# On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus

Authors: Saima Rashid; Aasma Khalid; Gauhar Rahman; Kottakkaran Sooppy Nisar; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operatorσΨqς=qς+1−qσς∈l1,l2,σ=l1+ω/1−q,0<q<1,ω≥0. As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory.


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## Body

## 1. Introduction

In 1882, Čebyšev discovered a fascinating and significantly valuable integral inequality as follows:(1)1l1−l2∫l1l2QxUxdx≤1l1−l2∫l1l2Qxdx1l1−l2∫l1l2Uxdx,if Q and U are two integrable and synchronous functions on l1,l2, where the functions Q and U are said to be synchronous on l1,l2 if
(2)Qx−QyUx−Uy≥0for all x,y∈l1,l2.It is well known that the Čebyšev inequality (1) has wild applications in the fields of pure and applied mathematics [1–10]. Recently, the generalizations and variants for the Čebyšev inequality (1) have attracted the attention of many researchers [11–20].Quantum difference operators are receiving an increase of interest due to their applications [21, 22]. Roughly speaking, quantum calculus can substitute the classical derivative by a difference operator, which allows to deal nondifferentiation functions.Letq∈0,1, ω≥0, I⊆ℝ be an interval such that ω0=ω/1−q∈I, and H1:I⟶ℝ be a real-valued function. Then, the Hahn difference operator Dq,ω [23] is defined by
(3)Dq,ωH1ς=H1qς+ω−H1ςςq−1+ω,ς≠ω0,H1′ω0,ς=ω0,if H1 is differentiable at ω0.The Hahn difference operator (3) unifies (in the limit) the Jackson q-difference derivative Dq [24] for q∈0,1 and the forward difference Dω for q⟶1, which are defined by
(4)DqH1ς=H1ς−H1qςς1−q,ς≠0,H1′0,ς=0,if H1′0 exists for ω=0, and
(5)DωH1ς=H1ς+ω−H1ςωfor ω>0.The Hahn difference operator has been applied successfully in the construction of families of orthogonal polynomials as well as in approximation problems [25–28].In [29], the authors introduced some concepts of fractional quantum calculus in terms of a q-shifting operator σΨqς=qς+1−qσ.LetI=l1,l2⊆ℝ be an interval. Then, the point σ of Hahn calculus on the interval l1,l2 generated by the quantum numbers 0<q<1 and ω≥0 is given by
(6)σ=l1+ω1−q.We state thatσ∈l1,l2 for all consequences of our investigation; the quantum Hahn shifting operator is defined by
(7)σΨqς=qς+1−qσ,ς∈l1,l2,and the iterated κ-times quantum shifting is given by
(8)σΨqκς=σΨqκ−1σΨqς=qκς+1−qκσ,with σΨq0ς=ς for ς∈l1,l2.Let us recall the basic knowledge of quantum Hahn calculus on an intervall1,l2 (see [30]).Definition 1.
LetH1 be a function defined on l1,l2. Then, the quantum Hahn difference operator is defined by
(9)l1Dq,ωH1ς=H1ς−H1σΨqςς−σΨqς,ς≠σ,H1′σ,ς=σ,if H1 is differentiable at σ.Definition 2.
LetH1:l1,l2⟶ℝ be a given function and x,y∈l1,l2. Then, the q,ω-quantum Hahn integral of H1 from x to y is defined by
(10)∫xyH1sl1dq,ωs=∫σyH1sl1dq,ωs−∫σxH1sl1dq,ωs,where
(11)∫σςH1sl1dq,ωs=ς−σΨqς∑i=0∞qiH1σΨqiςfor ς∈l1,l2 provided that the series converge at ς=x and ς=y. The function H1 is said to be q,ω-integrable on l1,l2 if (11) exists for all ς∈l1,l2.Before approaching the main definitions of fractional quantum Hahn calculus onl1,l2, we present the σ-power function which is stated as
(12)n−mσ0=1,n−mσκ=∏i=0κ−1n−σΨqim,κ∈ℕ∪∞.Precisely, ifα∈ℝ, then
(13)n−mσα=∏i=0∞n−σΨqimn−σΨqα+im,with σΨqςm=qςm+1−qςσ for ς∈ℝ.Theq-gamma function is defined as
(14)Γqη=1−q0η−11−qη−1for η∈ℝ\0,−1,−2,⋯.Obviously,Γqη+1=αqΓqη, where cq=1−qc/1−qc∈ℝ and q is the quantum number.Now, we introduce the concepts of fractional quantum Hahn derivative and integral of Riemann-Liouville type [31].Definition 3 (see [31]).
Suppose thatα≥0 and H1:l1,l2⟶ℝ is a real-valued function. Then, the fractional quantum Hahn derivative l1Dq,ωαH1ς of the Riemann-Liouville type of order α is defined by
(15)l1Dq,ωαH1ς=1Γqn−αl1Dq,ωn∫l1ςς−σΨqsσn−α−1H1sl1dq,ω,where n is the smallest integer greater than or equal to α.Definition 4 (see [31]).
Letα≥0 and H1:l1,l2⟶ℝ be a real-valued function. Then, the fractional quantum Hahn integral l1Iq,ωαH1ς of the Riemann-Liouville type of order α is defined by
(16)l1Iq,ωαH1ς=1Γqα∫l1ςς−σΨqsσα−1H1sl1dq,ω,ς∈l1,l2,with l1Iq,ωl1H1ς=H1ς if the right-hand side exists.Theorem 5 (see [31]).
Letα,β∈ℝ+, ϑ∈−1,∞, and σ∈l1,l2. Then, one has
(17)l1Iq,ωαx−l1σϑς=Γqϑ+1Γqα+ϑ+1ς−l1σα+ϑ,l1Dq,ωαx−l1σϑς=Γqϑ+1Γqα−ϑ+1ς−l1σα−ϑ.Fractional calculus is invariably important in almost all fields of mathematics and applied sciences. Also, the fractional differential equations can provide adequate models for many physical problems in areas such as heat equation, wave equation, Poisson equation and Laplace equation, fluid mechanics, biological populations, viscoelasticity, advection-diffusion, and signal processing [32, 33].Inequality plays an irreplaceable role in the development of mathematics. Very recently, many new inequalities such as Hermite-Hadamard type inequality [34–38], Petrović type inequality [39], Pólya-Szegö type inequality [40], Ostrowski type inequality [41], reverse Minkowski inequality [42], Jensen type inequality [43, 44], Bessel function inequality [45], trigonometric and hyperbolic function inequalities [46], fractional integral inequality [47–51], complete and generalized elliptic integral inequalities [52–57], generalized convex function inequality [58–60], and mean value inequality [61–63] have been discovered by many researchers. In particular, the applications of integral inequalities have gained considerable importance among researchers for fixed-point theorems; the existence and uniqueness of solutions for differential equations [64–68] and numerous numerical and analytical methods have been recommended for the advancement of integral inequalities [69–75].Asawasamrit et al. [76] expounded the concept of q-derivative over the interval l1,l2⊂ℝ and derived several inequalities on quantum analogues, for example, q-Cauchy-Schwarz inequality, q-Grüss-Čebyšev integral inequality, q-Grüss inequality, and other integral inequalities, by use of the convexity theory.The main purpose of the article is to provide the novel versions of the generalized Čebyšev inequalities and present the associated variants via quantum Hahn’s fractional integral operator.To end this section, we give the definition of the one-sided fractional quantum Hahn integral in the Riemann-Liouville sense.Definition 6.
Letα≥0 and H1:l1,l2⟶ℝ be a real-valued function. Then, the one-sided fractional quantum Hahn integral 0+Iq,ωαH1ς of Riemann-Liouville type of order α is defined by
(18)0+Iq,ωαH1ς=1Γqα∫0ςς−σΨqsσα−1H1sl1dq,ω,s>0.


## 2. Certain Extended Weighted Čebyšev Fractional Quantum Hahn Integral Operator

In this section, we provide several new generalizations for the weighted extensions of Čebyšev functionals via a quantum Hahn integral operator.Theorem 7.
Lets,s1,u,u1,r,r1>1 with s−1+s1−1=r−1+r1−1=u−1+u1−1=1, q∈0,1, ω≥0, H1 be a positive q,ω-integrable function defined on 0,∞, and Q and U be two q,ω-differentiable functions defined on 0,∞ such that Q′∈Ls0,∞ and U′∈Lr0,∞. Then, the inequalities
(19)20+Iq,ωαH1ς0+Iq,ωαQUς−0+Iq,ωαH1Qς0+Iq,ωαH1Uς≤∥Q′∥suΓquα∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1Φ′xΦ′yH1xH1yx−y1/s1+1/r1dq,ωxdq,ωy1/u×∥U′∥ru1Γqu1α∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yx−y1/s1+1/r1dq,ωxdq,ωy1/u1≤∥Q′∥su∥U′∥ru1Γqα2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yx−y1/s1+1/r1dq,ωxdq,ωyhold for all ς>0.Proof.
Letx,y∈0,ς and
(20)Gx,y=Qx−QyUx−Uy.
Then,Gx,y can be written as
(21)Gx,y=QxUx−QxUy−QyUx−UyQy.
Multiplying both sides of (21) by ς−σΨqxσα−1/ΓqαH1x and then performing the q,ω-integration with respect to x over 0,ς, we have
(22)∫0ςς−σΨqxσα−1ΓqαH1xGx,ydq,ωx=∫0ςς−σΨqxσα−1ΓqαH1xQxUxdq,ωx−∫0ςς−σΨqxσα−1ΓqαH1xQxUydq,ωx−∫0ςς−σΨqxσα−1ΓqαH1xQyUxdq,ωx−UyQy∫0ςς−σΨqxσα−1ΓqαH1xdq,ωx.
Inequality (22) can be rewritten as
(23)∫0ςς−σΨqxσα−1ΓqαH1xGx,ydq,ωx=0+Iq,ωαH1QUς−Uy0+Iq,ωαH1Qς−Qy0+Iq,ωαH1Uς+QyUy0+Iq,ωαH1ς.
Multiplying both sides of (23) by ς−σΨqyσα−1/ΓqαH1y and then performing the q,ω-integration with respect to y over 0,ς, we get
(24)1Γqα2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yGx,ydq,ωxdq,ωy=20+Iq,ωαH1ς0+Iq,ωαQUς−0+Iq,ωαH1Qς0+Iq,ωαH1Uς.
Similarly, we have(25)Gx,y=∫xy∫xyQ′θU′ϑdθdϑ.
Taking into account the Hölder inequality, we have(26)∣Qx−Qy∣≤x−y1/s1∣∫xy∣Q′θsdθ1/s,(27)∣Ux−Uy∣≤x−y1/r1∣∫xy∣U′ϑrdϑ1/r.
It follows from (26) and (27) that
(28)∣Gx,y∣≤∣Qx−QyUx−Uy∣≤x−y1/s1+1/r1∣∫xy∣Q′θsdθ1/s∣∫xy∣U′ϑrdϑ1/r.
From (24) and (28), we obtain
(29)2∣0+Iq,ωαH1ς0+Iq,ωαQUς−0+Iq,ωαH1Qς0+Iq,ωαH1Uς∣=1Γqα2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1y∣Gx,y∣dq,ωxdq,ωy≤1Γqα2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1y×x−y1/s1+1/r1∣∫xy∣Q′θsdθ1/s∣∫xy∣U′ϑrdϑ1/rdq,ωxdq,ωy.
Making use of the Hölder inequality for bivariate integral, we have(30)2∣0+Iq,ωαH1ς0+Iq,ωαQUς−0+Iq,ωαH1Qς0+Iq,ωαH1Uς∣≤1Γqα2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1y×x−y1/s1+1/r1∣∫xy∣Q′θsdθu/sdq,ωxdq,ωy1/u×∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1y×x−y1/s1+1/r1∣∫xy∣U′ϑrdϑu1/rdq,ωxdq,ωy1/u1.
It follows from (30) and the inequalities
(31)∣∫xy∣Q′θsdθ1/s≤Q′s,∣∫xy∣U′ϑrdϑ1/r≤a∥U′∥rthat
(32)2∣0+Iq,ωαH1ς0+Iq,ωαQUς−0+Iq,ωαH1Qς0+Iq,ωαH1Uς∣≤∥Q′∥suΓquα∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1Φ′xΦ′yH1xH1yx−y1/s1+1/r1dq,ωxdq,ωy1/u×∥U′∥ru1Γqu1α∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yx−y1/s1+1/r1dq,ωxdq,ωy1/u1.
Therefore, we get the desired inequality(33)2∣0+Iq,ωαH1ς0+Iq,ωαQUς−0+Iq,ωαH1Qς0+Iq,ωαH1Uς∣≤∥Q′∥su∥U′∥ru1Γq1α2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yx−y1/s1+1/r1dq,ωxdq,ωy.Letω=1. Then, Theorem 7 leads to Corollary 8 which provide a new result for q-fractional integral operator.hold for all ς>0.Corollary 8.
Lets,s1,u,u1,r,r1>1 with s−1+s1−1=r−1+r1−1=u−1+u1−1=1, q∈0,1, H1 be a positive q,1-integrable function defined on 0,∞, and Q and U be two q,1-differentiable functions defined on 0,∞ such that Q′∈Ls0,∞ and U′∈Lr0,∞. Then, the inequalities
(34)20+Iq,1αH1ς0+Iq,1αQUς−0+Iq,1αH1Qς0+Iq,1αH1Uς≤∥Q′∥suΓquα∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1Φ′xΦ′yH1xH1yx−y1/s1+1/r1dqxdqy1/u×∥U′∥ru1Γqu1α∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yx−y1/s1+1/r1dqxdqy1/u1≤∥Q′∥su∥U′∥ru1Γqα2∫0ς∫0ςς−σΨqxσα−1ς−σΨqyσα−1H1xH1yx−y1/s1+1/r1dqxdqyhold for all ς>0.Remark 9.
Ifq=ω=1, then Theorem 7 reduces to the result for the Riemann-Liouville fractional integral operator given in [77]. Some results given in the literature [13, 78] can also be obtained from Theorem 7 immediately.Theorem 10.
Lets,s1,u,u1,r,r1>1 with s−1+s1−1=r−1+r1−1=u−1+u1−1=1, qi∈0,1, ωi≥0i=1,2, H1 and H2 be the positive qi,ωi-integrable functions defined on 0,∞, and Q and U be the qi,ωi-differentiable functions defined on 0,∞ such that Q′∈Ls0,∞ and U′∈Lr0,∞. Then, the inequality
(35)∣0+Iq2,ω2βH2ς0+Iq1,ω1αH1QUς−0+Iq2,ω2βH1Uς0+Iq1,ω1αH1Qς−0+Iq2,ω2βH2Qς0+Iq1,ω1αH1Uς+0+Iq2,ω2βH2QUς0+Iq1,ω1αH1ς∣≤∥Q∥s∥U∥rΓq1αΓq2β∫0ς∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1×x−y1/s1+1/r1H1xH2ydq1,ω1xdq2,ω2yholds for all ς>0.Proof.
Multiplying both sides of (23) by ς−σ2Ψq2yσ2β−1/Γq2βH2y and performing q2,ω2-integration with respect to y over 0,ς, we have
(36)1Γq1αΓq2β∫0ς∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1H1xH2yGx,ydq1,ω1xdq2,ω29y=0+Iq2,ω2βH2ς0+Iq1,ω1αH1QUς−0+Iq2,ω2βH1Uς0+Iq1,ω1αH1Qς−0+Iq2,ω2βH2Qς0+Iq1,ω1αH1Uς+0+Iq2,ω2βH2QUς0+Iq1,ω1αH1ς.
Taking modulus on both sides of (36), one has
(37)∣0+Iq2,ω2βH2ς0+Iq1,ω1αH1QUς−0+Iq2,ω2βH1Uς0+Iq1,ω1αH1Qς−0+Iq2,ω2βH2Qς0+Iq1,ω1αH1Uς+0+Iq2,ω2βH2QUς0+Iq1,ω1αH1ς∣=1Γq1αΓq2β∫0ς∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1H1xH2y∣Gx,y∣dq1,ω1xdq2,ω2y≤1Γq1αΓq2β∫0ς∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1×x−y1/s1+1/r1∣∫xy∣Q′θsdθ1/s∣∫xy∣U′ϑrdϑ1/rH1xH2ydq1,ω1xdq2,ω2y=∥Q∥s∥U∥rΓq1αΓq2β∫0ς∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1×x−y1/s1+1/r1H1xH2ydq1,ω1xdq2,ω2y.Letωi=1 for i=1,2. Then, Theorem 10 leads to Corollary 11 which provide a new result for qi-fractional integral operator.Corollary 11.
Lets,s1,u,u1,r,r1>1 with s−1+s1−1=r−1+r1−1=u−1+u1−1=1, qi∈0,1i=1,2, H1 and H2 be the positive qi-integrable functions defined on 0,∞, and Q and U be the qi-differentiable functions defined on 0,∞ such that Q′∈Ls0,∞ and U′∈Lr0,∞. Then, the inequality
(38)∣0+Iq2βH2ς0+Iq1αH1QUς−0+Iq2βH1Uς0+Iq1αH1Qς−0+Iq2βH2Qς0+Iq1αH1Uς+0+Iq2βH2QUς0+Iq1αH1ς∣≤∥Q∥s∥U∥rΓq1αΓq2β∫0ς∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1×x−y1/s1+1/r1H1xH2ydq1xdq2yholds for all ς>0.Remark 12.
Letqi=ωi=1. Then, Theorem 10 becomes Theorem 14 of [77].


## 3. Some New Generalizations by Fractional Quantum Hahn Integral Operator

Theorem 13.
Letr,s>1 with 1/r+1/s=1, qi∈0,1 and ωi≥0 for i=1,2, and Q and U be the the positive qi,ωi-integral functions. Then the following inequalities
(A1)1/r0+Iq1,ω1αQrς0+Iq2,ω2βUrς+1/s0+Iq1,ω1αUsς0+Iq2,ω2βQsς≥0+Iq1,ω1αQUς0+Iq2,ω2βQUς
(A2)1/r0+Iq1,ω1αQrς0+Iq2,ω2βUsς+1/s0+Iq1,ω1αUsς0+Iq2,ω2βQrς≥0+Iq1,ω1αQUς0+Iq2,ω2βQr−1Us−1ς
(A3)1/r0+Iq1,ω1αQrς0+Iq2,ω2βU2ς+1/s0+Iq1,ω1αQ2ς0+Iq2,ω2βUsς≥0+Iq1,ω1αQUς0+Iq2,ω2βQ2/sU2/rς
(A4)1/r0+Iq1,ω1αQ2ς0+Iq2,ω2βQrς+1/s0+Iq1,ω1αU2ς0+Iq2,ω2βUsς≥0+Iq1,ω1αQ2/rU2/sς0+Iq2,ω2βQUς
hold for allς>0.Proof.
Considering the well-known Young inequality(39)1rρ1r+1sρ2s≥ρ1ρ2for all ρ1,ρ2≥0 and r,s>1 with r−1+s−1=1.
Substitutingρ1=QxUy and ρ2=QyUxx,y>0 into (39) gives
(40)1rQrxUry+1sQsyUsx≥QxUyQyUx.
Multiplying both sides of (40) by
(41)ς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βand then performing the qi,ωi-integration with respect to x and y over 0,ς lead to the conclusion that
(42)1r∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βQrxUrydq1,ω1xdq2,ω2y+1s∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βQsyUsxdq1,ω1xdq2,ω2y≥∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βQxUyQyUxdq1,ω1xdq2,ω2y.
Consequently, we have(43)1r0+Iq1,ω1αQrς0+Iq2,ω2βUrς+1s0+Iq1,ω1αUsς0+Iq2,ω2βQsς≥0+Iq1,ω1αQUς0+Iq2,ω2βQUς,which implies (A1). The remaining inequalities can be derived by adopting the similar argument and accompanying the selection of parameters in Young inequality as follows:
(A2)ρ1=Qx/Qy,ρ2=Ux/Uy,
(A3)ρ1=QxU2/ry,ρ2=Q2/sxUy,
(A4)ρ1=Q2/rxQy,ρ2=U2/sxUy.Theorem 14.
Letr,s>0 with r+s=1, qi∈0,1 and ωi≥0 for i=1,2, and Q and U be the positive qi,ωi-integral functions defined on 0,∞. Then, the inequalities.
(A5)r0+Iq1,ω1αQς0+Iq2,ω2βUς+s0+Iq1,ω1αUς0+Iq2,ω2βQς≥0+Iq1,ω1αQrUsς0+Iq2,ω2βQsUrς
(A6)r0+Iq1,ω1αQς0+Iq2,ω2βUς+s0+Iq1,ω1αUς0+Iq2,ω2βQς≥0+Iq1,ω1αQrUsς0+Iq2,ω2βQr−1Us−1ς
(A7)r0+Iq1,ω1αQς0+Iq2,ω2βU2/rς+s0+Iq1,ω1αUς0+Iq2,ω2βQ2/sς≥0+Iq1,ω1αQrUsς0+Iq2,ω2βQ2U2ς
(A8)r0+Iq1,ω1αQ2/rς0+Iq2,ω2βQς+s0+Iq1,ω1αU2/sς0+Iq2,ω2βUς≥0+Iq1,ω1αQ2U2ς0+Iq2,ω2βQsUrς
hold for allς>0.Proof.
Considering the well-known weightedAM−GM inequality
(44)rρ1+sρ2≥ρ1rρ2sρ1,ρ2≥0,r,s>0,r+s=1.
Substitutingρ1=QxUy and ρ2=QyUxx,y>0 yields
(45)rQxUy+sQyUx≥QrxUryQsyUsx.
Conducting product on both sides of (45) by
(46)ς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2β,and then performing the qi,ωi-integration with respect to x and y over 0,ς, we obtain
(47)r∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βQxUydq1,ω1xdq2,ω2y+s∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βQyUxdq1,ω1xdq2,ω2y≥∫0ςς−σ1Ψq1xσ1α−1ς−σ2Ψq2yσ2β−1Γq1αΓq2βQrxUryQsyUsxdq1,ω1xdq2,ω2y.
Consequently, we have(48)r0+Iq1,ω1αQς0+Iq2,ω2βUς+s0+Iq1,ω1αUς0+Iq2,ω2βQς≥0+Iq1,ω1αQrUsς0+Iq2,ω2βQsUrς,which implies (A5). The rest of variants can be derived by adopting the similar strategy and accompanying the selection of parameters in AM−GM inequality as follows:
(A6)ρ1=Qx/Qy,ρ2=Ux/Uy,
(A7)ρ1=QxU2/ry,ρ2=Q2/sxUy,
(A8)ρ1=Q2/rxQy,
ρ2=U2/sxUy.Theorem 15.
Letq∈0,1, ω≥0, Q and U be the positive q,ω-integrable functions defined on 0,∞, and
(49)e=min0≤x≤ςQςUς,(50)E=max0≤x≤ςQςUς.
Then, the inequalities
(A9) 0≤0+Iq,ωαQ2ς0+Iq,ωαU2ς≤e+E2/4eE0+Iq,ωαQU2ς
(A10) 0≤0+Iq,ωαQ2ς0+Iq,ωαU2ς−0+Iq,ωαQUς≤E−e2/2eE0+Iq,ωαQUς
(A11) 0≤0+Iq,ωαQ2ς0+Iq,ωαU2ς−0+Iq,ωαQU2ς≤E−e2/4eE0+Iq,ωαQU2ς
hold for allς>0.Proof.
From (49) and (50), we clearly see that
(51)QxUx−eE−QxUxU2x≥00≤x<ς.
Conducting product on both sides of (51) by ς−σΨqxσα−1/Γqα and then performing the q,ω-integration with respect to x over 0,ς yield
(52)0+Iq,ωαQ2ς+eE0+Iq,ωαU2ς≤e+E0+Iq,ωαQUς.
It follows fromeE>0 and
(53)0+Iq,ωαQ2ς−eE0+Iq,ωαU2ς≥0that
(54)20+Iq,ωαQ2ςeE0+Iq,ωαU2ς≤0+Iq,ωαQ2ς+eE0+Iq,ωαU2ς.
From (52) and (54), we conclude that
(55)0+Iq,ωαQ2ς0+Iq,ωαU2ς≤e+E24eE0+Iq,ωαQU2ς,which implies (A9). By making few changes in (A9), we can get (A10) and (A11).Theorem 16.
Letq∈0,1, ω≥0, and Q and U be the positive q,ω-integrable functions defined on 0,∞ such that
(56)0<δ1≤Qx≤δ2<∞,(57)0<Δ1≤Ux≤Δ2<∞.
Then, the inequalities
(A12)0≤0+Iq,ωαQ2ς0+Iq,ωαU2ς≤δ1Δ1+δ2Δ22/4δ1Δ1δ2Δ20+Iq,ωαQU2ς
(A13)0≤0+Iq,ωαQ2ς0+Iq,ωαU2ς−0+Iq,ωαQUς≤δ2Δ2−δ1Δ12/2δ1Δ1δ2Δ20+Iq,ωαQUς
(A14)0≤0+Iq,ωαQ2ς0+Iq,ωαU2ς−0+Iq,ωαQU2ς≤δ2Δ2−δ1Δ12/4δ1Δ1δ2Δ20+Iq,ωαQU2ς(A14)
hold for allς>0.Proof.
It follows from (56) and (57) that
(58)δ1Δ2≤QxUx≤δ2Δ1.
The proof can be derived by following Theorem15.Theorem 17.
Letr,s>0 with r+s=1, q∈0,1, ω≥0, P,Q and U be the positive q,ω-integrable functions defined on 0,∞ such that (56) and (57) hold. Then, the inequalities
(A15)0+Iq,ωαPQsς0+Iq,ωαP/Qrς≤rδ1+sδ2/δ1δ2r0+Iq,ωαPς
(A16)0+Iq,ωαPQsς0+Iq,ωαPUrς≤rδ1Δ1+sδ2Δ2/δ1δ2rΔ1Δ2s0+Iq,ωαPQUsς(A16)
hold for allς>0.Proof.
It follows fromsQx−rδ1Qx−δ2≤0 that
(59)sQ2x−rδ1+sδ2Qx+rδ1δ2≤0.
Multiplying both sides of (59) by Px/Qx, we have
(60)sPxQx+rδ1δ2PxQx≤rδ1+sδ2Px.
Taking into account the well-knownAM−GM inequality for (60), we get
(61)∫0ςς−σΨqxσα−1ΓqαPxQxdq,ωxs∫0ςς−σΨqxσα−1ΓqαPxQxdq,ωxr=1δ1δ2r∫0ςς−σΨqxσα−1ΓqαPxQxdq,ωxsδ1δ2∫0ςς−σΨqxσα−1ΓqαPxQxdq,ωxr≤1δ1δ2rsΓqα∫0ςς−σΨqxσα−1PxQxdq,ωx+rδ1δ2Γqα∫0ςς−σΨqxσα−1PxQxdq,ωx≤rδ1+sδ2δ1δ2r1Γqα∫0ςς−σΨqxσα−1Pxdq,ωx,which implies (A15).
Replacing, respectively,P and Q by PQU and Q/U in (61) and (56), we attain the required inequality (A16):
(62)∫0ςς−σΨqxσα−1ΓqαPxQxdq,ωxs∫0ςς−σΨqxσα−1ΓqαPxUxdq,ωxr≤rδ1Δ1+sδ2Δ2δ1δ2rΔ1Δ2s∫0ςς−σΨqxσα−1ΓqαPxQxUxdq,ωxs.Theorem 18.
Letq∈0,1, ω≥0, and P,Q and U be the q,ω-integrable functions defined on 0,∞ such that Px≥0 and (56) holds. Then, for all ς>0, we have
(A17)δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPQUς≤∣δ1Δ1+δ2Δ2∣0+Iq,ωαPQ2ς+0+Iq,ωαPU2ς(A17)
(A18)δ1δ2/Δ1Δ20+Iq,ωαPU2ς+Δ1Δ2/δ1δ20+Iq,ωαPQ2ς≤δ1Δ1/δ2Δ2+δ2Δ2/δ1Δ10+Iq,ωαPQUς(A18)
(A19)0+Iq,ωαPU2ς0+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ2/2δ1Δ1δ2Δ220+Iq,ωαPQUς(A19)Proof.
We clearly see that(63)Pxδ2Ux−Δ1QxΔ2Qx−δ1Ux≥0x≥0.
Inequality (63) can be written as
(64)δ1δ2PxU2x+Δ1Δ2PxQ2x≤δ1Δ1+δ2Δ2PxQxUx.
Conducting product on both sides of (64) by ς−σΨqxσα−1/Γqα and then performing the q,ω-integration with respect to x over 0,ς yield
(65)δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPQUς.
Also, by Cauchy inequality, we get(66)δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPQUς≤∣δ1Δ1+δ2Δ2∣0+Iq,ωαPQ2ς+0+Iq,ωαPU2ς.
Multiplying both sides of the inequality (66) by 1/δ1δ2Δ1Δ2, we get A17.
Alternately, it follows fromδ1δ2Δ1Δ2>0 and
(67)δ1δ20+Iq,ωαPU2ς−Δ1Δ20+Iq,ωαPQ2ς2≥0that
(68)2δ1δ20+Iq,ωαPU2ςΔ1Δ20+Iq,ωαPQ2ς≤δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς.
FromA17 and (68), we get
(69)4δ1δ2Δ1Δ20+Iq,ωαPU2ς0+Iq,ωαPQ2ς≤δ1Δ2+δ2Δ10+Iq,ωαPUQ2ς,which conclude A19. Similarly, we can prove the inequality A18.Theorem 19.
Letq∈0,1, ω≥0, and P,Q and U be the q,ω-integrable functions defined on 0,∞ such that Px≥0 and
(70)δ2Ux−Δ1QyΔ2Qy−δ1Ux≥0for all x,y>0. Then, for all ς>0, we have
(71)δ1δ10+Iq,ωαPς0+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPς0+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPQς0+Iq,ωαPUς,(72)δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPς0+Iq,ωαPQUς(73)δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPQς0+Iq,ωαPQUς.Proof.
Under the given assumption, we have(74)PxPyδ2Ux−Δ1QyΔ2Qy−δ1Ux≥0,which implies that
(75)δ1δ1PxPyU2x+Δ1Δ2PxPyQ2y≤δ1Δ1PxPyQyUx+δ2Δ2PxPyQyUx.
Conducting product on both sides of (75) by
(76)ς−σΨqxσα−1ς−σΨqyσα−1Γq2α,and then performing the q,ω-integration with respect to x and y over 0,ς, we get (71).
From (65), (71), and the Cauchy inequality
(77)0+Iq,ωαPQ2ς≤0+Iq,ωαPς0+Iq,ωαPQ2ς,0+Iq,ωαPU2ς≤0+Iq,ωαPς0+Iq,ωαPU2ς,we get
(78)δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1δ20+Iq,ωαPς0+Iq,ωαPQ2ς+Δ1Δ20+Iq,ωαPς0+Iq,ωαPU2ς≤δ1Δ1+δ2Δ20+Iq,ωαPς0+Iq,ωαPQUς,δ1δ20+Iq,ωαPU2ς+Δ1Δ20+Iq,ωαPQ2ς≤δ1Δ1+δ2Δ20+Iq,ωαPQς0+Iq,ωαPUς,which completes the proof of (72) and (73).


## 4. Conclusion

We have discovered several generalizations for the generalized Čebyšev type inequality via quantum fractional Hahn’s integral operator by using the quantum shift operatorσΨqς=qς+1−qσς∈l1,l2,σ=l1+ω/1−q,0<q<1,ω≥0, provided some associated variants to show the efficiency of quantum Hahn’s integral operator, and compared our obtained results and proposed technique with the previously known results and existing technique. The outcome shows that the proposed plans are extremely important and computationally appealing to deal with several sorts of differential equations. As a future research course of this paper, the new techniques obtained in the present paper can be prolonged to attain analytical solutions of quantum mechanics introduced in different works distributed currently connected with high-dimensional fractional equations.



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*Source: 8262860-2020-07-14.xml*