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

Department of Mathematics, Government College University, Faisalabad, Pakistan Department of Mathematics, University of Jhang, Jhang, Pakistan Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawser 11991, Saudi Arabia Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia Department of Medical Research, China Medical University, Taiwan 40402 Taichung, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

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. ose 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 function U: I ⟶ R is said to be convex function, if the following inequality holds: for all τ 1 , τ 2 ∈ 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, Hardytype inequality, Ostrowski-type inequality, and Gagliardo-Nirenberg-type inequality, but the most celebrated and significant inequality is the Hermite-Hadamard-type inequality [26][27][28][29], which is defined as 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][31][32][33][34][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. e 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 that C ⌣ ebys ⌣ ev and Gr € uss 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 Gr € uss [39], our aim is to show more general versions of C ⌣ ebys ⌣ ev and Gr € uss type inequalities. [38] introduced the well-known celebrated functional and is defined as follows: where U and V are two integrable functions on [τ 1 , τ 2 ]. If U and V are synchronous, i.e., for any λ, ω ∈ [τ 1 , τ 2 ], then s(U, 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][41][42]).
e 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 Gr € uss 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.

Preliminaries
In this sequel, we introduce a few notations and definitions of fractional calculus and present initial results wished in our proofs later.

Fractional Derivative in the ϑ-Hilfer Sense Operators
In this section, we present several C ⌣ ebys ⌣ ev type inequalities for Caputo fractional derivative in the ϑ-Hilfer sense operator defined in (15).
Proof. Since U (n) and V (n) are synchronous on [0, ∞), we have If we multiply both sides of inequality (55) by Journal of Mathematics Further integrating both sides with respect to r over (0, λ) gives Consequently, it follows that If we multiply both sides of inequality (60) by we arrive at Now, integrating over (0, λ) reveals

Corollary 5. Setting ϑ(λ) � λ, then under the assumption of eorem 4, we have a new result for the Caputo fractional derivative operator:
Proof. Using inequality (61) and multiplying both sides by Furthermore, integrating both sides with respect to s over (0, λ) leads to Hence, this completes the proof.

Corollary 6. Setting ϱ � δ � 1, then under the assumptions of eorem 5, we have
Theorem 6. For n ∈ N, ϱ ≥ 1, and let U (n) j for 1 ≤ j ≤ κ be real-valued increasing functions defined on [0, ∞). Also, assume that ϑ be differentiable and strictly increasing on Proof. To prove the present theorem, we use mathematical induction on κ ∈ N. Clearly, the case κ � 1 of (75) holds. For κ � 2, since U (n) 1 , U (n) 2 are increasing, we have Now, the left part of inequality (75) for κ � 2 is the same as that of eorem 4.
Suppose that inequality (82) holds for some κ ≥ 3. We observe that, since U (n) κ+1 . en, applying the case κ � 2 to the function U and V produces in which the induction hypothesis for κ is used inside the deduction of second inequality. e proof of eorem 6 is complete.
Corollary 11. If we choose ϑ(λ) � λ, then under the assumption of eorem 7, we have a new result for Caputo fractional derivative operator: where I(λ) is the identity function.

Gr € uss Type Inequalities for the Caputo Fractional Derivative in the ϑ-Hilfer Sense Operators
In this section, we prove some Gr € uss type inequalities involving the Caputo fractional derivative in the ϑ-Hifer sense operator defined in (15).

Conclusion
e main objective of this paper will be a motivation source for future studies. We established some new generalizations for Hermite-Hadamard type pertaining nth-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-andČebyšev-type inequalities for two synchronous functions via the Caputo fractional derivative in the ϑ-Hilfer sense. ese 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.

Data Availability
No data used in this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.