On New Unified Bounds for a Family of Functions via Fractional q -Calculus Theory

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Introduction
Fractional differential equations were executed to demonstrate tremendous innovations for different issues in the physical sciences [1][2][3][4][5][6][7][8][9][10][11][12][13][14][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][21][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 of q-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 fractional q-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 timefractional 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.
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-fractional q-calculus operator. Also, we present the results concerning for a class of family of nðn ∈ ℕÞ continuous positive decreasing functions on ½ς 1 , ς 2 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.

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 scale T t 0 is defined by If there is no confusion concerning t 0 , we will denote T t 0 by T .
Definition 2. The q-factorial function is defined in the following way Definition 3. The q-derivative of the q-factorial function with respect to ζ is defined by and the q-derivative of the q-factorial function with respect to s is defined by Definition 4. The q-exponential function is defined as Definition 5. The q-Gamma function is defined by Remark 6. We observe that and ½β q = 1 − q β /1 − q: 2 Journal of Function Spaces Definition 7. The fractional q-integral is defined as Remark 8. Let ΨðζÞ = 1. Then Definition 7 gives

Main Results
Now we demonstrate the left fractional q integral operator on an arbitrary time scale T to derive the generalization of some classical inequalities.
RðβÞ > 0, and Ψ be a continuous positive decreasing function defined on T t 0 .

Journal of Function Spaces
Now, we demonstrate the fractional q-integral to derive some inequalities for a class of n-decreasing positive functions.

Conclusion
In this note, we have derived certain variants by using the time-fractional q-calculus operator related to a class of n positive continuous, and decreasing functions on the interval ½ς 1 , ς 2 are elaborated. In [66], Liu et al. investigated thoughtprovoking integral inequalities for continuous functions on ½ς 1 , ς 2 . 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.

Data Availability
No data were used to support this study.