# Estimation of Integral Inequalities Using the Generalized Fractional Derivative Operator in the Hilfer Sense

Authors: Saima Rashid; Rehana Ashraf; Kottakkaran Sooppy Nisar; Thabet Abdeljawad
Journal: Journal of Mathematics
 (2020)

---

## Abstract

With the great progress of fractional calculus, integral inequalities have been greatly enriched by fractional operators; users and researchers have formed a real-world phenomenon in the production of the evaluation process, which results in convexity. Monotonicity and inequality theory has a strong relationship, whichever we work on, and we can apply it to the other one due to the strong correlation produced between them, especially in the past few years. In this article, we introduce some estimations of left and right sides of the generalized Caputo fractional derivatives of a function fornth order differentiability via convex function, and related inequalities have been presented. Monotonicity and convexity of functions are used with some usual and straightforward inequalities. Moreover, we establish some new inequalities for C⌣ eby s⌣ ev and Gru¨ss type involving the generalized Caputo fractional derivative operators. The finding provides the theoretical basis and practical significance for the establishment of fractional calculus in convexity. It also introduces new ways of thinking and methods for innovative scientific research.


---

## Body

## 1. Introduction

The field of fractional calculus deals with the integrals and differentiation of arbitrary noninteger order. In the last three centuries, this field has been considered as the most powerful tool in describing anomalous kinetics and its wide applications in diverse domains. Numerous phenomena such as mathematics, statistics, engineering, physics, chemistry, and biology can be modeled by utilizing ordinary differential equations involving fractional derivatives. Many mathematicians and physicists have contributed to the development of the theories of fractional calculus [1–11]. In practical applications, various types of fractional integrals and derivative operators such as Riemann–Liouville, Caputo, Riesz, Hilfer, Hadamard, Erdelyi–Kober, Saigo, and Marichev–Saigo–Maeda were extensively studied by various researchers, see [12–17].Later on, the mathematicians introduced the notion of fractional conformable integrals and derivatives which are cited therein. Khalil et al. [18] introduced fractional conformable derivatives operators with some shortcomings. Abdeljawad [19] investigated the properties of the fractional conformable derivative operators. Jarad et al. [20] defined generalized fractional conformable integral and derivative operators. In [21], Abdeljawad and Baleanu gave certain monotonicity results for fractional difference operators with discrete exponential kernels. Almeida [22] proposed Caputo fractional derivative in the sense of another function ϑ, and in [1], the authors contemplated the idea of Riemann–Liouville fractional integrals in the sense of another function ϑ. In [23], Atangana and Baleanu defined new fractional derivative operator with the nonlocal and nonsingular kernel.Inequalities concerning functions of two or several independent variables play an essential role in the continuous development of the theory and applications of differential and integral equations. Currently, distinctive versions of such inequalities had been developed which can be useful in the study of various classes of differential and integral equations. Those inequalities act as a far-reaching tool to study plasma physics, robotics, automatic control and many other branches of pure and applied sciences, and differential and integral equations [24, 25].Convex functions are very useful in the mathematical analysis due to their fascinating properties and convenient characterizations.Definition 1.
A functionU:ℐ⟶ℝ is said to be convex function, if the following inequality holds:(1)Uζτ1+1−ζτ2≤ζUτ1+1−ζUτ2,for all τ1,τ2∈ℐ and ζ∈0,1. If inequality (1) holds in the reverse order, then the function U is called concave function.
For convex functions, many equalities or inequalities have been established by many authors; for example, Hardy-type inequality, Ostrowski-type inequality, and Gagliardo–Nirenberg-type inequality, but the most celebrated and significant inequality is the Hermite–Hadamard-type inequality [26–29], which is defined as(2)Uτ1+τ22≤1τ2−τ1∫τ1τ2Uxdx≤Uτ1+Uτ22.
A number of mathematicians in the field of applied and pure mathematics have dedicated their efforts to extend, generalize, counterpart, and refine the Hermite–Hadamard inequality (2) for different classes of convex functions. For more recent results obtained on inequality (2), we refer the reader to references [30–35].
Inspired by the aforementioned development, we propose a famous approach of generalized fractional derivative investigated in [1, 22], especially Caputo fractional derivative in the ϑ-Hilfer sense is being utilized widely and furthermore, effectively utilized in numerous parts of sciences and engineering, see [36, 37]. Our concern is to utilize the convexity property of functions and use the absolute of their derivatives in obtaining the bounds for generalized Caputo fractional derivative presented by Definition 2.3. The new derivative is used to model the world, and we are capable of seeing that the choice of the generalized Caputo fractional derivative operator is essential for the efficiency of the numerical methods, fractional differential equations, and fractional integrodifferential equations.
It is widely recognized thatC⌣ebys⌣ev and Gru¨ss type inequalities in continuous and discrete cases which play a significant role in studying the qualitative conduct of differential and difference equations, respectively, in addition to many other areas of mathematics. Inspired by C⌣ebys⌣ ev [38] and Gru¨ss [39], our aim is to show more general versions of C⌣ebys⌣ ev and Gru¨ss type inequalities.
C⌣ebys⌣ ev [38] introduced the well-known celebrated functional and is defined as follows:(3)ℑU,V=1τ2−τ1∫τ1τ2UλVλdλ−1τ2−τ1∫τ1τ2Uλdλ1τ2−τ1∫τ1τ2Vλdλ,where U and V are two integrable functions on τ1,τ2. If U and V are synchronous, i.e.,(4)Uλ−UωVλ−Vω≥0,for any λ,ω∈τ1,τ2, then sU,V≥0.
Functional (3) has vast applications in probability, numerical analysis, quantum, and statistical theory. Alongside facet with numerous applications, the functional (3) has gained plenty of interest to yield a variety of fundamental inequalities (see, for example, [40–42]).
Another interesting and fascinating aspect of the theory of inequalities is the Gru¨ss type inequality [39] stated as follows:(5)sU,V≤Q−qR−r4,where two integrable functions U and V on τ1,τ2, and fulfill the following:(6)q≤Uλ≤Q,r≤Vλ≤R,for all λ∈τ1,τ2 and for some q,Q,r,R∈ℝ.
Many famous versions mentioned in the literature are direct effects of the numerous applications in optimizations and transform theory, see [24, 25, 42–51].
The principal aim of the present paper is to establish new bounds of some of the left-sided and right-sided Caputo fractional derivatives in Hilfer sense via convex functions that have been established. Some related inequalities via convexity and monotonicity of used functions have been proved. Moreover, the novel version of Gru¨ss and C⌣ebys⌣ ev types integral inequalities associated with Caputo fractional derivative operators in Hilfer sense are established for nth order differentiability of functions. We provide innovative special cases using a Caputo fractional derivative operator in Hilfer sense related to (3) and (5). Consequently, the effects furnished on this research paper are more generalized and may be useful in the study of fractional integral operators.


## 2. Preliminaries

In this sequel, we introduce a few notations and definitions of fractional calculus and present initial results wished in our proofs later.Definition 2.
(see [1, 2]). A function Uλ is said to be in Lp,r0,∞ space if(7)Lp,r0,∞=U:ULp,r0,∞=∫υ1υ2Uλpλrdλ1/p<∞,1≤p<∞,r≥0.
Forr=0,(8)Lp0,∞=U:ULp0,∞=∫υ1υ2Uλpdλ1/p<∞,1≤p<∞.Definition 3.
(see [52]). Let U∈L10,∞ and ϑ be an increasing and positive monotone function on 0,∞ and also derivative ϑ′ is continuous on 0,∞ and ϑ0=0. The space χϑp0,∞1≤p<∞ of those real-valued Lebesgue measurable functions U on 0,∞ for which(9)Uχϑp=∫0∞Uλpϑ′λdλ1/p<∞,1≤p<∞,and for the case p=∞(10)Uχϑ∞=esssup0≤λ<∞ϑ′λUλ.
In particular, whenϑλ=λ,1≤p<∞, the space χϑp0,∞ coincides with the Lp0,∞-space, and furthermore, if we take ϑλ=lnλ,1≤p<∞, the space χϑp0,∞ concurs with Lp,u1,∞-space.Definition 4.
(see [1, 22]). Let τ1,τ2−∞≤τ1<τ2≤∞ be a finite or infinite real interval and ϱ>0. Let ϑζ be an increasing and positive monotone function on τ1,τ2. Then, the left Caputo fractional derivative in the ϑ-Hilfer sense of order ϱ>0 is given by(11)Dτ1+ϱ,ϑcUλ≔Iτ1+n−ϱ,ϑ1ϑ′λddλnUλ,and the right Caputo fractional derivative in the ϑ-Hilfer sense of U by(12)Dτ2−ϱ,ϑcUλ≔Iτ2−n−ϱ,ϑ−1ϑ′λddλnUλ,where n=ϱ+1 for ϱ∉ℕ,n=ϱ for ϱ∈ℕ and if ϱ∉ℕ; then,(13)Dτ1+ϱ,ϑcUλ=1Γn−ϱ∫τ1λϑ′ζϑλ−ϑζn−ϱ−1Uϑnζdζ,(14)Dτ2−ϱ,ϑcUλ=−1nΓn−ϱ∫λτ2ϑ′ζϑζ−ϑλn−ϱ−1Uϑnζdζ,where Γz=∫0∞ζz−1e−ζdζ is the Euler gamma function.Remark 1.
It can be easily noticed that(1)
Whenϑζ=ζ, then (13) and (14) are the classical Caputo derivative [1](2)
Whenϑζ=lnζ, then (13) and (14) are the Caputo–Hadamard fractional derivative [43](3)
whenϑζ=ζρ/ρ, then (13) and (14) are the Caputo modification of the left and right generalized fractional derivatives in the sense of [53](4)
Whenϑζ=ζ−aρ/ρ, then (13) and (14) are the fractional conformable derivative in the sense of [20]
Now, we present a one-sided fractional operator which is known as the generalized Caputo fractional derivative operator.Definition 5.
Letτ1,τ2−∞≤τ1<τ2≤∞ be a finite or infinite real interval and ϱ>0. Let ϑζ be an increasing and positive monotone function on tau1,tau2. Then, the left-sided and right-sided Caputo fractional derivative in the ϑ-Hilfer sense of order ϱ is defined as follows:(15)D0+ϑ,ϱUλ=1Γn−ϱ∫0λϑ′ζϑλ−ϑζn−ϱ−1Uζdζ.


## 3. Hermite–Hadamard-Type Inequalities for Caputo Fractional Derivative in theϑ-Hilfer Sense

Theorem 1.
Forn∈ℕ,ϱ,δ≥1, and let there be a real-valued n-times differentiable function U:ℐ=τ1,τ2⟶ℝ defined on ℐ. Also, assume that ϑ be differentiable and strictly increasing such that with ϑ′∈L1τ1,τ2. If ϑn is a convex function on ℐ, for all τ1,τ2∈ℐ and τ1<τ2, then(16)Γn−ϱ+1Dτ1+ϱ−1,ϑcUλ+Γn−δ+1Dτ2−ϱ−1,ϑcUλ≤ϑλ−ϑτ1n−ϱλ−τ1λ−τ1Unλϑλ−Unτ1ϑτ1−Unλ−Unτ1∫τ1λϑζdζ+ϑτ2−ϑλn−δτ2−λτ2−λUnτ2ϑτ2−Unλϑλ−Unτ2−Unλ∫λτ2ϑζdζ.Proof.
Utilizing the given hypothesis, we have(17)ϑλ−ϑζn−ϱ≤ϑλ−ϑτ1n−ϱ,where λ∈τ1,τ2 and ζ∈τ1,τ2, ϱ≥1, and ϑ′ζ>0. Hence, the following inequality holds true:(18)ϑ′ζϑλ−ϑζn−ϱ≤ϑ′ζϑλ−ϑτ1n−ϱ.
By the convexity ofUn, we have(19)Unζ≤λ−ζλ−τ1Unτ1+ζ−τ1λ−τ1Unλ.
From (18) and (19), one has(20)∫τ1λϑλ−ϑζn−ϱUnζϑ′ζdζ≤ϑλ−ϑτ1n−ϱλ−τ1Unτ1∫τ1λλ−ζ+Unλ∫τ1λζ−τ1ϑ′ζdζ.
Using (13) from Definition 4, we obtain(21)Γn−ϱ+1Dτ1+ϱ−1,ϑcUλ≤ϑλ−ϑτ1n−ϱλ−τ1λ−τ1Unλϑλ−Unτ1ϑτ1−Unλ−Unτ1∫τ1λϑζdζ.
Now, forλ∈τ1,τ2,ζ∈λ,τ2, and δ≥1, the following inequality holds true:(22)ϑ′ζϑζ−ϑλn−δ≤ϑ′ζϑτ2−ϑλn−δ.
Utilizing convexity ofUn, it follows that(23)Unζ≤ζ−λτ2−λUnτ2+τ2−ζτ2−λUnλ.
Repeating the same procedure as we have done for (18) and (19), one can acquire from (22) and (23); then,(24)Γn−δ+1Dτ2−δ−1,ϑcUλ≤ϑτ2−ϑλn−δτ2−λ×τ2−λUnτ2ϑτ2−Unλϑλ−Unτ2−Unλ∫λτ2ϑζdζ.
From inequalities (21) and (24), we get (16) which is required.Corollary 1.
If we takeα=δ in (16), we get the result for generalized Caputo fractional derivative operator:(25)Γn−α+1Dτ1+α−1,ϑcUλ+Dτ2−α−1,ϑcUλ≤ϑλ−ϑτ1n−αλ−τ1λ−τ1Unλϑλ−Unτ1ϑτ1−Unλ−Unτ1∫τ1λϑζdζ+ϑτ2−ϑλn−ατ2−λτ2−λUnτ2ϑτ2−Unλϑλ−Unτ2−Unλ∫λτ2ϑζdζ.Theorem 2.
Forn∈ℕ,ϱ,δ≥1, and let there be a real-valued n-times differentiable function U:ℐ=τ1,τ2⟶ℝ defined on U:ℐ=τ1,τ2⟶ℝ. Also, assume that ϑ be differentiable and strictly increasing such that with ϑ′∈L1τ1,τ2. If ϑn+1 is a convex function on ℐ, for all τ1,τ2∈ℐ and τ1<τ2, then(26)Γn−ϱ+1Dτ1+ϱ,ϑcUλ+Γn−δ+1Dτ2−ϱ,ϑcUλ−ϑλ−ϑτ1n−ϱUnτ1+ϑτ2−ϑλn−δUnτ2≤ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1+ϑτ2−ϑλn−δτ2−λUn+1τ22+Un+1λϑλ−ϑτ1n−ϱλ−τ1+ϑτ2−ϑλn−δτ2−λ2.Proof.
From convexity ofUn+1, we have(27)Un+1ζ≤λ−ζλ−τ1Un+1τ1+ζ−τ1λ−τ1Un+1λ.
From (27), one obtains(28)Un+1ζ≤λ−ζλ−τ1Un+1τ1+ζ−τ1λ−τ1Un+1λ.
Since the functionϑ is differentiable and strictly increasing, therefore we have the following inequality:(29)ϑλ−ϑζn−ϱ≤ϑλ−ϑτ1n−ϱ,where λ∈τ1,τ2, ζ∈τ1,λ, and ϱ>0.
From (28) and (29), one has(30)ϑλ−ϑζn−ϱUn+1ζ≤ϑλ−ϑτ1n−ϱλ−τ1λ−ζUn+1τ1+ζ−τ1Un+1λ.
Integrating overτ1,λ, we have(31)∫τ1λϑλ−ϑζn−ϱUn+1ζdζ≤ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1∫τ1λλ−ζdζ+Un+1λ∫τ1λζ−τ1dζ=ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1+Un+1λ2,(32)∫τ1λϑλ−ϑτ1n−ϱUn+1ζdζ=Unζϑλ−ϑτ1n−ϱτ1λ+n−ϱ∫τ1λϑλ−ϑτ1n−ϱ−1ϑ′ζUnζdζ=−Unτ1ϑλ−ϑτ1n−ϱ+Γn−ϱ+1Dτ1+ϱ,ϑcUλ.
From (31), it follows that(33)Γn−ϱ+1Dτ1+ϱ,ϑcUλ−Unτ1ϑλ−ϑτ1n−ϱ≤ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1+Un+1λ2.
Also, from (27), one has(34)Un+1ζ≥λ−ζλ−τ1Un+1τ1+ζ−τ1λ−τ1Un+1λ.
Repeating the same procedure as we did for (28), we have(35)Unτ1ϑλ−ϑτ1n−ϱ−Γn−ϱ+1Dτ1+ϱ,ϑcUλ≤ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1+Un+1λ2.
From (33) and (35), we obtain(36)Γn−ϱ+1Dτ1+ϱ,ϑcfλ−Unτ1ϑλ−ϑτ1n−ϱ≤ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1+Un+1λ2.
From convexity ofUn+1, one obtains(37)Un+1ζ≤ζ−λτ2−λUn+1τ2+τ2−λτ2−λUn+1λ.
Now, forλ∈τ1,τ2 and ζ∈λ,τ2 and δ>0, the following inequality holds true:(38)ϑζ−ϑλn−δ≤ϑτ2−ϑλn−δ.
If we proceed in a similar way as we did for (28), (29), and (34), one can get from (37) and (38) the following inequality:(39)Γn−δ+1Dτ2−δ,ϑcUλ−Unτ2ϑτ2−ϑλn−δ≤ϑδ−ϑλn−δτ2−λUn+1λ+Un+1τ22.
From inequalities (36) and (39) via triangular inequality, we get (26) which is required.Corollary 2.
If we takeϱ=δ in (26), then we get the following inequality for generalized Caputo fractional derivative operator:(40)Γn−ϱ+1Dτ1+ϱ,ϑcUλ+Dτ2−ϱ,ϑcUλ−ϑλ−ϑτ1n−ϱUnτ1+ϑτ2−ϑλn−ϱUnτ2≤ϑλ−ϑτ1n−ϱλ−τ1Un+1τ1+ϑτ2−ϑλn−ϱτ2−λUn+1τ22+Un+1λϑλ−ϑτ1n−ϱλ−τ1+ϑτ2−ϑλn−ϱτ2−λ2.Lemma 1.
(see [28]). Suppose that U:ℐ⟶ℝ is a convex function which is symmetric about τ1+τ2/2; then, the following inequality holds:(41)Uτ1+τ22≤Uλ,λ∈τ1,τ2.Theorem 3.
Forn∈ℕ,ϱ,δ≥1, and let there be a real-valued n-times differentiable function U:ℐ=τ1,τ2⟶ℝ defined on ℐ, where n is a positive integer such that Un be positive convex and symmetric about τ1+τ2/2. Also, assume that ϑ be differentiable and strictly increasing such that with ϑ′∈L1τ1,τ2 for all τ1,τ2∈ℐ and τ1<τ2; then,(42)121n−ϱ+1+1n−δ+1Unτ1+τ22≤Γn−ϱ+1Dτ1+ϱ−1,ϑcUτ22ϑτ2−ϑτ1n−ϱ+1+Γn−δ+1Dτ2−δ−1,ϑcUτ12ϑτ2−ϑτ1n−δ+1≤1ϑτ2−ϑτ1τ2−τ1τ2−τ1Unτ2ϑτ2−Unτ1ϑτ1−Unτ2−Unτ1∫τ1τ2ϑλdλ.Proof.
Utilizing the given hypothesis, we have(43)ϑλ−ϑτ1n−δ≤ϑτ2−ϑτ1n−δ,where λ∈τ1,τ2,δ>0, and ϑ′λ>0. Hence, the following inequality holds true:(44)ϑ′λϑλ−ϑτ1n−δ≤ϑ′ϑτ2−ϑτ1n−δ.
By using the convexity ofUn, we have(45)Unλ≤λ−τ1τ2−τ1Unτ2+τ2−λτ2−τ1Unτ1.
From (44) and (45), one has(46)∫τ1τ2ϑλ−ϑτ1n−δUnλϑ′λdλ≤ϑτ2−ϑτ1n−δτ2−τ1Unτ2∫τ1τ2x−τ1ϑ′λdλ+Unτ1∫τ1τ2τ2−xϑ′λdλ.
Using (13) from Definition 4, we obtain(47)Γn−δ+1Dτ2−δ−1,ϑcfτ1≤ϑτ2−ϑτ1n−δτ2−τ1τ2−τ1Unτ2ϑτ2−Unτ1ϑτ1−Unτ2−Unτ1∫τ1τ2ϑλdλ.
Now forλ∈τ1,τ2, ζ∈λ,τ2, and ϱ>0, the following inequality holds true:(48)ϑ′λϑτ2−ϑλn−ϱ≤ϑ′λϑτ2−ϑτ1n−ϱ.
By a similar argument which we have done for (44) and (45), one can get from (45) and (48) the following inequality:(49)Γn−ϱ+1Dτ1+ϱ−1,ϑcUτ2≤ϑτ2−ϑτ1n−ϱτ2−τ1τ2−τ1Unτ2ϑτ2−Unτ1ϑτ1−Unτ2−Unτ1∫τ1τ2ϑλdλ.
SinceUn is convex and symmetric about τ1+τ2/2 using Lemma 1 and multiplying (41) with ϑλ−ϑτ1n−δϑ′λ and integrating over τ1,τ2, we obtain(50)Unτ1+τ22∫τ1τ2ϑλ−ϑτ1n−δϑ′λdλ≤∫τ1τ2ϑλ−ϑτ1n−δϑ′λUnλdλ.
Using (14) from Definition 4, we obtain(51)Unτ1+τ2/22n−δ+1≤Γn−δ+1Dτ2−δ−1,ϑcUτ12ϑτ2−ϑτ1n−δ+1.
Similarly, using Lemma1 and multiplying (41) with ϑτ2−ϑλn−ϱϑ′λ and then integrating over τ1,τ2, we have(52)Unτ1+τ2/22n−ϱ+1≤Γn−ϱ+1Dτ1+ϱ−1,ϑcUτ22ϑτ2−ϑτ1n−ϱ+1.
Adding (51) and (52), we get the first inequality (42).Corollary 3.
If we takeϱ=δ in (42), then we get the following inequality for generalized Caputo fractional derivative operator:(53)1n−ϱ+1Unτ1+τ22≤Γn−ϱ+1Dτ1+ϱ−1,ϑcUτ2+Dτ2−ϱ−1,ϑcUτ12ϑτ2−ϑτ1n−ϱ+1≤τ2−τ1Unτ2ϑτ2−Unτ1ϑτ1−Unτ2−Unτ1∫τ1τ2ϑλdλϑτ2−ϑτ1τ2−τ1.


## 4. C⌣ebys⌣ev Type Inequalities for the Caputo Fractional Derivative in the ϑ-Hilfer Sense Operators

In this section, we present severalC⌣ ebys⌣ev type inequalities for Caputo fractional derivative in the ϑ-Hilfer sense operator defined in (15).Theorem 4.
Forn∈ℕ,ϱ≥1, and let there be two absolutely continuous functions U and V which are synchronous on 0,∞. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0. Then, for all λ>0, we have(54)D0+ϱ,ϑcUVλ≥Γn−ϱ+1ϑλn−ϱD0+ϱ,ϑcUλD0+ϱ,ϑcVλ.Proof.
SinceUn and Vn are synchronous on 0,∞, we have(55)UnrVnr+UnsVns≥UnrVns+UnsVnr.
If we multiply both sides of inequality (55) by(56)1Γn−ϱϑ′rϑλ−ϑrn−ϱ−1,λ∈ℝ,results in(57)1Γn−ϱϑ′rϑλ−ϑrn−ϱ−1UnrVnr+UnsVns1Γn−ϱϑ′rϑλ−ϑrn−ϱ−1≥Vns1Γn−ϱϑ′rϑλ−ϑrn−ϱ−1Unr+Uns1Γn−ϱϑ′rϑλ−ϑrn−ϱ−1Vnr.
Further integrating both sides with respect tor over 0,λ gives(58)1Γn−ϱ∫0λϑ′rϑλ−ϑrn−ϱ−1UnrVnrdr+UnsVns1Γn−ϱ∫0λϑ′rϑλ−ϑrn−ϱ−1dr≥Vns1Γn−ϱ∫0λϑ′rϑλ−ϑrn−ϱ−1Unrdr+Uns1Γn−ϱ∫0λϑ′rϑλ−ϑrn−ϱ−1Vnrdr.
Consequently, it follows that(59)D0+ϱ,ϑcUVλ+UnsVns1Γn−ϱ∫0λϑ′rϑλ−ϑrn−ϱ−1dr≥VnsD0+ϱ,ϑcUλ+UnsD0+ϱ,ϑcVλ,(60)D0+ϱ,ϑcUVλUnsVnsϑλn−ϱΓn−ϱ+1≥VnsD0+ϱ,ϑcUλ+UnsD0+ϱ,ϑcVλ,where(61)1Γn−ϱ∫0λϑ′rϑλ−ϑrn−ϱ−1dr=ϑλn−ϱn−ϱΓn−ϱ.
If we multiply both sides of inequality (60) by(62)1Γn−ϱϑ′sϑλ−ϑsn−ϱ−1,λ∈ℝ,we arrive at(63)D0+ϱ,ϑcUVλ1Γn−ϱϑ′sϑλ−ϑsn−ϱ−1+1Γn−ϱϑ′sϑλ−ϑsn−ϱ−1UnsVnsϑλn−ϱΓn−ϱ+1≥D0+ϱ,ϑcUλ1Γn−ϱϑ′sϑλ−ϑsn−ϱ−1Vns+D0+ϱ,ϑcVλ1Γn−ϱϑ′sϑλ−ϑsn−ϱ−1Uns.
Now, integrating over0,λ reveals(64)D0+ϱ,ϑcUVλ1Γn−ϱ∫0λϑ′sϑλ−ϑsn−ϱ−1ds+1Γn−ϱ∫0λϑ′sϑλ−ϑsn−ϱ−1UnsVnsdsϑλn−ϱΓn−ϱ+1≥D0+ϱ,ϑcUλ1Γn−ϱ∫0λϑ′sϑλ−ϑsn−ϱ−1Vnsds+D0+ϱ,ϑcVλ1Γn−ϱ∫0λϑ′sϑλ−ϑsn−ϱ−1Unsds.
Therefore, we have(65)ϑλn−ϱΓn−ϱ+1D0+ϱ,ϑcUVλ+D0+ϱ,ϑcUVλϑλn−ϱΓn−ϱ+1≥D0+ϱ,ϑcUλD0+ϱ,ϑcVλ+D0+ϱ,ϑcVλD0+ϱ,ϑcUλ.
The proof of Theorem4 is complete.Corollary 4.
Settingϱ=1, then under the assumptions of Theorem 4, we have(66)D0+ϑcUVλ≥Γnϑλn−1D0+ϑcUλD0+ϑcVλ.Corollary 5.
Settingϑλ=λ, then under the assumption of Theorem 4, we have a new result for the Caputo fractional derivative operator:(67)D0+ϱcUVλ≥Γn−ϱ+1λn−ϱD0+ϱcUλD0+ϱcVλ.Theorem 5.
Forn∈ℕ,ϱ,δ≥1, and let there be two absolutely continuous functions U and V which are synchronous on 0,∞. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0. Then, for all λ>0, we have(68)D0+ϱ,ϑcUVλϑλn−δΓn−δ+1+ϑλn−D0+δ,ϑcUVλΓn−ϱ+1≥D0+ϱ,ϑcUλD0+δ,ϑcVλ+D0+ϱ,ϑcVλD0+δ,ϑcUλ.Proof.
Using inequality (61) and multiplying both sides by(69)1Γn−δϑ′sϑλ−ϑsn−δ−1,λ∈ℝ,yields(70)D0+ϱ,ϑcUVλ1Γn−δϑ′sϑλ−ϑsn−δ−1+1Γn−δϑ′sϑλ−ϑsn−δ−1UnsVnsϑλn−ϱΓn−ϱ+1≥D0+ϱ,ϑcUλ1Γn−δϑ′sϑλ−ϑsn−δ−1Vns+D0+ϱ,ϑcVλ1Γn−δϑ′sϑλ−ϑsn−δ−1Uns.
Furthermore, integrating both sides with respect tos over 0,λ leads to(71)D0+ϱ,ϑcUVλΓn−δ∫0λϑ′sϑλ−ϑsn−δ−1ds+ϑλn−ϱΓn−ϱ+11Γn−δ∫0λϑ′sϑλ−ϑsn−δ−1UnsVnsds≥D0+ϱ,ϑcUλΓn−δ∫0λϑ′sϑλ−ϑsn−δ−1Vnsds+D0+ϱ,ϑcVλΓn−δ∫0λϑ′sϑλ−ϑsn−δ−1Unsds.
Therefore, we have(72)D0+ϱ,ϑcUVλϑλn−δn−δ+1+ϑλn−D0+δ,ϑcUVλΓn−ϱ+1≥D0+ϱ,ϑcUλD0+δ,ϑcVλ+D0+ϱ,ϑcVλD0+δ,ϑcUλ.
Hence, this completes the proof.Remark.
Applying Theorem5 to ϱ=δ results in Theorem 4.Corollary 6.
Settingϱ=δ=1, then under the assumptions of Theorem 5, we have(73)D0+ϑcUVλ≥Γnϑλn−1D0+ϑcUλD0+ϑcVλ.Corollary 7.
Settingϑλ=λ, then under the assumption of Theorem 5, we have inequality for Caputo fractional derivative:(74)λn−δD0+ϱcUVλΓn−δ+1+λn−D0+δcUVλΓn−ϱ+1≥D0+ϱcUλD0+δcVλ+D0+ϱcVλD0+δcUλ.Theorem 6.
Forn∈ℕ,ϱ≥1, and let Ujn for 1≤j≤κ be real-valued increasing functions defined on 0,∞. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0. Then, for all λ>0, we have(75)D0+ϱ,ϑc∏j=1κUjλ≥Γn−ϱ+1ϑλn−κ−1∏j=1κD0+ϱ,ϑcUjλ.Proof.
To prove the present theorem, we use mathematical induction onκ∈ℕ. Clearly, the case κ=1 of (75) holds.
Forκ=2, since U1n,U2n are increasing, we have(76)U1nλ−U1nω,U2nλ−U2nω≥0.
Now, the left part of inequality (75) for κ=2 is the same as that of Theorem 4.
Suppose that inequality (82) holds for some κ≥3. We observe that, since Ujn is increasing, U=∏j=1κUjn is increasing. Let V=Uκ+1n. Then, applying the case κ=2 to the function U and V produces(77)D0+ϱ,ϑc∏j=1κUjUκ+1λ≥Γn−ϱ+1ϑλn−ϱD0+ϱ,ϑc∏j=1κUjD0+ϱ,ϑcUκ+1λ≥Γn−ϱ+1ϑλn−ϱκ∏j=1κ+1D0+ϱ,κcUjλ,in which the induction hypothesis for κ is used inside the deduction of second inequality. The proof of Theorem 6 is complete.Corollary 8.
LetUjn for 1≤j≤κ be real-valued increasing functions defined on 0,∞. For ϱ>0, we have(78)D0+ϑc∏j=1κUjλ≥Γnϑλn−1κ−1∏j=1κD0+ϑcUjλ.Proof.
This follows from takingϱ=1 in Theorem 6.Corollary 9.
If we chooseϑλ=λ, then under the assumptions of Theorem 6, we have a new result for Caputo fractional derivative operator:(79)D0+ϱc∏j=1κUjλ≥Γn−ϱ+1λn−ϱκ−1∏j=1κD0+ϱcUjλ.Theorem 7.
Forn∈ℕϱ≥1, let U,V be two absolutely continuous mappings on 0,∞ such that Un is increasing, Vn is differentiable, and Vn+1 is a lower bound c=infμ∈0,∞Vn+1μ. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0. Then, for all λ>0, we have(80)D0+ϱ,ϑcUVλ≥Γn−ϱ+1ϑλn−ϱD0+ϱ,ϑcUλD0+ϱ,ϑcVλ−θλϑλn−ϱΓn−ϱ+1D0+ϱ,ϑcUλ+θD0+ϱ,ϑcℐUλ,where ℐλ is the identity function.Proof.
Lethλ=Vλ−θλ. We shall show that h is differentiable and increasing on 0,∞. As we did in the proof of Theorem 6, for clarity, let Υλ=θλ, and we find(81)D0+ϱ,ϑcUV−Υλ≥Γn−ϱ+1ϑλn−ϱD0+ϱ,ϑcUλD0+ϱ,ϑcV−Υλ=Γn−ϱ+1ϑλn−ϱD0+ϱ,ϑcUλD0+ϱ,ϑcVλ−Γn−ϱ+1ϑλn−ϱD0+ϱ,ϑcUλD0+ϱ,ϑcΥλ,where(82)D0+ϱ,ϑcUV−Υλ=D0+ϱ,ϑcUVλ−θD0+ϱ,ϑcℐUλ,(83)D0+ϱ,ϑcΥλ=θλϑλn−ϱΓn−ϱ+1.
Substituting (82) and (83) into (81) leads to the desired results.Corollary 10.
If we chooseϱ=1, then under the assumption of Theorem 7, we have(84)D0+ϑcUVλ≥Γnϑλn−1D0+ϑcUλD0+ϑcVλ−θλϑλnΓnD0+ϑcUλ+θD0+ϑcℐUλ,where ℐλ is the identity function.Corollary 11.
If we chooseϑλ=λ, then under the assumption of Theorem 7, we have a new result for Caputo fractional derivative operator:(85)D0+ϱcUVλ≥Γn−ϱ+1λn−D0+ϱcUλD0+ϱcVλ−θλn−ϱ+1Γn−ϱ+1D0+ϱcUλ+θD0+ϱcℐUλ,where ℐλ is the identity function.


## 5. Gru¨ss Type Inequalities for the Caputo Fractional Derivative in the ϑ-Hilfer Sense Operators

In this section, we prove some Gru¨ss type inequalities involving the Caputo fractional derivative in the ϑ-Hifer sense operator defined in (15).Theorem 8.
Forn∈ℕ,ϱ,δ>0, and let there be an absolutely continuous function U defined on 0,∞. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0. Suppose that there exist two integrable functions Υ1n,Υ2n on 0,∞ such that(86)Υ1nλ≤Unλ≤Υ2nλ,∀λ∈0,∞.Then, we obtain the following inequality for the generalized Caputo fractional integral operator:(87)D0+ϑ,δcΥ1λcD0+ϑ,Uλ+D0+ϑ,ϱcΥ2λD0+ϑ,δcUλ≥D0+ϑ,ϱcΥ2λD0+ϑ,δcΥ1λ+D0+ϑ,ϱcUλD0+ϑ,δcUλ.Proof.
From (86), for all r≥0,s≥0, we have(88)Υ2nr−UnrUns−Υ1ns≥0,Υ2nrUns+Υ1nsUnrΥ1nsΥ2nr+UnrUns.
If we multiply both sides of (86) by ϑλ−ϑrn−ϱ−1ϑ′r/Γn−ϱ and integrating with respect to r on 0,λ, we obtain(89)Uns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rΥ2nrdr+Υ1ns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rUnrdr≥Υ1ns1Γn−ϱ∫0λ∫0λϑλ−ϑrn−ϱ−1ϑ′rΥ2nrdr+Uns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rUnrdr,which can be written as follows:(90)UnsD0+ϑ,ϱcΥ2λ+Υ1nsD0+ϑ,ϱcUλ≥Υ1nsD0+ϑ,ϱcΥ2λ+UnsD0+ϑ,ϱcUλ.
If we multiply both sides of (90) bs ϑλ−ϑsn−δ−1ϑ′s/Γn−δ and integrating with respect to s on 0,λ, we obtain(91)D0+ϑ,δcΥ1λD0+ϑ,ϱcUλ+D0+ϑ,ϱcΥ2λD0+ϑ,δcUλ≥D0+ϑ,ϱcΥ2λD0+ϑ,δcΥ1λ+D0+ϑ,ϱcUλD0+ϑ,δcUλ.
Hence, this completes the proof.Corollary 12.
Letϑλ=λ in Theorem 8; then, we have the inequality for Caputo-type fractional derivative operator:(92)D0+δcΥ1λD0+ϱcUλ+D0+ϱcΥ2λD0+δcUλ≥D0+ϱcΥ2λD0+δcΥ1λ+D0+ϱcUλD0+δcUλ.Corollary 13.
LetU be an absolutely continuous on 0,∞. Suppose that q≤Unλ≤Q, for all λ∈0,∞ and q,Q∈ℛ. Then, for λ>0,ϱ>0,δ>0, we have(93)qϑn−δλΓn−δ+1D0+ϑ,ϱcUλ+Qϑn−ϱλΓn−ϱ+1D0+ϑ,δcUλ≥qQϑn−δλΓn−δ+1ϑn−λΓn−ϱ+1+D0+ϑ,ϱcUλD0+ϑ,δcUλ.Theorem 9.
Forn∈ℕ,ϱ,δ>0, and let there be an absolutely continuous function U defined on 0,∞. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0. Suppose that (86) holds, and moreover assume that there exist φ1n and φ2n integrable functions on 0,∞ such that(94)φ1nλ≤Vnλ≤φ2nλ,∀λ∈0,∞.Then, the following inequalities hold for generalized Caputo fractional derivative operator:(95)aD0+ϑ,δcφ1λD0+ϑ,ϱcUλ+D0+ϑ,ϱcΥ2λD0+ϑ,δcVλ≥D0+ϑ,δcφ1λD0+ϑ,ϱcΥ2λ+D0+ϑ,ϱcUλD0+ϑ,δcVλ,bD0+ϑ,δcΥ1λD0+ϑ,ϱcVλ+D0+ϑ,ϱcφ2λD0+ϑ,δcUλ≥D0+ϑ,δcΥ1λD0+ϑ,ϱcφ2λ+D0+ϑ,δcUλD0+ϑ,ϱcVλ,cD0+ϑ,ϱcΥ2λD0+ϑ,δcφ2λ+D0+ϑ,ϱcUλcD0+ϑ,δVλ≥D0+ϑ,ϱcΥ2λD0+ϑ,δcVλ+D0+ϑ,δcφ2λD0+ϑ,ϱcUλ,dD0+ϑ,ϱcΥ1λD0+ϑ,δcφ1λ+D0+ϑ,ϱcUλD0+ϑ,δcVλ≥D0+ϑ,ϱcΥ1λD0+ϑ,δcVλ+cD0+ϑ,δφ1λD0+ϑ,ϱcUλ.Proof.
From (86) and (94) for all r≥0,s≥0, we have(96)Υ2nr−UnrVns−φ1ns≥0,then(97)Υ2nrVns+φ1nsUnr≥φ1nsΥ2nr+UnrVns.
If we multiply both sides of (97) by ϑλ−ϑrn−ϱ−1ϑ′r/Γn−ϱ, and integrating with respect to r on 0,λ, we have(98)Vns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rΥ2nrdr+φ1ns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rUnrdr≥φ1ns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rΥ2nrdr+Vns1Γn−ϱ∫0λϑλ−ϑrn−ϱ−1ϑ′rUnrdr.
It can be written as(99)VnsD0+ϑ,ϱcΥ2λ+φ1nsD0+ϑ,ϱcUλ≥φ1nsD0+ϑ,ϱcΥ2λ+VnsD0+ϑ,ϱcUλ.
If we multiply both sides of (99) by ϑλ−ϑsn−δ−1ϑ′s/Γn−δ and integrating with respect to s on 0,λ, we obtain(100)D0+ϑ,δcφ1λD0+ϑ,ϱcUλ+D0+ϑ,ϱcΥ2λD0+ϑ,δcVλ≥D0+ϑ,δcφ1λD0+ϑ,ϱcΥ2λ+D0+ϑ,ϱcUλD0+ϑ,δcVλ.
This provesa.
To proveb−d, we use the following inequalities:(101)bφ2nr−VnrUnsΥ1ns≥0,cΥ2nr−UnrVns−φ2ns≤0,dΥ1nr−UnrVns−φ1ns≤0.
The following inequalities are special cases of Theorem9.Corollary 14.
LetU and V be two absolutely continuous on 0,∞,λ,ϱ,δ>0. Suppose that there exist real constants q,Q,p,P such that(102)q≤Unλ≤Q,p≤Vnλ≤P,∀λ∈0,∞.Then, we have(103)ipϑn−δλΓn−δ+1D0+ϑ,ϱcUλ+Qϑn−ϱλΓn−ϱ+1D0+ϑ,δcVλ≥pQϑn−δλΓn−δ+1ϑn−ϱλΓn−ϱ+1+D0+ϑ,ϱcUλD0+ϑ,δcVλ,iiqϑn−δλΓn−δ+1D0+ϑ,ϱcVλ+Pϑn−ϱλΓn−ϱ+1D0+ϑ,δcUλ≥qPϑn−δλΓn−δ+1ϑn−ϱλΓn−ϱ+1+D0+ϑ,δcUλD0+ϑ,ϱcVλ,iiiPQϑn−δλΓn−δ+1ϑn−ϱλΓn−ϱ+1+D0+ϑ,ϱcUλD0+ϑ,δcVλ≥Qϑn−ϱλΓn−ϱ+1D0+ϑ,δcVλ+Pϑn−δλΓn−δ+1D0+ϑ,ϱcUλ,ivpqϑn−δλΓn−δ+1ϑn−ϱλΓn−ϱ+1+D0+ϑ,ϱcUλD0+ϑ,δcVλ≥qϑn−ϱλΓn−ϱ+1D0+ϑ,δcVλ+pϑn−δλΓn−δ+1D0+ϑ,ϱcUλ.Corollary 15.
LetU,V∈L10,∞ and ϑλ=λ,λ>0,ϱ>0,δ>0. Suppose that there exist real constants q,Q,p,P, such that(104)q≤Unλ≤Q,p≤Vnλ≤P,∀λ∈0,∞.Then, we have new inequalities for Caputo fractional derivative operator:(105)aD0+δcφ1λD0+ϱcUλ+D0+ϱcΥ2λD0+δcVλ≥D0+δcφ1λD0+ϱcΥ2λ+D0+ϱcUλD0+δcVλ,bD0+δcΥ1λD0+ϱcVζ+D0+ϱcφ2λD0+δcUλ≥D0+δcΥ1λD0+ϱcφ2λ+D0+δcUλD0+ϱcVλ,cD0+ϱcΥ2λD0+δcφ2λ+D0+ϱcUλD0+δcVλ≥D0+ϱcΥ2λD0+δcVλ+D0+δcφ2λD0+ϱcUλ,dD0+ϱcΥ1λD0+δcφ1λ+D0+ϱcUλD0+δcλ≥D0+ϱcΥ1λD0+δcVλ+D0+δcφ1λD0+ϱcUλ.Example 1.
Forn∈ℕ,ϱ,δ>0, and let there be an absolutely continuous function U and V defined on 0,∞. Also, assume that ϑ be differentiable and strictly increasing on 0,∞ with ϑ0=0, and p,q>0 satisfying p+q=1. Then, for λ>0, one has(106)a∗pD0+ϑ,δcUλD0+ϑ,ϱcVλ+qD0+ϑ,ϱcUλD0+ϑ,δcVγ≥D0+ϑ,δcUpλVqλD0+ϑ,ϱcUqλVpλ,b∗pD0+ϑ,δcUp−1λD0+ϑ,ϱcUλVqλ+qD0+ϑ,ϱcVq−1λD0+ϑ,δcUqλVλ≥D0+ϑ,δcVqλD0+ϑ,ϱcUpλ,c∗pD0+ϑ,δcUλD0+ϑ,ϱcV2/pλ+qD0+ϑ,δcVλD0+ϑ,ϱcU2/pλ≥D0+ϑ,δcUpλVλD0+ϑ,ϱcVqλU2λ,d∗pD0+ϑ,δcU2/pλVqλD0+ϑ,ϱcVp−1λ+qD0+ϑ,δcVq−1λD0+ϑ,ϱcU2/pλVpλ≥D0+ϑ,δcU2λD0+ϑ,ϱcV2λ.Proof.
From the well-known weightedAM−GM inequality,(107)pa+qb≥apbq,∀a,b≥0,p,q>0,p+q=1.
By settinga=UnrVns and b=UnsVnr,s,r>1, we have(108)pUnrVns+qUnsVnr≥UnrVnspUnsVnrq.
Multiplying both sides of (108) by ϑ′sϑλ−ϑsn−δ−1ϑ′rϑλ−ϑrn−ϱ−1/Γn−δΓn−ϱ, which is positive because r,s∈0,λ,λ>0, and integrating the resulting identity from 0 to λ, we have(109)pΓn−δΓn−ϱ∫0λ∫0λϑ′sϑλ−ϑsn−δ−1ϑ′rϑλ−ϑrn−ϱ−1UnrVnsdrds+qΓn−δΓn−ϱ∫0λ∫0λϑ′sϑλ−ϑsn−δ−1ϑ′rϑλ−ϑrn−ϱ−1UnsVnrdrds≥1Γn−δΓn−ϱ∫0λ∫0λϑ′sϑλ−ϑsn−δ−1ϑ′rϑλ−ϑrn−ϱ−1×UnrVnspUnsVnrqdsdr,
We conclude that(110)pD0+ϑ,δUλD0+ϑ,ϱVλ+qD0+ϑ,ϱUλD0+ϑ,δVλ≥D0+ϑ,δUpλVqλD0+ϑ,ϱUqλVpλ,which implies a∗. The rest of inequalities can be shown in a similar way by the following choice of parameters in AM−GM inequality:(111)b∗a=UnsUnr,b=VnrVns,Unr,Vns≠0,c∗a=UnrVn2/qs,b=Un2/qsVnr,d∗a=Un2/prVns,b=Un2/qsVnr,Vnr,Vnr≠0.


## 6. Conclusion

The main objective of this paper will be a motivation source for future studies. We established some new generalizations for Hermite–Hadamard type pertainingnth-order differentiability for convex functions via Caputo fractional operator in the ϑ-Hilfer sense. To this date, this is the novel version of the Grüss- and Čebyšev-type inequalities for two synchronous functions via the Caputo fractional derivative in the ϑ-Hilfer sense. These estimates, bounds, and inequalities hold for all fractional operators mentioned in Remark 1. We conclude this paper by emphasizing, again, that our main result here, being of a very general in nature, can be specialized to yield numerous interesting fractional integral inequalities. Furthermore, they are expected to find some applications for establishing the uniqueness of solutions in fractional boundary value problems in the fractional partial differential equations.



---

*Source: 1626091-2020-10-27.xml*