# On New Unified Bounds for a Family of Functions via Fractionalq-Calculus Theory

Authors: Li Xu; Yu-Ming Chu; Saima Rashid; A. A. El-Deeb; Kottakkaran Sooppy Nisar
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The present article deals with the new estimates inq-calculus and fractional q-calculus on a time scale Tt0=0∪t:t=t0qn,nisanonnegativeinteger, where t0∈ℝ and 0<q<1. The role of fractional time scale q-calculus can be found as one of the prominent techniques to generate some variants for a class of positive functions nn∈ℕ.Finally, our work will provide foundation and motivation for further investigation on time-fractional q-calculus systems that have an intriguing application in quantum theory and special relativity theory.


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## Body

## 1. Introduction

Fractional differential equations were executed to demonstrate tremendous innovations for different issues in the physical sciences [1–15]. Since most frameworks involve recollections, the scientists are agreeing with the nonlocality of the fractional operators make it progressively functional in demonstrating the classical derivatives. Recently, nonlocal fractional derivatives without the singular kernel have been exhibited and contemplated [16, 17]. However, there are no solid numerical defenses of the new sorts of fractional derivatives; their applications were demonstrated by numerous analysts [18, 19]. Furthermore, presently we have the utilization of fractional calculus in fields like science, material science, and building and among different zones. It is a stunner of the fractional calculus that we have such a large number of valuable meanings of differential and integral operators, for instance, Saigo, conformable, Riemann-Liouville, Katugampola, Hadamard, Erdélyi-Kober, Liouville, local, and Weyl types. These operators are having their significance and applications in picture handling, science, hydrodynamics, and viscoelastic. For a detailed depiction of the origination of fractional calculus, advancement, and applications, we refer the interested readers to the notable books and research articles [20–22].Hilger [23] began the theories of time scales in his doctoral dissertation and combined discrete and continuous analysis [24, 25]. At that time, this theory has received a lot of attention. In the book written by Bohner and Peterson [26] on the issues of time scale, a brief summary is given and several time calculations are performed. Over the past decade, many analysts working in special applications have proved a reasonable number of dynamic inequalities on a time scale [27, 28]. Several researchers have created various results relating to fractional calculus on time scales to obtain the corresponding dynamic variants [29].In the eighteenth century (1707–1783), Euler initiated calculus with no limits refer to as quantum calculus. Jackson began a deliberate investigation ofq-calculus and presented the q-definite integrals. Additionally, he was the first to create q-calculus in an efficient manner. Few selected branches of pure and applied mathematics, such as combinatorics, Gauss hypergeometric functions, orthogonal polynomials, dynamic, and quantum theory, have been enhanced by the exploration work of different researchers.Motivated, by what we mentioned above, we extend the idea of fractionalq-calculus type operators with a time scale to arbitrary positive order, provide several bounds for a family of n∈ℕ, and finally prove several variants for time-fractional q-calculus theory. These new results have utilities in the monotonicity for this nabla continuous fractional operator with singular and nonsingular kernel and compare them to the discrete classical ones. The time-fractional q-calculus under consideration in this paper have kernels different from classical nabla fractional differences with kernels depending on the rising factorial powers, and we believe that they bring new kernels with new memories, which may be of different interest for applications. The idea is quite new and seems to have opened new doors of investigation towards various scientific fields of research including engineering, fluid dynamics, meteorology, analysis, and aerodynamics.Inequalities have wild applications in pure and applied mathematics [30–33]. 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 functions inequalities [46], fractional integral inequality [47–51], complete and generalized elliptic integrals inequalities [52–57], generalized convex function inequality [58–60], and mean values inequality [61–63] have been discovered by many researchers.Variants regarding fractional integral operators are the use of noteworthy significant strategies amongst researchers and accumulate fertile functional applications in various areas of science [64, 65]. We state some of them, that is, the variants of Minkowski, Hardy, Opial, Hermite-Hadamard, Grüss, Lyenger, Ostrowski, C˘ ebyšev, and Pólya-Szegö, and others. Such applications of fractional integral operators compelled us to show the generalization by using a family of n positive functions involving time-fractional q-calculus integrals operators.Owing to the above phenomena, the key aim of this research is to demonstrate the notations and primary definitions of our noteworthy time-fractionalq-calculus operator. Also, we present the results concerning for a class of family of nn∈ℕ continuous positive decreasing functions on ς1,ς2 by employing a time-fractional q-calculus operator. Finally, it is emphasized that combining these two approaches, q-fractional calculus and time scale analysis, could be the most efficient way of incorporating inequalities into both times and q-components for quantum theory and special relativity theory.


## 2. Preliminaries

Let us recall some necessary definitions and preliminary results that are used for further discussion. For more details, we may refer to [33].Definition 1 (See [33]).
The particular time scaleTt0 is defined by
(1)Tt0=t:t=t0qn,nisanonnegativeinteger∪0,0<q<1.
If there is no confusion concerningt0, we will denote Tt0 by T.Definition 2.
Theq-factorial function is defined in the following way
(2)ζ−φn_=ζ−φζ−qφζ−qφ⋯ζ−qnφ,n∈ℕ,ζ−φn_=ζn∏κ=0∞1−φ/ζqκ1−φ/ζqn+κ,n∉ℕ.Definition 3.
Theq-derivative of the q-factorial function with respect toζ is defined by
(3)∇qζ−φn_=1−qn1−qζ−φn−1¯,and the q-derivative of the q-factorial function with respect to s is defined by
(4)∇qζ−φn_=−1−qn1−qζ−qφn−1¯.Definition 4.
Theq-exponential function is defined as
(5)eqζ=∏κ=0∞1−qκζ,eq0=1.Definition 5.
Theq-Gamma function is defined by
(6)Γqβ=11−q∫01ζ1−qβ−1eqqζ∇ζ,β∈ℝ+.Remark 6.
We observe that(7)Γqβ+1=βqΓqβ,β∈ℝ+,and βq=1−qβ/1−q.Definition 7.
The fractionalq-integral is defined as
(8)∇q−βΨζ=1Γqβ∫ς1ζζ−qφβ−1¯Ψφ∇φ.Remark 8.
LetΨζ=1. Then Definition 7 gives
(9)∇q−β1=1Γqβq−1qβ−1ζβ¯=1Γqβ+1ζβ¯.


## 3. Main Results

Now we demonstrate the left fractionalq integral operator on an arbitrary time scale T to derive the generalization of some classical inequalities.Theorem 9.
Letα>0, η≥δ>0, β∈ℂ with Rβ>0, and Ψ be a continuous positive decreasing function defined on Tt0. Then, one has
(10)∇ς1+,q−βΨηζ∇ς1+,q−βΨδζ≥∇ς1+,q−βζ−ς1αΨηζ∇ς1+,q−βζ−ς1αΨδζ.Proof.
Using the hypothesis given in Theorem9, we have
(11)ω−ς1α−φ−ς1αΨη−δφ−Ψη−δω≥0,where α>0,η≥δ>0, and φ,ω∈ς1,ζ.It follows from (11) that
(12)ω−ς1αΨη−δφ−φ−ς1αΨη−δω−ω−ς1αΨη−δω+φ−ς1αΨη−δφ≥0.Multiplying (12) by 1/Γqβζ−qφβ−1/Ψδφ,φ∈ς1,ζ, we have
(13)1Γqβζ−qφβ−1_ω−ς1αΨη−δφ−φ−ς1αΨη−δω−ω−ς1αΨη−δω+φ−ς1αΨη−δφΨδφ=ω−ς1α1Γqβζ−qφβ−1_ΨδφΨη−δφ−φ−ς1α1Γqβζ−qφβ−1_ΨδφΨη−δω−ω−ς1α1Γqβζ−qφβ−1_ΨδφΨη−δω+φ−ς1α1Γqβζ−qφβ−1_ΨδφΨη−δφ≥0.Integrating on both sides of (13) for φ over ς1,ζ, we have
(14)ω−ς1α∫ς1ζ1Γqβζ−qφβ−1_ΨδφΨη−δφ∇φ−φ−ς1α∫ς1ζ1Γqβζ−qφβ−1_ΨδφΨη−δω∇φ−ω−ς1α∫ς1ζ1Γqβζ−qφβ−1_ΨδφΨη−δω∇φ+φ−ς1α∫ς1ζ1Γqβζ−qφβ−1_ΨδφΨη−δφ∇φ≥0,that is
(15)ω−ς1α∇ς1+,q−βΨηζ+Ψη−δω∇ς1+,q−βζ−ς1αΨδζ−ω−ς1αΨη−δω∇ς1+,q−βΨδζ−ζ−ς1αΨηζ≥0.Multiplying (15) by 1/Γqβζ−qωβ−1/Ψδω,ω∈ς1,ζ, and integrating for ω over ς1,ζ shows
(16)∇ς1+,q−βΨηζ∇ς1+,q−βζ−ς1αΨδζ−∇ς1+,q−βζ−ς1αΨηζΨδζ≥0.Dividing the above inequality by∇ς1+,q−βζ−ς1αΨδζ∇ς1+,q−βΨδζ, we get the desired inequality (10).Theorem 10.
Let
α
>
0
, and
η
≥
δ
>
0, β,λ∈ℂ with Rβ>0 and Rλ>0, and Ψ be a continuous positive decreasing function defined on Tt0. Then the time-fractional q-integral satisfies the inequality
(17)∇ς1+,q−βΨηζ∇ς1+,q−λζ−ς1αΨδζ+∇ς1+,q−λΨηζ∇ς1+,q−βζ−ς1αΨδζ∇ς1+,q−λΨδζ∇ς1+,q−βζ−ς1αΨηζ+∇ς1+,q−βΨδζ∇ς1+,q−λζ−ς1αΨηζ≥1.Proof.
Multiplying both sides of (15) by 1/Γqλζ−qωλ−1/Ψδω,ω∈ς1,ζ and integrating for ω over ς1,ζ shows
(18)∇ς1+,q−βΨηζ∇ς1+,q−λζ−ς1αΨδζ+∇ς1+,q−λΨηζ∇ς1+,q−βζ−ς1αΨδζ−∇ς1+,q−λΨδζ∇ς1+,q−βζ−ς1αΨηζ−∇ς1+,q−βΨδζ∇ς1+,q−λζ−ς1αΨηζ≥0.Dividing (18) by
(19)∇ς1+,q−λΨδζ∇ς1+,q−βζ−ς1αΨηζ−∇ς1+,q−βΨδζ∇ς1+,q−λζ−ς1αΨηζ,we get the desired inequality (17).Theorem 11.
Letα>0, η≥δ>0, β∈ℂ with Rβ>0, Ψ be a continuous positive decreasing function defined on Tt0, and ℏ be a continuous positive increasing function on Tt0. Then the time-fractional q-integral satisfies the inequality
(20)∇ς1+,q−βΨηζ∇ς1+,q−βΨδζℏαζ∇ς1+,q−βΨδζ∇ς1+,q−βΨηζℏαζ≥1.Proof.
Using the hypothesis given in Theorem11, we have
(21)ℏαω−ℏαφΨη−δφ−Ψη−δω≥0,where α>0,η≥δ>0, and φ,ω∈ς1,ζ. From (21), we have
(22)ℏαωΨη−δφ−ℏαφΨη−δω+ℏαωΨη−δω−ℏαφΨη−δφ≥0.Taking product of (22) by 1/Γqβζ−qφβ−1/Ψδφ,φ∈ς1,ζ, we get
(23)1Γqβζ−qφβ−1_ΨδφℏαωΨη−δφ−ℏαφΨη−δω+ℏαωΨη−δω−ℏαφΨη−δφ=ℏαω1Γqβζ−qφβ−1_Ψηφ−ℏαφ1Γqβζ−qφβ−1_Ψη−δωΨδφ+ℏαω1Γqβζ−qφβ−1_Ψη−δωΨδφ−ℏαφ1Γqβζ−qφβ−1_Ψηφ≥0.Integrating (23) for φ over ς1,ζ, we obtain
(24)ℏαωΓqβ∫ς1ζζ−qφβ−1_Ψηφ∇φ−ℏαφΓqβ∫ς1ζζ−qφβ−1_Ψη−δωΨδφ∇φ+ℏαωΓqβ∫ς1ζζ−qφβ−1_Ψη−δωΨδφ∇φ−ℏαφΓqβ∫ς1ζζ−qφβ−1_Ψηφ∇φ≥0.It follows that(25)ℏαω∇ς1+,q−βΨηζ+Ψη−δω∇ς1+,q−βℏαζΨδζ−ℏαωΨη−δω∇ς1+,q−βΨδζ−∇ς1+,q−βℏαζΨδζ≥0.Again, taking the product (15) by 1/Γqβζ−qωβ−1/Ψδω,ω∈ς1,ζ, and integrating for ω over ς1,ζ gives
(26)∇ς1+,q−βΨηζ∇ς1+,q−βΨδζℏαζ−∇ς1+,q−βΨδζ∇ς1+,q−βΨηζℏαζ≥0,which completes the proof of the desired inequality (20).Theorem 12.
Letα>0, η≥δ>0, Rλ,Rβ>0 with Rλ,Rβ>0,Ψ be a continuous positive decreasing function defined on Tt0 and ℏ be a continuous positive increasing function on Tt0. Then, one has
(27)∇ς1+,q−βΨηζ∇ς1+,q−λℏαζΨδζ+∇ς1+,q−λΨηζ∇ς1+,q−βℏθζΨδζ∇ς1+,q−βℏαζΨηζ∇ς1+,q−λΨδζ+∇ς1+,q−λℏαζΨηζ∇ς1+,q−λΨδζ≥1.Proof.
Multiplying both sides of (25) by 1/Γqβζ−qωβ−1/Ψδω,ω∈ς1,ζ, and integrating for ω over ς1,ζ leads to the conclusion that
(28)∇ς1+,q−βΨηζ∇ς1+,q−λℏαζΨδζ+∇ς1+,q−λΨηζ∇ς1+,q−βℏθζΨδζ−∇ς1+,q−βℏαζΨηζ∇ς1+,q−λΨδζ−∇ς1+,q−λℏαζΨηζ∇ς1+,q−λΨδζ≥0.It follows that(29)∇ς1+,q−βΨηζ∇ς1+,q−λℏαζΨδζ+∇ς1+,q−λΨηζ∇ς1+,q−βℏθζΨδζ≥∇ς1+,q−βℏαζΨηζ∇ς1+,q−λΨδζ+∇ς1+,q−λℏαζΨηζ∇ς1+,q−λΨδζ.Dividing above inequality by(30)∇ς1+,q−βℏαζΨηζ∇ς1+,q−λΨδζ+∇ς1+,q−λℏαζΨηζ∇ς1+,q−λΨδζ,we get the desired inequality (27).Now, we demonstrate the fractionalq-integral to derive some inequalities for a class of n-decreasing positive functions.Theorem 13.
Letα>0, η≥δκ>0 for any fixed κ∈1,2,3,⋯,n, β∈ℂ with Rβ>0, and Ψj,j=1,2,3,⋯,n be a sequence of continuous positive decreasing functions defined on Tt0. Then, the time-fractional q-integral satisfies the inequality
(31)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ≥∇ς1+,q−βζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−βζ−ς1α∏j=1nΨjδjζ.Proof.
SinceΨj,j=1,2,3,⋯,n is a sequence of continuous positive decreasing functions on ς1,ζ, we have
(32)ω−ς1α−φ−ς1αΨκη−δκφ−Ψκη−δκω≥0,for any fixed κ∈1,2,3,⋯,n,α>0, η≥δκ>0 and φ,ω∈ς1,ζ.It follows from (32) that
(33)ω−ς1αΨκη−δκφ+φ−ς1αΨκη−δκω≥ω−ς1αΨκη−δκω+φ−ς1αΨκη−δκφ.Taking the product of (22) by 1/Γqβζ−qφβ−1/∏j=1nΨjδjφ,φ∈ς1,ζ, and integrating for φ over ς1,ζ, we have
(34)1Γqβζ−qφβ−1_ω−ς1αΨκη−δκφ+φ−ς1αΨκη−δκω−ω−ς1αΨκη−δκω−φ−ς1αΨκη−δκφ∏j=1nΨjδjφ=ω−ς1α1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ+φ−ς1α1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω−ω−ς1α1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω−φ−ς1α1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ≥0.Integrating (34) for φ over ς1,ζ, we get
(35)ω−ς1α1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ∇φ+φ−ς1α1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω∇φ−ω−ς1α1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω∇φ−φ−ς1α1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ∇φ≥0.It follows from (35) that
(36)ω−ς1α∇ς1+,q−β∏j≠κnΨjδjΨκηζ+Ψκη−δκω∇ς1+,q−βζ−ς1α∏j=1nΨjδjζ≥ω−ς1αΨκη−δκω∇ς1+,q−β∏j=1nΨjδjζ+∇ς1+,q−βζ−ς1α∏j≠κnΨjδjΨκηζ.Again, taking the product of (36) by 1/Γqβζ−qωβ−1/∏j=1nΨjδjω,ω∈ς1,ζ, and integrating for ω over ς1,ζ, we obtain
(37)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−βζ−ς1α∏j=1nΨjδjζ≥∇ς1+,q−βζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ,which gives the desired inequality (31).Theorem 14.
Letα>0, η≥δκ>0 for any fixed κ∈1,2,3,⋯,n, β,λ∈ℂ with Rβ>0,Rλ>0, and Ψj,j=1,2,3,⋯,n be a sequence of continuous positive decreasing functions defined on Tt0. Then, we have the inequality
(38)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−λζ−ς1α∏j=1nΨjδjζ+∇ς1+,q−λ∏j≠κnΨjδjΨκηζ∇ς1+,q−βζ−ς1α∏j=1nΨjδjζ/∇ς1+,q−βζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−λ∏j=1nΨjδjζ+∇ς1+,q−λζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ≥1.Proof.
Taking product on both sides of (36) by 1/Γqλζ−qθλ−1/∏j=1nΨjδjω,ω∈ς1,ζ, and integrating for ω over ς1,ζ, we get
(39)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−λζ−ς1α∏j=1nΨjδjζ+∇ς1+,q−λ∏j≠κnΨjδjΨκηζ∇ς1+,q−βζ−ς1α∏j=1nΨjδjζ≥∇ς1+,q−βζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−λ∏j=1nΨjδjζ+∇ς1+,q−λζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ.Dividing the above inequality by(40)∇ς1+,q−βζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−λ∏j=1nΨjδjζ+∇ς1+,q−λζ−ς1α∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ,gives the desired inequality (38).Theorem 15.
Letα>0, η≥δκ>0 for any fixed κ∈1,2,3,⋯,n, β∈ℂ with Rβ>0, and ℏ and Ψjj=1,2,3,⋯,n be the continuous positive decreasing functions defined on Tt0. Then, the time-fractional q-integral satisfies the inequality
(41)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−βℏαζ∏j=1nΨjδjζ∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ≥1.Proof.
It follows from the given hypothesis that(42)ℏαω−ℏαφΨκη−δκφ−Ψκη−δκω≥0,for any fixed κ∈1,2,3,⋯,n,α>0,η≥δκ>0, and φ,ω∈ς1,ζ.
Inequality (42) leads to
(43)ℏαωΨκη−δκφ+ℏαφΨκη−δκω−ℏαωΨκη−δκω−ℏαφΨκη−δκφ≥0.
Taking the product on both sides of (43) by 1/Γqβζ−qφβ−1/∏j=1nΨjδjφ,φ∈ς1,ζ, and integrating for φ over ς1,ζ, we obtain
(44)ℏαω1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ+ℏαφ1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω−ℏαω1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω−ℏαφ1Γqβζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ≥0.
Integrating (44) for φ over ς1,ζ, we have
(45)ℏαω1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ∇φ+ℏαφ1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω∇φ−ℏαω1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκω∇φ−ℏαφ1Γqβ∫ς1ζζ−qφβ−1_∏j=1nΨjδjφΨκη−δκφ∇φ≥0.
From (43), we clearly see that
(46)ℏαω∇ς1+,q−β∏j≠κnΨjδjΨκηζ+Ψκη−δκζ∇ς1+,q−βℏαζ∏j=1nΨjδjζ−ℏαωΨκη−δκω∇ς1+,q−β∏j=1nΨjδjζ−∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ≥0.
Again, taking the product on both sides of (46) by 1/Γqβζ−qθ∏j=1nΨjδjω,ω∈ς1,ζ, and integrating for ω over ς1,ζ, we have
(47)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−βℏαζ∏j=1nΨjδjζ−∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ≥0,which completes the proof of the desired inequality (41).Theorem 16.
Letα>0,η≥δκ>0 for any fixed κ∈1,2,3,⋯,n, β,λ∈ℂ, with Rβ>0,Rλ>0, Ψj,j=1,2,3,⋯,n be a sequence of continuous positive decreasing functions defined on Tt0 and ℏ be a continuous positive increasing functions defined on Tt0. Then
(48)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−λℏαζ∏j=1nΨjδjζ+∇ς1+,q−βℏαζ∏j=1nΨjδjζ∇ς1+,q−λ∏j≠κnΨjδjΨκηζ/∇ς1+,q−λ∏j=1nΨjδjζ∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ+∇ς1+,q−λℏαζ∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ≥1.Proof.
Multiplying both sides of (46) by 1/Γqλζ−qωλ−1/∏j=1nΨjδjω,ω∈ς1,ζ, and integrating for ω over ς1,ζ, we have
(49)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−λℏαζ∏j=1nΨjδjζ+∇ς1+,q−βℏαζ∏j=1nΨjδjζ∇ς1+,q−λ∏j≠κnΨjδjΨκηζ−∇ς1+,q−λ∏j=1nΨjδjζ∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ−∇ς1+,q−λℏαζ∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ≥0.
It follows that(50)∇ς1+,q−β∏j≠κnΨjδjΨκηζ∇ς1+,q−λℏαζ∏j=1nΨjδjζ+∇ς1+,q−βℏαζ∏j=1nΨjδjζ∇ς1+,q−λ∏j≠κnΨjδjΨκηζ≥∇ς1+,q−λ∏j=1nΨjδjζ∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ+∇ς1+,q−λℏαζ∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ.
Dividing both sides of the above inequality by(51)∇ς1+,q−λ∏j=1nΨjδjζ∇ς1+,q−βℏαζ∏j≠κnΨjδjΨκηζ+∇ς1+,q−λℏαζ∏j≠κnΨjδjΨκηζ∇ς1+,q−β∏j=1nΨjδjζ,gives desired inequality (48).


## 4. Conclusion

In this note, we have derived certain variants by using the time-fractionalq-calculus operator related to a class of n positive continuous, and decreasing functions on the interval ς1,ς2 are elaborated. In [66], Liu et al. investigated thought-provoking integral inequalities for continuous functions on ς1,ς2. Recently, Dahmani [67] has presented the more generalizations of the work of [66] by utilizing the Riemann-Liouville fractional integral operators. If we take into account T=ℝ and q=1, then our findings are the special cases of the results proposed by Dahmani [67]. The established relationship highlighted the importance of selecting appropriate combinations and validated q-fractional time scale approaches for special relativity theory and quantum mechanics. From the existence and uniqueness viewpoint, it is found that the q-fractional order controls potentially provide the tools to better represent measured that cannot be fit to the classical model.



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*Source: 4984612-2020-07-16.xml*