# On Discrete Fractional Integral Inequalities for a Class of Functions

Authors: Saima Rashid; Hijaz Ahmad; Aasma Khalid; Yu-Ming Chu
Journal: Complexity
 (2020)

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## Abstract

Discrete fractional calculusℱC is proposed to depict neural systems with memory impacts. This research article aims to investigate the consequences in the frame of the discrete proportional fractional operator. ℏ-discrete exponential functions are assumed in the kernel of the novel generalized fractional sum defined on the time scale ℏℤ. The nabla ℏ-fractional sums are accounted in particular. The governing high discretization of problems is an advanced version of the existing forms that can be transformed into linear and nonlinear difference equations using appropriately adjusted transformations invoking property of observing the new chaotic behaviors of the logistic map. Based on the theory of discrete fractional calculus, explicit bounds for a class of positive functions nn∈ℕ concerned are established. These variants can be utilized as a convenient apparatus in the qualitative analysis of solutions of discrete fractional difference equations. With respect to applications, we can apply the introduced outcomes to explore boundedness, uniqueness, and continuous reliance on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations.


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## Body

## 1. 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 [1–3]. 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. They 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 [4–6]). An assortment of consequences that facilitated in emergence of the theory of discrete ℱC is presented in [7–9]. 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 ℏℤ 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, 14–21]). 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 ℏ-discrete calculus. Regarding utilities in various fields of mathematics, we refer to [29, 30]. Additionally, we notice that ℏ-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 nn∈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 [37–53]. Inspired by the discretization process, the principal purpose of this investigation is to present a new research area for mathematicians in the frame of ℏ-discrete fractional and discrete ℏ-proportional fractional sums. Taking into account the concepts of our noteworthy discrete fractional operators, we demonstrate the generalizations associating ℏ-analogue for a class of family of continuous positive decreasing functions on ℕa1,ℏ by the proposed discrete ℏ-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.


## 2. Preliminaries

Particular imperative characterizations and deductions in discreteℱC are mentioned as follows [10]. For the accessibility, for a1,b1∈ℝ and ℏ>0, we symbolize ℕa1,ℏ=a1,a1+ℏ,a1+2ℏ,…, and ℕb1,ℏ=b1,b1−ℏ,b1−2ℏ,…,.Definition 1.
(see [10]). The consequent equalities are valid:(i)
Letη be a natural number; then, the η rising factorial of t is expressed as(1)tη¯=∏κ=0η−1t+κ,t0¯=1.(ii)
For any real number, theβ rising function becomes(2)tβ¯=Γt+βΓt,such thatt∈C∖…,−2,−1,0,0β¯=0.Additionally, we have(3)∇tβ¯=βtβ−1¯.Hence,tβ¯ is increasing on ℕ0.(iii)
LetΓ denote the usual special gamma function and recall the notation that is known as the falling factorial power:(4)tβ¯=Γt+1Γt+β−1.
Throughout, we assume that ift+β−1∈…,−2,−1,0, then tβ¯=0.
Now,∇˜ℏϒt=ϒt−ϒt−ℏ/ℏ represents the backward difference operator on ℏℤ and Δ˜ℏϒt=ϒt+ℏ−ϒt/ℏ is the backward difference operator on ℏℤ.
Also, we can achieve the backward and forward difference operators∇˜ϒt=ϒt−ϒt−1 and Δ˜ϒt=ϒt+1−ϒt, respectively. Moreover, σℏt=t+ℏ and ρℏt=t−ℏ are the forward and backward jumping operator on time scale ℏℤ, respectively.Definition 2.
For arbitraryt,β∈ℝ and ℏ>0, and the nabla ℏ-factorial function is defined by(5)tℏβ¯=ℏβΓt/ℏ+βΓt/ℏ.
Specifically, forℏ=1, we obtain identity (2).
A forthright confirmation prompts(6)∇˜ℏtℏβ¯=βtℏβ−1¯.Definition 3.
(see [54]). (Nabla ℏ-fractional sums). For backward jump operator ρt=t−ℏ with ℏ>0, let a function ϒ:ℕa1,ℏ=a1,a1+ℏ,a1+2ℏ,…,↦ℝ be the nabla left ℏ-fractional sum of order β>0, stated as follows:(7)∇˜ℏ−βa1ϒt=1Γβ∫a1tt−ρℏϕℏβ−1¯ϒϕ∇˜ℏϕ,=1Γβ∑κ=a1/ℏ+1t/ℏt−ρℏκℏℏβ−1¯ϒκℏℏ,t∈ℕa1+ℏ,ℏ.
Let a functionϒ:ℕb1,ℏ=b1,b1−ℏ,b1−2ℏ,…,↦ℝ be the nabla right ℏ-fractional sum of order β>0ending atb1, stated as follows:(8)∇˜b1−βℏϒt=1Γβ∫tb1ϕ−ρℏtℏβ−1¯ϒϕΔ˜ℏϕ,=1Γβ∑κ=t/ℏb1/ℏ−1κℏ−ρℏtℏβ−1¯ϒκℏℏ.Definition 4.
(see [54]). (Nabla ℏ-Riemann–Liouville fractional differences). The nabla left- and right-sided difference of order β>0 (starting from a1) has the form(9)∇˜ℏβa1ϒt=∇˜ℏn∇˜ℏ−n−βa1ϒt,=∇˜ℏnΓn−β∑κ=a1/ℏ+1t/ℏt−ρℏκℏℏn−β−1¯ϒκℏℏ,t∈ℕa1+ℏ,ℏ,(10)∇˜b1βℏϒt=−1nΔ˜ℏn∇˜b1−n−βℏϒt,=−1nΔ˜ℏnΓn−β∑κ=t/ℏb1/ℏ−1κℏ−ρℏtℏn−β−1¯ϒκℏℏ,t∈b1,ℏ,ℏℕ.
Now, we demonstrate the proportional fractional sum with memory depending on the proportional difference, which is mainly due to Abdeljawad et al. [54].Definition 5.
(see [54]). For ρ∈0,1 and β∈C,ℜβ>0; we state the left (proportional) fractional sum of ϒ by(11)∇˜ℏ−β,ρa1ϒt=1ρβΓβ∫a1te^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ϒϕ∇˜ℏϕ,=1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒκℏℏ,(12)∇˜b1−β,ρℏϒt=1ρβΓβ∫tb1e^νϕ−t+βℏ,0ϕ−ρℏtℏβ−1¯ϒϕΔ˜ℏϕ,=1ρβΓβ∑κ=t/ℏb1/ℏ−1e^νκℏ−t+βℏ,0κℏ−ρℏtℏβ−1¯ϒκℏℏ,where ν=ρ−1/ρ.Remark 1.
In view of Definition5, if we choose ρ↦1, then we acquire the left and right nabla ℏ-fractional sums of order β, respectively.Definition 6.
(see [54]). For ρ∈0,1 and α∈C,ℜα>0; we define the left (proportional) fractional difference of ϒ by(13)∇˜ℏα,ρa1ϒt=∇˜ℏn,ρa1∇˜ℏ−n−α,ρϒt,=∇˜ℏ,tn,ρρn−αΓn−α∑κ=a1/ℏ+1t/ℏe^νt−κℏ+n−αℏ,0t−ρℏκℏℏn−α−1¯ϒκℏℏ,and the right (proportional) fractional difference ending at b1 is stated as follows:(14)∇˜b1β,ρℏϒt=∇˜ℏn,ρ⊖∇˜b1−n−α,ρℏϒt,=∇˜ℏ,tn,ρ⊖ρn−βΓn−β∑κ=t/ℏb1/ℏ−1e^νκℏ−t+n−βℏ,0κℏ−ρℏtℏn−β−1¯ϒκℏℏ,where n=ℜβ+1 and ν=ρ−1/ρ.


## 3. New Estimations within Proportional Fractional Sums

This segment is dedicated to giving our fundamental consequences of this paper. We define new forms for a class of a family ofnn∈ℕ continuous positive decreasing functions on ℕa1,ℏ in the settings of discrete proportional fractional operator.Theorem 1.
Forα>0,σ≥ς>0, and let there be a continuous positive decreasing function ϒ defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(15)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−β,ρa1ϒζt≥∇˜ℏ−β,ρa1t−a1αϒσt∇˜ℏ−β,ρa1t−a1αϒςt,where ν=ρ−1/ρ,ρ∈0,1,β∈C, and ℜβ>0.Proof.
By means of the given assumption in Theorem1, we have(16)θ−a1α−ϕ−a1αϒσ−ςϕ−ϒσ−ςθ≥0,where α>0,σ≥ς>0, and ϕ,θ∈a1,a1+ℏ,a1+2ℏ,…,.
By (16), we have(17)θ−a1αϒσ−ςϕ−ϕ−a1αϒσ−ςθ−θ−a1αϒσ−ςθ+ϕ−a1αϒσ−ςϕ≥0.
Therefore, multiplying (17) by e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯/ρβΓβϒςϕ,t∈ℕa1,ℏβ, and ϕ∈a1,a1+ℏ,a1+2ℏ,…,, we obtain(18)e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβθ−a1αϒσ−δϕ−ϕ−a1αϒσ−ςθ−θ−a1αϒσ−ςθ+ϕ−a1αϒσ−ςϕϒςϕ,=θ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒδϕϒσ−ςϕ,−ϕ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒςϕϒσ−ςθ,−θ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒςϕϒσ−ςθ,+ϕ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒςϕϒσ−ςϕ≥0,and again, summing both sides of (18) for ϕ∈a1,a1+ℏ,a1+2ℏ,…,, we obtain(19)θ−a1α1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒδκℏℏϒσ−ςκℏ,−1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒςκℏℏϒσ−ςθκℏ−a1α,−θ−a1α1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒςκℏℏϒσ−ςθ,+1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒςκℏℏϒσ−ςκℏℏκℏ−a1α≥0.
It follows that(20)θ−a1α∇˜ℏ−β,ρa1ϒσt+Ψσ−ςθ∇˜ℏ−β,ρa1t−a1αϒδt−θ−a1αϒσ−ςθ∇˜ℏ−β,ρa1ϒδt−∇˜ℏ−β,ρa1t−a1αϒσt≥0.
Multiplying (20) by e^νt−θ+βℏ,0t−ρℏθℏβ−1¯/ρβΓβϒςθ and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,…, show(21)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−β,ρa1t−a1αϒςt−∇˜ℏ−β,ρa1t−a1αϒσt∇˜ℏ−β,ρa1ϒςt≥0.
Dividing the above inequality by∇˜ℏ−β,ρa1t−a1αϒςt∇˜ℏ−β,ρa1ϒςt, we obtain desired inequality (15).Theorem 2.
Forα>0,σ≥ς>0, and let there be a continuous positive decreasing function ϒ defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(22)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−λ,ρa1t−a1αϒςt+∇˜ℏ−λ,ρa1ϒσt∇˜ℏ−β,ρa1t−a1αΨςt∇˜ℏ−λ,ρa1ϒςt∇˜ℏ−β,ρa1t−a1αϒσt+∇˜ℏ−β,ρa1ϒςt∇˜ℏ−λ,ρa1t−a1αϒσt≥1.Proof.
Multiplying both sides of (20) by e^νt−θ+λℏ,0t−ρℏθℏλ−1¯/ρλΓλϒςθ and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,…, show(23)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−λ,ρa1t−a1αϒςt+∇˜ℏ−λ,ρa1ϒσt∇˜ℏ−β,ρa1t−a1αϒςt−∇˜ℏ−λ,ρa1ϒςt∇˜ℏ−β,ρa1t−a1αϒσt−∇˜ℏ−β,ρa1ϒςt∇˜ℏ−λ,ρa1t−a1αϒσt≥0.
Hence, dividing (23) by(24)∇˜ℏ−λ,ρa1ϒςt∇˜ℏ−β,ρa1t−a1αϒσt−∇˜ℏ−β,ρa1ϒσt∇˜ℏ−λ,ρa1t−a1αϒσt,we obtain desired inequality (23).Theorem 3.
Forα>0,σ≥ς>0, and let there be a continuous positive decreasing function ϒ defined on ℕa1,ℏ. Also, we have a continuous positive increasing function Φ defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(25)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−β,ρa1ϒςtΦαζ∇˜ℏ−β,ρa1ϒςt∇˜ℏ−β,ρa1ϒσtΦαt≥1,where ν=ρ−1/ρ,ρ∈0,1,β∈C,and ℜβ>0.Proof.
Using the hypothesis given in Theorem3, we have(26)Φαθ−Φαϕϒσ−ςϕ−ϒσ−ςθ≥0,where α>0,σ≥ς>0, and ϕ,θ∈a1,a1+ℏ,a1+2ℏ,…,. From (26), we have(27)Φαθϒσ−ςϕ−Φαϕϒσ−ςθ+Φαθϒσ−ςθ−Φαϕϒσ−ςϕ≥0.
Taking product of (27) by e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯/ρβΓβϒςϕ,t∈ℕa1,ℏβ, and ϕ∈a1,a1+ℏ,a1+2ℏ,…,, we have(28)e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒςϕϕαθΥσ−ςϕ−Φαϕϒσ−ςθ+Φαθϒσ−ςθ−Φαϕϒσ−ςϕ,=e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβΦαθϒσϕ,−Φαϕe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒσ−ςθϒςϕ,+Φαθe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒσ−ςθϒςϕ,−Φαϕe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβϒσϕ≥0.
Summing both sides of (28) for ϕ∈a1,a1+ℏ,a2+2ℏ,…,, we have(29)ΦαθρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒσκℏℏ,−1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒσ−ςθϒςκℏℏΦακℏℏ,+ΦαθρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒσ−ςθϒςκℏℏ,−1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯ϒσκℏℏΦακℏℏ≥0.
From (30), it follows that(30)Φαθ∇˜ℏ−β,ρa1ϒσt+ϒσ−ςθ∇˜ℏ−β,ρa1Φαtϒςt−Φαθϒσ−ςθ∇˜ℏ−β,ρa1ϒςt−∇˜ℏ−β,ρa1Φαtϒςt≥0.
Again, taking product (20) by e^νt−θ+βℏ,0t−ρℏθℏβ−1¯/ρβΓβϒςθ and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,…, show(31)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−β,ρa1ϒςtΦαt−∇˜ℏ−β,ρa1ϒςt∇˜ℏ−β,ρa1ϒσtΦαt≥0,and we obtain desired inequality (25).Theorem 4.
Forα>0,σ≥ς>0 and let there be a continuous positive decreasing function ϒ defined on ℕa1,ℏ. Also, we have a continuous positive increasing function Φ defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(32)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−λ,ρa1Φαtϒςt+∇˜ℏ−λ,ρa1ϒσt∇˜ℏ−λ,ρa1Φαtϒςt∇˜ℏ−β,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt+∇˜ℏ−λ,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt≥1,where ν=ρ−1/ρ,ρ∈0,1,β,λ∈C, and ℜβℜλ>0.Proof.
Multiplying both sides of (30) by e^νt−θ+λℏ,0t−ρℏθℏλ−1¯/ρλΓλϒςθ and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,…, show(33)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−λ,ρa1Φαtϒςt+∇˜ℏ−λ,ρa1ϒσt∇˜ℏ−β,ρa1Φαtϒςt−∇˜ℏ−β,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt−∇˜ℏ−λ,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt≥0.
It follows that(34)∇˜ℏ−β,ρa1ϒσt∇˜ℏ−λ,ρa1Φαtϒςt+∇˜ℏ−λ,ρa1ϒσt∇˜ℏ−β,ρa1Φαtϒςt≥∇˜ℏ−β,ρa1Φαζϒσt∇˜ℏ−λ,ρa1ϒςt+∇˜ℏ−λ,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt.
Dividing the above inequality by(35)∇˜ℏ−β,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt+∇˜ℏ−λ,ρa1Φαtϒσt∇˜ℏ−λ,ρa1ϒςt,we obtain the desired inequality (32).


## 4. Certain Bounds for a Sequence of Decreasing Functions within Proportional Fractional Sums

Now, we demonstrate the discreteℏ-proportional fractional sum to derive some inequalities for a class of n-decreasing positive functions.Theorem 5.
Forα>0,σ≥ςϑ>0, for any ϑ∈1,2,3,…,n, and let there be a sequence of continuous positive decreasing functions ϒl,l=1,2,3,…,n defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(36)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt≥∇˜ℏ−β,ρa1t−a1α∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1t−a1α∏l=1nϒlςlt,where ν=ρ−1/ρ,ρ∈0,1,β∈C, and ℜβ>0.Proof.
Sinceϒl,l=1,2,3,…,n is a sequence of continuous positive decreasing functions on a1,a1+ℏ,a1+2ℏ,…,, we have(37)θ−a1α−ϕ−a1αϒϑσ−ςϑϕ−ϒϑσ−ςϑθ≥0,for any fixed ϑ∈1,2,3,…,n,α>0,σ≥ςϑ>0, and ϕ,θ∈a1,a1+ℏ,a1+2ℏ,…,.
By (37), we have(38)θ−a1αϒϑσ−ςϑϕ+ϕ−a1αϒϑσ−ςϑθ≥θ−a1αϒϑσ−ςϑθ+ϕ−a1αϒϑσ−ςϑϕ.
Taking product of (38) by e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯/ρβΓβ∏l=1nϒlςlϕ,t∈ℕa1,ℏβ, and ϕ∈a1,a1+ℏ,a1+2ℏ,…,, we have(39)e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβθ−a1αϒϑσ−ςϑϕ+ϕ−a1αϒϑσ−ςϑθ−θ−a1αϒϑσ−ςϑθ,−ϕ−a1αϒϑσ−ςϑϕ∏l=1nϒlςlϕ,=θ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑϕ,+ϕ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑθ,−θ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑθ,−ϕ−a1αe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑϕ≥0,(40)θ−a1α1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlκℏℏϒϑσ−ςϑκℏℏ,+1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlℏκℏϒϑσ−ςϑθκℏ−a1α,−θ−a1α1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlκℏℏΥϑσ−ςϑθ,−1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nκℏ−a1αϒlςlκℏℏϒϑσ−ςϑκℏℏ≥0,(41)θ−a1α∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt+ϒϑσ−ςϑθ∇˜ℏ−β,ρa1t−ς1α∏l=1nϒlςlt≥θ−a1αϒϑσ−ςϑθ∇˜ℏ−β,ρa1∏l=1nϒlςlt+∇˜ℏ−β,ρa1t−ς1α∏l≠ϑnϒlςlϒϑσt,
e^νt−θ+βℏ,0t−ρℏθℏβ−1¯/ρβΓβ∏l=1nϒlςlθ, and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,…, shows(42)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlΥϑσt∇˜ℏ−β,ρa1t−ς1α∏l=1nϒlςlt≥∇˜ℏ−β,ρa1t−ς1α∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt,which gives the desired inequality (36).Theorem 6.
Forα>0,σ≥ςϑ>0, for any ϑ∈1,2,3,…,n, and let there be a sequence of continuous positive decreasing functions Υl,l=1,2,3,…,n defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality: (43)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1t−a1α∏l=1nϒlςlt+∇˜ℏ−λ,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1t−a1α∏l=1nϒlςlt∇˜ℏ−β,ρa1t−a1α∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1∏l=1nϒlςlt+∇˜ℏ−λ,ρa1t−a1α∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt≥1,where ν=ρ−1/ρ,ρ∈0,1,β,λ∈C, and ℜβ,ℜλ>0.Proof.
Taking product on both sides of (41) by e^νt−θ+λℏ,0t−ρℏθℏλ−1¯/ρλΓλ∏l=1nϒlςlθ and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,…, show(44)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1t−a1α∏l=1nϒlςlt,+∇˜ℏ−λ,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1t−a1α∏l=1nϒlςlt,≥∇˜ℏ−β,ρa1t−a1α∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1∏l=1nϒlςlt,+∇˜ℏ−λ,ρa1t−a1α∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt.
Dividing the above inequality by(45)∇˜ℏ−β,ρa1t−a1α∏l≠ϑnϒlςlϒϑσ∇˜ℏ−λ,ρa1∏l=1nϒlςlt+∇˜ℏ−λ,ρa1t−a1α∏l≠ϑnϒlςlϒϑσ∇˜ℏ−β,ρa1∏l=1nϒlςltgives the desired inequality (43).Theorem 7.
Forα>0,σ≥ςϑ>0 for any ϑ∈1,2,3,…,n and let there be a sequence of continuous positive decreasing functions ϒl,l=1,2,3,…,n defined on ℕa1,ℏ. Also, there is a continuous positive increasing function Φ defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(46)∇˜ℏ−β,ρa1∏l≠θnΨlςlΨθσt∇˜ℏ−β,ρa1Φαt∏l=1nϒlςlt∇˜ℏ−β,ρa1Φαt∏l≠θnΨlςlΨϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt≥1,where ν=ρ−1/ρ,ρ∈0,1,β∈C, and ℜβ>0.Proof.
Under the given hypothesis, we have(47)Φαθ−Φαϕϒϑσ−ςϑϕ−ϒϑσ−ςϑθ≥0,for any fixed ϑ∈1,2,3,…,n,α>0,σ≥ςϑ>0, and ϕ,θ∈a1,a1+ℏ,a1+2ℏ,….
From (47), we have(48)Φαθϒϑσ−ςϑϕ+Φαϕϒϑσ−ςϑθ−Φαθϒϑσ−ςϑθ−Φαϕϒϑσ−ςϑϕ≥0.
Taking product on both sides of (50) by e^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯/ρβΓβ∏l=1nϒlςlϕ,t∈ℕa1,ℏβ and ϕ∈a1,a1+ℏ,a1+2ℏ,…, we have(49)Φαθe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑϕ,+Φαϕe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑθ,−Φαθe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑθ,−Φαϕe^νt−ϕ+βℏ,0t−ρℏϕℏβ−1¯ρβΓβ∏l=1nϒlςlϕϒϑσ−ςϑϕ≥0,(50)Φαθ1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlκℏℏϒϑσ−ςϑκℏℏ,+1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlκℏℏϒϑσ−ςϑκℏℏ,−Φαθ1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlκℏℏϒϑσ−ςϑθ,−1ρβΓβ∑κ=a1/ℏ+1t/ℏe^νt−κℏ+βℏ,0t−ρℏκℏℏβ−1¯∏l=1nϒlςlκℏℏϒϑσ−ςϑκℏℏΦακℏℏ≥0.
From (50), it follows that(51)Φαθ∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt+ϒϑσ−ςϑt∇˜ℏ−β,ρa1Φαt∏l=1nϒlςlt−Φαθϒϑσ−ςϑθ∇˜ℏ−β,ρa1∏l=1nϒlςlt−∇˜ℏ−β,ρa1Φαt∏l≠ϑnΨlςlϒϑσt≥0,
e^νt−θ+βℏ,0t−ρℏθℏβ−1¯/ρβΓβ∏l=1nϒlςlθ, and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,… shows(52)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1Φαt∏l=1nϒlςlt−∇˜ℏ−β,ρa1Φαt∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt≥0,and we get the desired inequality (46).Theorem 8.
Forα>0,σ≥ςϑ>0 for any ϑ∈1,2,3,…,n and let there be a sequence of continuous positive decreasing functions ϒl,l=1,2,3,…,n defined on ℕa1,ℏ. Also, there is a continuous positive increasing function Φ defined on ℕa1,ℏ. Then, the discrete ℏ-proportional fractional sum satisfies the following inequality:(53)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1Φαt∏l=1nϒlςlt+∇˜ℏ−β,ρa1Φαt∏l=1nϒlςlt∇˜ℏ−λ,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1∏l=1nϒlςlt∇˜ℏ−β,ρa1Φαt∏l≠ϑnϒlςlϒϑσt+∇˜ℏ−λ,ρa1Φαt∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt≥1.Proof.
Multiplying both sides of (51) by e^νt−θ+λℏ,0t−ρℏθℏλ−1¯/ρλΓλ∏l=1nϒlςlθ, and summing both sides for θ∈a1,a1+ℏ,a1+2ℏ,… show(54)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1Φαt∏l=1nϒlςlt,+∇˜ℏ−β,ρa1Φαt∏l=1nϒlςlt∇˜ℏ−λ,ρa1∏l≠ϑnϒlςlϒϑσt,−∇˜ℏ−λ,ρa1∏l=1nϒlςlt∇˜ℏ−β,ρa1Φαt∏l≠ϑnϒlςlϒϑσt,−∇˜ℏ−λ,ρa1Φαt∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt≥0.
It follows that(55)∇˜ℏ−β,ρa1∏l≠ϑnϒlςlϒϑσt∇˜ℏ−λ,ρa1Φαt∏l=1nϒlςlt,+∇˜ℏ−β,ρa1Φαt∏l=1nϒlςlt∇˜ℏ−λ,ρa1∏l≠ϑnϒlςlϒϑσt,≥∇˜ℏ−λ,ρa1∏l=1nϒlςlt∇˜ℏ−β,ρa1Φαt∏l≠ϑnϒlςlϒϑσt,+∇˜ℏ−λ,ρa1Φαt∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt.
Dividing both sides by(56)∇˜ℏ−λ,ρa1∏l=1nϒlςlt∇˜ℏ−β,ρa1Φαt∏l≠ϑnϒlςlϒϑσt+∇˜ℏ−λ,ρa1Φαt∏l≠ϑnϒlςlϒϑσt∇˜ℏ−β,ρa1∏l=1nϒlςlt,we get the desired inequality (53).


## 5. Conclusion

DiscreteℱC has gained a lot of achievement in real-world phenomena, such as fractional chaotic maps, image encryption, and another discrete-time modelling. One of the supreme fundamental problems in the study of difference equations is to investigate the qualitative characteristics of the solutions of these aforesaid fields. Discrete fractional variants are noteworthy mechanisms that expedite discovering such properties. In the present note, we have proposed a novel proportional fractional sum to derive innovative descriptions for the noted class of a family of continuous positive decreasing functions on time scale ℏℤ. By considering the ℏ-proportional fractional sums within the nabla ℏ-fractional sum, we derived the extension of sequences of decreasing functions in the frame of time-scale domains. The noted consequences can also be extended to the weighted function case. One can straightforwardly make sense that the contemporary consequences simplify the ones formerly achieved in the literature. Certainly, the case ρ↦1 recaptures the outcomes of nabla ℏ-fractional sums. For indicating the strength of the offered fallouts, we employ them to investigate numerous initial value problems of fractional difference equations.



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