Certain Inequalities Involving Generalized ErdélyiKober Fractional qIntegral Operators.  
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PMID: 25295293 Owner: NLM Status: InDataReview 
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In recent years, a remarkably large number of inequalities involving the fractional qintegral operators have been investigated in the literature by many authors. Here, we aim to present some new fractional integral inequalities involving generalized ErdélyiKober fractional qintegral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kober type fractional qintegral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered. 
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Praveen Agarwal; Soheil Salahshour; Sotiris K Ntouyas; Jessada Tariboon 
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Type: Journal Article Date: 20140911 
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Title: TheScientificWorldJournal Volume: 2014 ISSN: 1537744X ISO Abbreviation: ScientificWorldJournal Publication Date: 2014 
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Created Date: 20141008 Completed Date:  Revised Date:  
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Languages: eng Pagination: 174126 Citation Subset: IM 
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Journal Information Journal ID (nlmta): ScientificWorldJournal Journal ID (isoabbrev): ScientificWorldJournal Journal ID (publisherid): TSWJ ISSN: 23566140 ISSN: 1537744X Publisher: Hindawi Publishing Corporation 
Article Information Download PDF Copyright © 2014 Praveen Agarwal et al. openaccess: Received Day: 21 Month: 6 Year: 2014 Accepted Day: 26 Month: 8 Year: 2014 Print publication date: Year: 2014 Electronic publication date: Day: 11 Month: 9 Year: 2014 Volume: 2014Elocation ID: 174126 PubMed Id: 25295293 ID: 4177081 DOI: 10.1155/2014/174126 
Certain Inequalities Involving Generalized ErdélyiKober Fractional qIntegral Operators  
Praveen Agarwal^{1}  http://orcid.org/0000000175568942 
Soheil Salahshour^{2}  http://orcid.org/0000000313903551 
Sotiris K. Ntouyas^{3}  
Jessada Tariboon^{4}*  http://orcid.org/0000000181853539 
^{1}Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India 

^{2}Department of Computer Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran 

^{3}Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece 

^{4}Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand 

Correspondence: *Jessada Tariboon: jessadat@kmutnb.ac.th [other] Academic Editor: Junesang Choi 
Let us start by considering the following functional (see [^{1}]):
(1)
T(f,g,p,q) =∫abq(x)dx∫abp(x)f(x)g(x)dx +∫abp(x)dx∫abq(x)f(x)g(x)dx −(∫abq(x)f(x)dx)(∫abp(x)g(x)dx) −(∫abp(x)f(x)dx)(∫abq(x)g(x)dx), 
(2)
(f(x)−f(y))(g(x)−g(y))≥0, 
(3)
T(f,g,p,q)≥0. 
(4)
(f(x)−f(y))(g(x)−g(y))≤0, 
If f and g are two differentiable and synchronous functions on [a, b] and p is a positive integrable function on [a, b] with f′(x)≥m and g′(x)≥r for x ∈ [a, b], then we have
(5)
T(f,g,p)=T(f,g,p,p)≥mrT(x−a,x−a,p)≥0. 
If f and g are asynchronous on [a, b], then we have
(6)
T(f,g,p)≤mrT(x−a,x−a,p)≤0. 
(7)
T(f,g,p)≤MRT(x−a,x−a,p)≤0. 
The study of the fractional integral and fractional qintegral inequalities has been of great importance due to the fundamental role in the theory of differential equations. In recent years, a number of researchers have done deep study, that is, the properties, applications, and different extensions of various fractional qintegral operators (see, e.g., [^{12}–^{16}]).
The purpose of this paper is to find qcalculus analogs of some classical integral inequalities. In particular, we will find qgeneralizations of the Chebyshev integral inequalities by using the generalized ErdélyiKober fractional qintegral operator introduced by Galué [^{17}]. The main objective of this paper is to present some new fractional qintegral inequalities involving the generalized ErdélyiKober fractional qintegral operator. We consider the case of synchronous functions as well as the case of functions bounded by integrable functions. Some of the known and new results are as follows, as special cases of our main findings. We emphasize that the results derived in this paper are more generalized results rather than similar published results because we established all results by using the generalized ErdélyiKober fractional qintegral operator. Our results are general in character and give some contributions to the theory qintegral inequalities and fractional calculus.
In the sequel, we required the following wellknown results to establish our main results in the present paper. The qshifted factorial (a; q)_{n} is defined by
(8)
(a;q)n:={1,(n=0)∏k=0n−1(1−aqk),(n∈N), 
The qshifted factorial for negative subscript is defined by
(9)
(a;q)−n≔1(1−aq−1)(1−aq−2)⋯(1−aq−n)(n∈N0). 
(10)
(a;q)∞≔∏k=0∞(1−aqk) (a,q∈C; q<1). 
(11)
(a;q)n=(a;q)∞(aqn;q)∞ (n∈Z), 
(12)
(a;q)α=(a;q)∞(aqα;q)∞ (α∈C; q<1), 
We begin by noting that F. J. Jackson was the first to develop qcalculus in a systematic way. For 0 < q < 1, the qderivative of a continuous function f on [0, b] is defined by
(13)
Dqf(t)≔dqdqtf(t)=f(t)−f(qt)(1−q)t, t∈(0,b], 
(14)
limq→1Dqf(t)=ddtf(t), 
The function F(t) is a qantiderivative of f(t) if D_{q}F(t) = f(t). It is denoted by
(15)
∫f(t)dqt. 
The Jackson integral of f(t) is thus defined, formally, by
(16)
∫f(t)dqt≔(1−q)t∑j=0∞qjf(qjt), 
(17)
∫f(t)dqg(t)=∑j=0∞f(qjt)(g(qjt)−g(qj+1t)). 
Suppose that 0 < a < b. The definite qintegral is defined as follows:
(18)
∫0bf(t)dqt:=(1−q)b∑j=0∞qjf(qjb), 
(19)
∫abf(t)dqt=∫0bf(t)dqt−∫0af(t)dqt. 
A more general version of (18) is given by
(20)
∫0bf(t)dqg(t)=∑j=0∞f(qjb)(g(qjb)−g(qj+1b)). 
The classical Gamma function Γ(z) (see, e.g., [^{18}, Section 1.1]) was found by Euler while he was trying to extend the factorial n! = Γ(n + 1)(n ∈ N_{0}) to real numbers. The qfactorial function [n]_{q}! (n ∈ N_{0}) of n! defined by
(21)
[n]q!≔{1,if n=0,[n]q[n−1]q⋯[2]q[1]q,if n∈N, 
(22)
(1−q)−n∏k=0∞(1−qk+1)(1−qk+1+n)=(q;q)∞(qn+1;q)∞(1−q)−n≔Γq(n+1) (0<q<1). 
(23)
Γq(a)≔(q;q)∞(qa;q)∞(1−q)1−a (0<q<1). 
The qanalogue of (t − a)^{n} is defined by the polynomial
(24)
(t−a)(n)≔{1,(n=0)(t−a)(t−qa)⋯(t−qn−1a),(n∈N),=tn(at;q)n (n∈N0). 
(25)
(t−a)(γ)≔tγ∏k=0∞1−(a/t)qk1−(a/t)qγ+k, t≠0. 
Definition 1 .
Let R(β), R(μ) > 0 and η ∈ C. Then a generalized ErdélyiKober fractional integral I_{q}^{α,β,η} for a realvalued continuous function f(t) is defined by (see, [^{17}])
(26)
Iqη,μ,β{f}(t) =βt−β(η+μ)Γq(μ)∫0t(tβ−τβq)(μ−1)τβ(η+1)−1f(τ)dqτ =β(1−q1/β)(1−q)μ−1∑k=0∞(qμ;q)k(q;q)kqk(η+1)f(tqk/β). 
Definition 2 .
A qanalogue of the Kober fractional integral operator is given by (see, [^{20}])
(27)
Iqη,μ{f}(t):=(Iqη,μ,1{f}(t))=t−η−μΓq(μ)∫0t(t−τq)(μ−1)τηf(τ)dqτ,(μ>0; η∈C; 0<q<1). 
Remark 3 .
It is easy to see that
(28)
Γq(μ)>0; (qμ;q)k>0, 
(29)
Iqη,μ,β{f}(t)≥0, 
On the same way each term in the series of Kober qintegral operator (27) is also nonnegative and thus
(30)
Iqη,μ{f}(t)≥0, 
This section begins by presenting two inequalities involving generalized ErdélyiKober qintegral operator (26) stated in Lemmas 4 and 5 below.
Lemma 4 .
Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:
(31)
Iqη,μ,β{u}(t)Iqη,μ,β{vfg}(t) +Iqη,μ,β{v}(t)Iqη,μ,β{ufg}(t) ≥Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) +Iqη,μ,β{vf}(t)Iqη,μ,β{ug}(t), 
Proof
Let f and g be two continuous and synchronous functions on [0, ∞). Then, for all τ, ρ ∈ (0, t) with t > 0, we have
(32)
(f(τ)−f(ρ))(g(τ)−g(ρ))≥0, 
(33)
f(τ)g(τ)+f(ρ)g(ρ)≥f(τ)g(ρ)+f(ρ)g(τ). 
Now, multiplying both sides of (33) by (βt^{−β(η+μ)}/Γ_{q}(μ))(t^{β} − τ^{β}q)^{(μ−1)}τ^{β(η+1)−1}u(τ), integrating the resulting inequality with respect to τ from 0 to t, and using (26), we get
(34)
Iqη,μ,β{ufg}(t)+f(ρ)g(ρ)Iqη,μ,β{u}(t) ≥g(ρ)Iqη,μ,β{uf}(t)+f(ρ)Iqη,μ,β{ug}(t). 
Next, multiplying both sides of (34) by (βt^{−β(η+μ)}/Γ_{q}(μ))(t^{β} − ρ^{β}q)^{(μ−1)}ρ^{β(η+1)−1}v(ρ), integrating the resulting inequality with respect to ρ from 0 to t, and using (26), we are led to the desired result (31).
Lemma 5 .
Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:
(35)
Iqζ,ν,δ{v}(t)Iqη,μ,β{ufg}(t) +Iqζ,ν,δ{vfg}(t)Iqη,μ,β{u}(t) ≥Iqζ,ν,δ{vg}(t)Iqη,μ,β{uf}(t) +Iqζ,ν,δ{vf}(t)Iqη,μ,β{ug}(t), 
Proof
Multiplying both sides of (34) by
(36)
δt−δ(ζ+ν)Γq(ν)(tδ−ρδq)(ν−1)ρδ(ζ+1)−1v(ρ), 
Theorem 6 .
Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:
(37)
2Iqη,μ,β{l}(t)[Iqη,μ,β{m}(t)Iqη,μ,β{nfg}(t)+Iqη,μ,β{n}(t)Iqη,μ,β{mfg}(t)] +2Iqη,μ,β{m}(t)Iqη,μ,β{n}(t)Iqη,μ,β{lfg}(t) ≥Iqη,μ,β{l}(t)[Iqη,μ,β{mf}(t)Iqη,μ,β{ng}(t)+Iqη,μ,β{nf}(t)Iqη,μ,β{mg}(t)] +Iqη,μ,β{m}(t)[Iqη,μ,β{lf}(t)Iqη,μ,β{ng}(t)+Iqη,μ,β{nf}(t)Iqη,μ,β{lg}(t)] +Iqη,μ,β{n}(t)[Iqη,μ,β{lf}(t)Iqη,μ,β{mg}(t)+Iqη,μ,β{mf}(t)Iqη,μ,β{lg}(t)], 
Proof
By setting u = m and v = n in Lemma 4, we get
(38)
Iqη,μ,β{m}(t)Iqη,μ,β{nfg}(t) +Iqη,μ,β{n}(t)Iqη,μ,β{mfg}(t) ≥Iqη,μ,β{mf}(t)Iqη,μ,β{ng}(t) +Iqη,μ,β{nf}(t)Iqη,μ,β{mg}(t). 
(39)
Iqη,μ,β{l}(t)[Iqη,μ,β{m}(t)Iqη,μ,β{nfg}(t)+Iqη,μ,β{n}(t)Iqη,μ,β{mfg}(t)] ≥Iqη,μ,β{l}(t)[Iqη,μ,β{mf}(t)Iqη,μ,β{ng}(t)+Iqη,μ,β{nf}(t)Iqη,μ,β{mg}(t)]. 
(40)
Iqη,μ,β{m}(t)[Iqη,μ,β{l}(t)Iqη,μ,β{nfg}(t)+Iqη,μ,β{n}(t)Iqη,μ,β{lfg}(t)] ≥Iqη,μ,β{m}(t)[Iqη,μ,β{lf}(t)Iqη,μ,β{ng}(t)+Iqη,μ,β{nf}(t)Iqη,μ,β{lg}(t)], 
(41)
Iqη,μ,β{n}(t)[Iqη,μ,β{l}(t)Iqη,μ,β{mfg}(t)+Iqη,μ,β{m}(t)Iqη,μ,β{lfg}(t)] ≥Iqη,μ,β{n}(t)[Iqη,μ,β{lf}(t)Iqη,μ,β{mg}(t)+Iqη,μ,β{mf}(t)Iqη,μ,β{lg}(t)]. 
Theorem 7 .
Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:
(42)
Iqη,μ,β{l}(t)[2Iqη,μ,β{m}(t)Iqζ,ν,δ{nfg}(t)+Iqη,μ,β{n}(t)Iqζ,ν,δ{mfg}(t)+Iqζ,ν,δ{n}(t)Iqη,μ,β{mfg}(t)] +Iqη,μ,β{lfg}(t)[Iqη,μ,β{m}(t)Iqζ,ν,δ{n}(t)+Iqη,μ,β{n}(t)Iqζ,ν,δ{m}(t)] ≥Iqη,μ,β{l}(t)[Iqη,μ,β{mf}(t)Iqζ,ν,δ{ng}(t)+Iqη,μ,β{mg}(t)Iqζ,ν,δ{nf}(t)] +Iqη,μ,β{m}(t)[Iqη,μ,β{lf}(t)Iqζ,ν,δ{ng}(t)+Iqη,μ,β{lg}(t)Iqζ,ν,δ{nf}(t)] +Iqη,μ,β{n}(t)[Iqη,μ,β{lf}(t)Iqζ,ν,δ{mg}(t)+Iqη,μ,β{lg}(t)Iqζ,ν,δ{mf}(t)], 
Proof
Setting u = m and v = n in (35), we have
(43)
Iqζ,ν,δ{n}(t)Iqη,μ,β{mfg}(t) +Iqζ,ν,δ{nfg}(t)Iqη,μ,β{m}(t) ≥Iqζ,ν,δ{ng}(t)Iqη,μ,β{mf}(t) +Iqζ,ν,δ{nf}(t)Iqη,μ,β{mg}(t). 
(44)
Iqη,μ,β{l}(t)[Iqζ,ν,δ{n}(t)Iqη,μ,β{mfg}(t)+Iqζ,ν,δ{nfg}(t)Iqη,μ,β{m}(t)] ≥Iqη,μ,β{l}(t)[Iqζ,ν,δ{ng}(t)Iqη,μ,β{mf}(t)+Iqζ,ν,δ{nf}(t)Iqη,μ,β{mg}(t)]. 
(45)
Iqη,μ,β{m}(t)[Iqζ,ν,δ{n}(t)Iqη,μ,β{lfg}(t)+Iqζ,ν,δ{nfg}(t)Iqη,μ,β{l}(t)] ≥Iqη,μ,β{m}(t)[Iqζ,ν,δ{ng}(t)Iqη,μ,β{lf}(t)+Iqζ,ν,δ{nf}(t)Iqη,μ,β{lg}(t)],Iqη,μ,β{n}(t)[Iqζ,ν,δ{m}(t)Iqη,μ,β{lfg}(t)+Iqζ,ν,δ{mfg}(t)Iqη,μ,β{l}(t)] ≥Iqη,μ,β{n}(t)[Iqζ,ν,δ{mg}(t)Iqη,μ,β{lf}(t)+Iqζ,ν,δ{mf}(t)Iqη,μ,β{lg}(t)]. 
Remark 8 .
It may be noted that inequalities (37) and (42) in Theorems 6 and 7, respectively, are reversed if the functions are asynchronous on [0, ∞). The special case of (42) in Theorem 7 when β = δ, η = ζ, and μ = ν is easily seen to yield inequality (37) in Theorem 6.
Remark 9 .
We remark further that we can present a large number of special cases of our main inequalities in Theorems 6 and 7. Here, we give only two examples: setting β = 1 in (37) and β = δ = 1 in (42), we obtain interesting inequalities involving ErdélyiKober fractional integral operator.
Corollary 10 .
Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:
(46)
2Iqη,μ{l}(t)[Iqη,μ{m}(t)Iqη,μ{nfg}(t)+Iqη,μ{n}(t)Iqη,μ{mfg}(t)] +2Iqη,μ{m}(t)Iqη,μ{n}(t)Iqη,μ{lfg}(t) ≥Iqη,μ{l}(t)[Iqη,μ{mf}(t)Iqη,μ{ng}(t)+Iqη,μ{nf}(t)Iqη,μ{mg}(t)] +Iqη,μ{m}(t)[Iqη,μ{lf}(t)Iqη,μ{ng}(t)+Iqη,μ{nf}(t)Iqη,μ{lg}(t)] +Iqη,μ{n}(t)[Iqη,μ{lf}(t)Iqη,μ{mg}(t)+Iqη,μ{mf}(t)Iqη,μ{lg}(t)], 
Corollary 11 .
Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, ∞), and let l, m, n : [0, ∞)→[0, ∞) be continuous functions. Then, the following inequality holds true:
(47)
Iqη,μ{l}(t)[2Iqη,μ{m}(t)Iqζ,ν{nfg}(t)+Iqη,μ{n}(t)Iqζ,ν{mfg}(t)+Iqζ,ν{n}(t)Iqη,μ{mfg}(t)] +Iqη,μ{lfg}(t)[Iqη,μ{m}(t)Iqζ,ν{n}(t)+Iqη,μ{n}(t)Iqζ,ν{m}(t)] ≥Iqη,μ{l}(t)[Iqη,μ{mf}(t)Iqζ,ν{ng}(t)+Iqη,μ{mg}(t)Iqζ,ν{nf}(t)] +Iqη,μ{m}(t)[Iqη,μ{lf}(t)Iqζ,ν{ng}(t)+Iqη,μ{lg}(t)Iqζ,ν{nf}(t)] +Iqη,μ{n}(t)[Iqη,μ{lf}(t)Iqγ,δ{mg}(t)+Iqη,μ{lg}(t)Iqζ,ν{mf}(t)], 
In this section we obtain some new inequalities involving ErdélyiKober fractional qintegral operator in the case where the functions are bounded by integrable functions and are not necessary increasing or decreasing as are the synchronous functions.
Theorem 13 .
Let 0 < q < 1, let f be an integrable function on [0, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Assume the following.

(H_{1})
There exist two integrable functions φ_{1}, φ_{2} on [0, ∞) such that
[Formula ID: eq48]
(48)φ1(t)≤f(t)≤φ2(t), ∀t∈[0,∞).
(49)
Iqη,μ,β{uφ2}(t)Iqη,μ,β{vf}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vφ1}(t) ≥Iqη,μ,β{uφ2}(t)Iqη,μ,β{vφ1}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vf}(t). 
Proof
From (H_{1}), for all τ ≥ 0 and ρ ≥ 0, we have
(50)
(φ2(τ)−f(τ))(f(ρ)−φ1(ρ))≥0. 
(51)
φ2(τ)f(ρ)+φ1(ρ)f(τ) ≥φ1(ρ)φ2(τ)+f(τ)f(ρ). 
(52)
Iqη,μ,β{uφ2}(t)f(ρ)+Iqη,μ,β{uf}(t)φ1(ρ) ≥Iqη,μ,β{uφ2}(t)φ1(ρ)+Iqη,μ,β{uf}(t)f(ρ). 
As special cases of Theorems 13, we obtain the following results.
Corollary 14 .
Let 0 < q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we have
(53)
MIqη,μ,β{u}(t)Iqη,μ,β{vf}(t) +mIqη,μ,β{uf}(t)Iqη,μ,β{v}(t) ≥mMIqη,μ,β{u}(t)Iqη,μ,β{v}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vf}(t). 
Corollary 15 .
Let 0 < q < 1, let f be an integrable function on [1, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Assume that there exists an integrable function φ(t) on [0, ∞) and a constant M > 0 such that
(54)
φ(t)−M≤f(t)≤φ(t)+M, 
(55)
Iqη,μ,β{uφ}(t)Iqη,μ,β{vf}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vφ}(t) +MIqη,μ,β{u}(t)Iqη,μ,β{vf}(t) +MIqη,μ,β{v}(t)Iqη,μ,β{uφ}(t) +M2Iqη,μ,β{u}(t)Iqη,μ,β{v}(t) ≥Iqη,μ,β{uφ}(t)Iqη,μ,β{vφ}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vf}(t) +MIqη,μ,β{u}(t)Iqη,μ,β{vφ}(t) +MIqη,μ,β{uf}(t)Iqη,μ,β{v}(t). 
Theorem 16 .
Let 0 < q < 1, let f be an integrable function on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ_{1}, θ_{2} > 0 satisfying 1/θ_{1} + 1/θ_{2} = 1. Suppose that (H_{1}) holds. Then, for t > 0, μ, β > 0, and η ∈ C, we have
(56)
1θ1Iqη,μ,β{v}(t)Iqη,μ,β{u(φ2−f)θ1}(t) +1θ2Iqη,μ,β{u}(t)Iqη,μ,β{v(f−φ1)θ2}(t) +Iqη,μ,β{uφ2}(t)Iqη,μ,β{vφ1}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vf}(t) ≥Iqη,μ,β{uφ2}(t)Iqη,μ,β{vf}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vφ1}(t). 
Proof
According to the wellknown Young inequality [^{3}]
(57)
1θ1xθ1+1θ2yθ2≥xy, ∀x,y≥0,θ1,θ2>0, 1θ1+1θ2=1, 
(58)
1θ1(φ2(τ)−f(τ))θ1+1θ2(f(ρ)−φ1(ρ))θ2 ≥(φ2(τ)−f(τ))(f(ρ)−φ1(ρ)). 
(59)
β2t−2β(η+μ)Γq2(μ)(tβ−τβq)(μ−1)(tβ−ρβq)(μ−1) ×(τρ)β(η+1)−1u(τ)v(ρ), 
Corollary 17 .
Let 0 < q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we have
(60)
(m+M)2Iqη,μ,β{u}(t)Iqη,μ,β{v}(t) +2Iqη,μ,β{uf}(t)Iqη,μ,β{vf}(t) +Iqη,μ,β{vf2}(t)(Iqη,μ,β{u}(t)+Iqη,μ,β{v}(t)) ≥2(m+M)(Iqη,μ,β{uf}(t)Iqη,μ,β{v}(t)+ Iqη,μ,β{u}(t)Iqη,μ,β{vf}(t)). 
Theorem 18 .
Let 0 < q < 1, let f be an integrable function on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ_{1}, θ_{2} > 0 satisfying θ_{1} + θ_{2} = 1. In addition, suppose that (H_{1}) holds. Then, for t > 0, μ, β > 0, and η ∈ C, we have
(61)
θ1Iqη,μ,β{uφ2}(t)Iqη,μ,β{v}(t) +θ2Iqη,μ,β{u}(t)Iqη,μ,β{vf}(t) ≥θ1Iqη,μ,β{uf}(t)Iqη,μ,β{v}(t) +θ2Iqη,μ,β{u}(t)Iqη,μ,β{vφ1}(t) +Iqη,μ,β{u(φ2−f)θ1}(t)Iqη,μ,β{v(f−φ1)θ2}(t). 
Proof
From the wellknown Weighted AMGM inequality [^{3}]
(62)
θ1x+θ2y≥xθ1yθ2, ∀x,y≥0, θ1,θ2>0, θ1+θ2=1, 
(63)
θ1(φ2(τ)−f(τ))+θ2(f(ρ)−φ1(ρ)) ≥(φ2(τ)−f(τ))θ1(f(ρ)−φ1(ρ))θ2. 
(64)
β2t−2β(η+μ)Γq2(μ)(tβ−τβq)(μ−1)(tβ−ρβq)(μ−1) × (τρ)β(η+1)−1u(τ)v(ρ), 
Corollary 19 .
Let 0 < q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we have
(65)
(M−m)Iqη,μ,β{u}(t)Iqη,μ,β{v}(t)+Iqη,μ,β{u}(t)Iqη,μ,β{vf}(t) ≥Iqη,μ,β{uf}(t)Iqη,μ,β{v}(t) +2Iqη,μ,β{uM−f}(t)Iqη,μ,β{vf−m}(t). 
Lemma 20 (see [^{22}]).
Assume that a ≥ 0, p ≥ q ≥ 0, and p ≠ 0. Then,
(66)
aq/p≤(qpk(q−p)/pa+p−qpkq/p), for any k>0. 
Theorem 21 .
Let 0 < q < 1, let f be an integrable function on [0, ∞), let u : [0, ∞)→[0, ∞) be a continuous function, and let constants p ≥ q ≥ 0, p ≠ 0. In addition, assume that (H_{1}) holds. Then, for any k > 0, t > 0, μ, β > 0, and η ∈ C, the following two inequalities hold:
(67)
(i) Iqη,μ,β{u(φ2−f)q/p}(t)+qpk(q−p)/pIqη,μ,β{uf}(t) ≤qpk(q−p)/pIqη,μ,β{uφ2}(t)+p−qpkq/pIqη,μ,β{u}(t), 
(68)
(ii) Iqη,μ,β{u(f−φ1)q/p}(t)+qpk(q−p)/pIqη,μ,β{uφ1}(t) ≤qpk(q−p)/pIqη,μ,β{uf}(t)+p−qpkq/pIqη,μ,β{u}(t). 
Proof
By condition (H_{1}) and Lemma 20, for p ≥ q ≥ 0, p ≠ 0, it follows that
(69)
(φ2(τ)−f(τ))q/p≤qpk(q−p)/p(φ2(τ)−f(τ))+p−qpkq/p, 
Corollary 22 .
Let 0 < q < 1, let f be an integrable function on [0, ∞) satisfying m ≤ f(t) ≤ M, for all t ∈ [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let m, M ∈ R. Then, for t > 0, μ, β > 0, and η ∈ C, we have
(70)
(i) 2Iqη,μ,β{uM−f}(t)+Iqη,μ,β{uf}(t) ≤(M+1)Iqη,μ,β{u}(t),(ii) 2Iqη,μ,β{uf−m}(t)+(m−1)Iqη,μ,β{u}(t) ≤Iqη,μ,β{uf}(t). 
Theorem 23 .
Let 0 < q < 1, let f and g be two integrable functions on [0, ∞), and let u, v : [0, ∞)→[0, ∞) be continuous functions. Suppose that (H_{1}) holds and moreover we assume the following.

(H_{2}) There exist ψ_{1} and ψ_{2} integrable functions on [0, ∞) such that
[Formula ID: eq71]
(71)ψ1(t)≤g(t)≤ψ2(t) ∀t∈[0,∞).
(72)
(i) Iqη,μ,β{uφ2}(t)Iqη,μ,β{vg}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vψ1}(t) ≥Iqη,μ,β{uφ2}(t)Iqη,μ,β{vψ1}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t),(ii) Iqη,μ,β{uψ2}(t)Iqη,μ,β{vf}(t) +Iqη,μ,β{ug}(t)Iqη,μ,β{vφ1}(t) ≥Iqη,μ,β{uψ2}(t)Iqη,μ,β{vφ1}(t) +Iqη,μ,β{ug}(t)Iqη,μ,β{vf}(t),(iii) Iqη,μ,β{uφ2}(t)Iqη,μ,β{vψ2}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) ≥Iqη,μ,β{uφ2}(t)Iqη,μ,β{vg}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vψ2}(t),(iv) Iqη,μ,β{uφ1}(t)Iqη,μ,β{vψ1}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) ≥Iqη,μ,β{uφ1}(t)Iqη,μ,β{vg}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vψ1}(t). 
Proof
To prove (i), from (H_{1}) and (H_{2}), we have for t ∈ [0, ∞) that
(73)
(φ2(τ)−f(τ))(g(ρ)−ψ1(ρ))≥0. 
(74)
φ2(τ)g(ρ)+ψ1(ρ)f(τ)≥ψ1(ρ)φ2(τ) +f(τ)g(ρ). 
(75)
g(ρ)Iqη,μ,β{uφ2}(t)+ψ1(ρ)Iqη,μ,β{uf}(t) ≥ψ1(ρ)Iqη,μ,β{uφ2}(t)+g(ρ)Iqη,μ,β{uf}(t). 
To prove (ii)–(iv), we use the following inequalities:
(76)
(ii) (ψ2(τ)−g(τ))(f(ρ)−φ1(ρ))≥0, (iii) (φ2(τ)−f(τ))(g(ρ)−ψ2(ρ))≤0, (iv) (φ1(τ)−f(τ))(g(ρ)−ψ1(ρ))≤0. 
Theorem 24 .
Let f and g be two integrable functions on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ_{1}, θ_{2} > 0 satisfying 1/θ_{1} + 1/θ_{2} = 1. Suppose that (H_{1}) and (H_{2}) hold. Then, for t > 0, μ, β > 0, and η ∈ C, the following inequalities hold:
(77)
(i) 1θ1Iqη,μ,β{u(φ2−f)θ1}(t)Iqη,μ,β{v}(t) +1θ2Iqη,μ,β{v(ψ2−g)θ2}(t)Iqη,μ,β{u}(t) +Iqη,μ,β{uφ2}(t)Iqη,μ,β{vg}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vψ2}(t) ≥Iqη,μ,β{uφ2}(t)Iqη,μ,β{vψ2}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t),(ii) 1θ1Iqη,μ,β{u(φ2−f)θ1}(t)Iqη,μ,β{v(ψ2−g)θ1}(t) +1θ2Iqη,μ,β{u(ψ2−g)θ2}(t)Iqη,μ,β{v(φ2−f)θ2}(t) ≥Iqη,μ,β{u(φ2−f)(ψ2−g)}(t) ×Iqη,μ,β{v(ψ2−g)(φ2−f)}(t),(iii) 1θ1Iqη,μ,β{u(f−φ1)θ1}(t)Iqη,μ,β{v}(t) +1θ2Iqη,μ,β{v(g−ψ1)θ2}(t)Iqη,μ,β{u}(t) +Iqη,μ,β{uf}(t)Iqη,μ,β{vψ1}(t) +Iqη,μ,β{uφ1}(t)Iqη,μ,β{vg}(t) ≥Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) +Iqη,μ,β{uφ1}(t)Iqη,μ,β{vψ1}(t),(iv) 1θ1Iqη,μ,β{u(f−φ1)θ1}(t)Iqη,μ,β{v(g−ψ1)θ1}(t) +1θ2Iqη,μ,β{u(g−ψ1)θ2}(t)Iqη,μ,β{v(f−φ1)θ2}(t) ≥Iqη,μ,β{u(f−φ1)(g−ψ1)}(t) ×Iqη,μ,β{v(g−ψ1)(f−φ1)}(t). 
Proof
The inequalities (i)–(iv) can be proved by choosing the parameters in the Young inequality [^{3}]:
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(i) x=φ2(τ)−f(τ), y=ψ2(ρ)−g(ρ),(ii) x=(φ2(τ)−f(τ))(ψ2(ρ)−g(ρ)),y=(ψ2(τ)−g(τ))(φ2(ρ)−f(ρ)),(iii) x=f(τ)−φ1(τ), y=g(ρ)−ψ1(ρ),(iv) x=(f(τ)−φ1(τ))(g(ρ)−ψ1(ρ)),y=(g(τ)−ψ1(τ))(f(ρ)−φ1(ρ)). 
Theorem 25 .
Let f and g be two integrable functions on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let θ_{1}, θ_{2} > 0 satisfying θ_{1} + θ_{2} = 1. Suppose that (H_{1}) and (H_{2}) hold. Then, for t > 0, μ, β > 0, and η ∈ C, the following inequalities hold:
(79)
(i) θ1Iqη,μ,β{uφ2}(t)Iqη,μ,β{v}(t) +θ2Iqη,μ,β{vψ2}(t)Iqη,μ,β{u}(t) ≥θ1Iqη,μ,β{uf}(t)Iqη,μ,β{v}(t) +θ2Iqη,μ,β{vg}(t)Iqη,μ,β{u}(t) +Iqη,μ,β{u(φ2−f)θ1}(t)Iqη,μ,β{v(ψ2−g)θ2}(t),(ii) θ1Iqη,μ,β{uφ2}(t)Iqη,μ,β{vψ2}(t) +θ1Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) +θ2Iqη,μ,β{uψ2}(t)Iqη,μ,β{vφ2}(t) +θ2Iqη,μ,β{ug}(t)Iqη,μ,β{vf}(t) ≥θ1Iqη,μ,β{uφ2}(t)Iqη,μ,β{vg}(t) +θ1Iqη,μ,β{uf}(t)Iqη,μ,β{vψ2}(t) +θ2Iqη,μ,β{uψ2}(t)Iqη,μ,β{vf}(t) +θ2Iqη,μ,β{ug}(t)Iqη,μ,β{vφ2}(t) +Iqη,μ,β{u(φ2−f)θ1(ψ2−g)θ2}(t) ×Iqη,μ,β{v(ψ2−g)θ1(φ2−f)θ2}(t),(iii) θ1Iqη,μ,β{uf}(t)Iqη,μ,β{v}(t) +θ2Iqη,μ,β{vg}(t)Iqη,μ,β{u}(t) ≥θ1Iqη,μ,β{uφ1}(t)Iqη,μ,β{v}(t) +θ2Iqη,μ,β{vψ1}(t)Iqη,μ,β{u}(t) +Iqη,μ,β{u(f−φ1)θ1}(t)Iqη,μ,β{v(g−ψ1)θ2}(t),(iv) θ1Iqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) +θ1Iqη,μ,β{uφ1}(t)Iqη,μ,β{vψ1}(t) +θ2Iqη,μ,β{ug}(t)Iqη,μ,β{vf}(t) +θ2Iqη,μ,β{uψ1}(t)Iqη,μ,β{vφ1}(t) ≥θ1Iqη,μ,β{uf}(t)Iqη,μ,β{vψ1}(t) +θ1Iqη,μ,β{uφ1}(t)Iqη,μ,β{vg}(t) +θ2Iqη,μ,β{ug}(t)Iqη,μ,β{vφ1}(t) +θ2Iqη,μ,β{uψ1}(t)Iqη,μ,β{vf}(t) +Iqη,μ,β{u(f−φ1)θ1(g−ψ1)θ2}(t) ×Iqη,μ,β{v(g−ψ1)θ1(f−φ1)θ2}(t). 
Proof
The inequalities (i)–(iv) can be proved by choosing the parameters in the Weighted AMGM [^{3}]:
(80)
(i) x=φ2(τ)−f(τ), y=ψ2(ρ)−g(ρ),(ii) x=(φ2(τ)−f(τ))(ψ2(ρ)−g(ρ)),y=(ψ2(τ)−g(τ))(φ2(ρ)−f(ρ)),(iii) x=f(τ)−φ1(τ), y=g(ρ)−ψ1(ρ),(iv) x=(f(τ)−φ1(τ))(g(ρ)−ψ1(ρ)),y=(g(τ)−ψ1(τ))(f(ρ)−φ1(ρ)). 
Theorem 26 .
Let f and g be two integrable functions on [0, ∞), let u, v : [0, ∞)→[0, ∞) be continuous functions, and let constants p ≥ q ≥ 0, p ≠ 0. Assume that (H_{1}) and (H_{2}) hold. Then, for any k > 0, t > 0, μ, β > 0, and η ∈ C, the following inequalities hold:
(81)
(i) Iqη,μ,β{u(φ2−f)q/p(ψ2−g)q/p}(t) +qpk(q−p)/pIqη,μ,β{uφ2g}(t) +qpk(q−p)/pIqη,μ,β{ufψ2}(t) ≤qpk(q−p)/pIqη,μ,β{uφ2ψ2}(t) +qpk(q−p)/pIqη,μ,β{ufg}(t) +p−qpkq/pIqη,μ,β{u}(t),(ii) Iqη,μ,β{u(φ2−f)q/p}(t)Iqη,μ,β{v(ψ2−g)q/p}(t) +qpk(q−p)/pIqη,μ,β{uφ2}(t)Iqη,μ,β{vg}(t) +qpk(q−p)/pIqη,μ,β{uf}(t)Iqη,μ,β{vψ2}(t) ≤qpk(q−p)/pIqη,μ,β{uφ2}(t)Iqη,μ,β{vψ2}(t) +qpk(q−p)/pIqη,μ,β{uf}(t)Iqη,μ,β{vg}(t)g(t) +p−qpkq/pIqη,μ,β{u}(t)Iqη,μ,β{v}(t),(iii) Iqη,μ,β{u(f−φ1)q/p(g−ψ1)q/p}(t) +qpk(q−p)/pIqη,μ,β{uψ1f}(t) +qpk(q−p)/pIqη,μ,β{uφ1g}(t) ≤qpk(q−p)/pIqη,μ,β{ufg}(t) +qpk(q−p)/pIqη,μ,β{uφ1ψ1}(t) +p−qpkq/pIqη,μ,β{u}(t),(iv) Iqη,μ,β{u(f−φ1)q/p}(t)Iqη,μ,β{v(g−ψ1)q/p}(t) +qpk(q−p)/pIqη,μ,β{uf}(t)Iqη,μ,β{vψ1}(t) +qpk(q−p)/pIqη,μ,β{uφ1}(t)Iqη,μ,β{vg}(t) ≤qpk(q−p)/pIqη,μ,β{uf}(t)Iqη,μ,β{vg}(t) +qpk(q−p)/pIqη,μ,β{uφ1}(t)Iqη,μ,β{vψ1}(t) +p−qpkq/pIqη,μ,β{u}(t)Iqη,μ,β{v}(t). 
Proof
The inequalities (i)–(iv) can be proved by choosing the parameters in Lemma 20:
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(i) a=(φ2(τ)−f(τ))(ψ2(τ)−g(τ)), (ii) a=(φ2(τ)−f(τ))(ψ2(ρ)−g(ρ)), (iii) a=(f(τ)−φ1(τ))(g(τ)−ψ1(τ)), (iv) a=(f(τ)−φ1(τ))(g(ρ)−ψ1(ρ)). 
We conclude our present investigation with the remark that the results derived in this paper are general in character and give some contributions to the theory of qintegral inequalities and fractional calculus. Moreover, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. In last, use of the generalized ErdélyiKober fractional qintegral operator due to Gaulué is the advantage of our results because after setting suitable parameter values in our main results, we get known results established by number of authors.
The research of J. Tariboon is supported by King Mongkut's University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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