderson HC (2003) Matrix vesicles and calcification. Curr Rheumatol Rep 5: 222-226. doi: 10.1007/s11926-003-0071-z
![]() |
[98] |
Bellido T, Plotkin LI, Bruzzaniti A (2019) Bone cells. Basic and Applied Bone Biology, 2 Eds Elsevier, 37-55. doi: 10.1016/B978-0-12-813259-3.00003-8
![]() |
[99] |
Weinstein RS, Jilka RL, Parfitt AM, et al. (1998) Inhibition of osteoblastogenesis and promotion of apoptosis of osteoblasts and osteocytes by glucocorticoids. Potential mechanisms of their deleterious effects on bone. J Clin Invest 102: 274-282. doi: 10.1172/JCI2799
![]() |
[100] |
Vezeridis PS, Semeins CM, Chen Q, et al. (2006) Osteocytes subjected to pulsating fluid flow regulate osteoblast proliferation and differentiation. Biochem Bioph Res Co 348: 1082-1088. doi: 10.1016/j.bbrc.2006.07.146
![]() |
[101] |
Lind M, Deleuran B, Thestrup-Pedersen K, et al. (1995) Chemotaxis of human osteoblasts: Effects of osteotropic growth factors. Apmis 103: 140-146. doi: 10.1111/j.1699-0463.1995.tb01089.x
![]() |
[102] |
Russo CR, Lauretani F, Seeman E, et al. (2006) Structural adaptations to bone loss in aging men and women. Bone 38: 112-118. doi: 10.1016/j.bone.2005.07.025
![]() |
[103] |
Ozcivici E, Luu YK, Adler B, et al. (2010) Mechanical signals as anabolic agents in bone. Nat Rev Rheumatol 6: 50-59. doi: 10.1038/nrrheum.2009.239
![]() |
[104] |
Rosa N, Simoes R, Magalhães FD, et al. (2015) From mechanical stimulus to bone formation: a review. Med Eng Phys 37: 719-728. doi: 10.1016/j.medengphy.2015.05.015
![]() |
[105] |
Noble BS, Peet N, Stevens HY, et al. (2003) Mechanical loading: biphasic osteocyte survival and targeting of osteoclasts for bone destruction in rat cortical bone. Am J Physiol-Cell Ph 284: C934-C943. doi: 10.1152/ajpcell.00234.2002
![]() |
[106] |
Robling AG, Castillo AB, Turner CH (2006) Biomechanical and molecular regulation of bone remodeling. Annu Rev Biomed Eng 8: 455-498. doi: 10.1146/annurev.bioeng.8.061505.095721
![]() |
[107] |
Qin QH, Mai YW (1999) A closed crack tip model for interface cracks inthermopiezoelectric materials. Int J Solids Struct 36: 2463-2479. doi: 10.1016/S0020-7683(98)00115-2
![]() |
[108] |
Yu SW, Qin QH (1996) Damage analysis of thermopiezoelectric properties: Part I—crack tip singularities. Theor Appl Fract Mec 25: 263-277. doi: 10.1016/S0167-8442(96)00026-2
![]() |
[109] |
Qin QH, Mai YW, Yu SW (1998) Effective moduli for thermopiezoelectric materials with microcracks. Int J Fracture 91: 359-371. doi: 10.1023/A:1007423508650
![]() |
[110] |
Jirousek J, Qin QH (1996) Application of hybrid-Trefftz element approach to transient heat conduction analysis. Comput Struct 58: 195-201. doi: 10.1016/0045-7949(95)00115-W
![]() |
[111] |
Qin QH (1995) Hybrid-Trefftz finite element method for Reissner plates on an elastic foundation. Comput Method Appl M 122: 379-392. doi: 10.1016/0045-7825(94)00730-B
![]() |
[112] |
Qin QH (1994) Hybrid Trefftz finite-element approach for plate bending on an elastic foundation. Appl Math Model 18: 334-339. doi: 10.1016/0307-904X(94)90357-3
![]() |
[113] |
Qin QH (2013) Mechanics of Cellular Bone Remodeling: Coupled Thermal, Electrical, and Mechanical Field Effects CRC Press. doi: 10.1201/b13728
![]() |
[114] | Wang H, Qin QH (2010) FE approach with Green's function as internal trial function for simulating bioheat transfer in the human eye. Arch Mech 62: 493-510. |
[115] | Qin QH (2003) Fracture analysis of cracked thermopiezoelectric materials by BEM. Electronic J Boundary Elem 1: 283-301. |
[116] |
Qin QH, Ye JQ (2004) Thermoelectroelastic solutions for internal bone remodeling under axial and transverse loads. Int J Solids Struct 41: 2447-2460. doi: 10.1016/j.ijsolstr.2003.12.026
![]() |
[117] |
Qin QH, Qu C, Ye J (2005) Thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads. Biomaterials 26: 6798-6810. doi: 10.1016/j.biomaterials.2005.03.042
![]() |
[118] |
Ducher G, Jaffré C, Arlettaz A, et al. (2005) Effects of long-term tennis playing on the muscle-bone relationship in the dominant and nondominant forearms. Can J Appl Physiol 30: 3-17. doi: 10.1139/h05-101
![]() |
[119] |
Robling AG, Hinant FM, Burr DB, et al. (2002) Improved bone structure and strength after long-term mechanical loading is greatest if loading is separated into short bouts. J Bone Miner Res 17: 1545-1554. doi: 10.1359/jbmr.2002.17.8.1545
![]() |
[120] |
Rubin J, Rubin C, Jacobs CR (2006) Molecular pathways mediating mechanical signaling in bone. Gene 367: 1-16. doi: 10.1016/j.gene.2005.10.028
![]() |
[121] |
Tatsumi S, Ishii K, Amizuka N, et al. (2007) Targeted ablation of osteocytes induces osteoporosis with defective mechanotransduction. Cell Metab 5: 464-475. doi: 10.1016/j.cmet.2007.05.001
![]() |
[122] |
Robling AG, Turner CH (2009) Mechanical signaling for bone modeling and remodeling. Crit Rev Eukar Gene 19: 319-338. doi: 10.1615/CritRevEukarGeneExpr.v19.i4.50
![]() |
[123] |
Galli C, Passeri G, Macaluso GM (2010) Osteocytes and WNT: the mechanical control of bone formation. J Dent Res 89: 331-343. doi: 10.1177/0022034510363963
![]() |
[124] |
Robling AG, Duijvelaar KM, Geevers JV, et al. (2001) Modulation of appositional and longitudinal bone growth in the rat ulna by applied static and dynamic force. Bone 29: 105-113. doi: 10.1016/S8756-3282(01)00488-4
![]() |
[125] |
Burr DB, Milgrom C, Fyhrie D, et al. (1996) In vivo measurement of human tibial strains during vigorous activity. Bone 18: 405-410. doi: 10.1016/8756-3282(96)00028-2
![]() |
[126] |
Sun W, Chi S, Li Y, et al. (2019) The mechanosensitive Piezo1 channel is required for bone formation. Elife 8: e47454. doi: 10.7554/eLife.47454
![]() |
[127] |
Goda I, Ganghoffer JF, Czarnecki S, et al. (2019) Topology optimization of bone using cubic material design and evolutionary methods based on internal remodeling. Mech Res Commun 95: 52-60. doi: 10.1016/j.mechrescom.2018.12.003
![]() |
[128] |
Goda I, Ganghoffer JF (2018) Modeling of anisotropic remodeling of trabecular bone coupled to fracture. Arch Appl Mech 88: 2101-2121. doi: 10.1007/s00419-018-1438-y
![]() |
[129] |
Louna Z, Goda I, Ganghoffer JF, et al. (2017) Formulation of an effective growth response of trabecular bone based on micromechanical analyses at the trabecular level. Arch Appl Mech 87: 457-477. doi: 10.1007/s00419-016-1204-y
![]() |
[130] |
Goda I, Ganghoffer JF (2017) Construction of the effective plastic yield surfaces of vertebral trabecular bone under twisting and bending moments stresses using a 3D microstructural model. ZAMM Z Angew Math Mech 97: 254-272. doi: 10.1002/zamm.201600141
![]() |
[131] |
Qin QH, Wang YN (2012) A mathematical model of cortical bone remodeling at cellular level under mechanical stimulus. Acta Mech Sinica-Prc 28: 1678-1692. doi: 10.1007/s10409-012-0154-z
![]() |
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