On Discrete Fractional Integral Inequalities for a Class of Functions

Department of Mathematics, Government College University, Faisalabad, Pakistan Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan Department of Mathematics, Government College Women University, Faisalabad, Pakistan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China


Introduction
Recently, FC and its concrete utilities have increased a great deal of significance in light of the fact that fractional operators have become a useful asset with more precise and victories in demonstrating a few complex marvels in various apparently differing and broad fields of science and numerous areas, for example, fluid flow, optics, chaos, image processing, virology, and financial economics [1][2][3]. A few decades ago, the fractional differential equations and dynamical frameworks have been substantiated as being important devices in displaying several phenomena in various branches of pure and applied sciences. ey attract incredible utility in research-oriented areas, for example, fluid mechanics, thermodynamics, vibration, groundwater flow with memory, and image processing (see the fundamental monograph and the fascinating paper [4][5][6]). An assortment of consequences that facilitated in emergence of the theory of discrete FC is presented in [7][8][9].
Atici and Eloe prudently provoked the enthusiasm for the theory of fractional difference operators [10]. Numerous researchers characterized fractional difference with various sorts of kernel having a discrete force law with discrete exponential and generalized Mittag-Leffler functions [11] and discrete exponential and Mittag-Leffler functions on generalized ZZ time scale [12] and kernel depending on the consequence of both power-law and exponential functions [13]. It is notable that the discretization cycle is one of the most requested devices for scientists who are intrigued in reproduction and computational examination. In arrangement with the reality that not all discrete operators acquire similar features as of the continuous ones, the exploration of the discrete alignment of FC has become squeezing prerequisites (see [4,7,[14][15][16][17][18][19][20][21]). Numerous authors devoted their attention to searching novel operators of arbitrary order. Definitely, the assortment of such tools provides analysts more opportunities to apply them to various models (see [22,23]). In [24,25], the authors presented and explored local-type derivatives and integrals with arbitrary order and without memory, called conformable derivatives and integrals, with the disadvantage that the function itself is not acquired in the restricting situation when the order of the derivative, α, approaches 0. Afterward, the researchers [26] proposed proportional-type derivatives that legitimately combine the function itself and its derivative as the parameter ρ tends to 0 and 1, respectively. We have comprehended that these local-type derivatives and integrals are valuable to produce new sorts of operators with memory through various kinds of kernels [27,28]. In recent years, the study which is promptly increasing the extent of incredible intrigue both from the hypothetical and applied perspective is the study of fractional Z-discrete calculus. Regarding utilities in various fields of mathematics, we refer to [29,30]. Additionally, we notice that Z-discrete FC is extremely significant in applied analysis, for example, financial mathematics, banking, and material sciences. Finally, the calculus emerging from the definition of discrete proportional fractional sums has become appealing to many authors and now it is a matter of strong research, in various directions: existence and uniqueness of solutions to discrete fractional equations modelling tumor growths [22], continuity of solutions with respect to initial conditions and also to the order α of the derivative [31], and the Euler-Lagrange equation and Legendre's optimality condition for the calculus of variations problems [32]. Discrete fractional variants have consistently been of excessive prominence for the advancement of remarkable momentous approaches midst investigators and accumulate productive purposeful demonstrations in several areas of science and technology. Certain distinctive examples are Ostrowski, Lyenger [33], Grownwall [34],Čebyšev [35], Hermite-Hadamard [36], and henceforth. Several researchers have devoted their concentrations for exploring the novel versions of fractional integral inequalities for a family of n(n ∈ N)-positive increasing functions. In this flow, we observe that some variants have been concerned with the qualitative investigation of solutions of discrete fractional difference equations arising in the theory of discrete FC. For current consequences on this trend, we refer the readers to [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Inspired by the discretization process, the principal purpose of this investigation is to present a new research area for mathematicians in the frame of Z-discrete fractional and discrete Z-proportional fractional sums. Taking into account the concepts of our noteworthy discrete fractional operators, we demonstrate the generalizations associating Z-analogue for a class of family of continuous positive decreasing functions on N a 1 ,Z by the proposed discrete Z-proportional fractional sums. Additionally, it is accentuated that mingling these two methodologies, discrete FC and integral inequalities, might be the supreme proficient approach of combining inequities into both times and fractional operator theory. Finally, our findings can deliver a prevailing instrument to illustrate the dynamics of discrete complex frameworks all the more profoundly.

Preliminaries
Particular imperative characterizations and deductions in discrete FC are mentioned as follows [10]. For the accessibility, for a 1 , b 1 ∈ R and Z > 0, we symbolize N a 1 ,Z � a 1 , a 1 + Z, a 1 + 2Z, . . . , Definition 1 (see [10]). e consequent equalities are valid: (i) Let η be a natural number; then, the η rising factorial of t is expressed as (ii) For any real number, the β rising function becomes Additionally, we have Hence, t β is increasing on N 0 . (iii) Let Γ denote the usual special gamma function and recall the notation that is known as the falling factorial power: .
Also, we can achieve the backward and forward differ-  For arbitrary t, β ∈ R and Z > 0, and the nabla Z-factorial function is defined by Specifically, for Z � 1, we obtain identity (2). A forthright confirmation prompts Definition 3 (see [54]). (Nabla Z-fractional sums). For backward jump operator . . , ↦R be the nabla left Z-fractional sum of order β > 0, stated as follows: Let a function Υ: . . , ↦R be the nabla right Z-fractional sum of order β > 0(ending at b 1 ), stated as follows: Definition 4 (see [54]). (Nabla Z-Riemann-Liouville fractional differences). e nabla left-and right-sided difference of order β > 0 (starting from a 1 ) has the form Now, we demonstrate the proportional fractional sum with memory depending on the proportional difference, which is mainly due to Abdeljawad et al. [54].

Remark 1.
In view of Definition 5, if we choose ρ↦1, then we acquire the left and right nabla Z-fractional sums of order β, respectively.

New Estimations within Proportional Fractional Sums
is segment is dedicated to giving our fundamental consequences of this paper. We define new forms for a class of a family of n(n ∈ N) continuous positive decreasing functions on N a 1 ,Z in the settings of discrete proportional fractional operator.

Certain Bounds for a Sequence of Decreasing Functions within Proportional Fractional Sums
Now, we demonstrate the discrete Z-proportional fractional sum to derive some inequalities for a class of n-decreasing positive functions.