Conformable Integral Operators Research Articles

Gronwall type inequality on generalized fractional conformable integral operators

Abstract In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 {0<\alpha\leq{1}} (Riemann approach), yielding Hadamard fractional integrals when α = 0 {\alpha=0} . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for generalized fractional operators are established, as well as solutions to initial value problems in several new weighted spaces of functions.

On Fractional Ostrowski-Mercer-Type Inequalities and Applications

The objective of this research is to study in detail the fractional variants of Ostrowski–Mercer-type inequalities, specifically for the first and second order differentiable s-convex mappings of the second sense. To obtain the main outcomes of the paper, we leverage the use of conformable fractional integral operators. We also check the numerical validations of the main results. Our findings are also validated through visual representations. Furthermore, we provide a detailed discussion on applications of the obtained results related to special means, q-digamma mappings, and modified Bessel mappings.

On some inequalities for uniformly convex mapping with estimations to normal distributions

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).

New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization

AbstractThe Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming∣f′∣| f^{\prime} |and∣f′∣q(q>1)| f^{\prime} {| }^{q}\hspace{0.33em}\left(q\gt 1)as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.

Several Integral Inequalities of Hermite–Hadamard Type Related to k-Fractional Conformable Integral Operators

In this paper, we present some ideas and concepts related to the k-fractional conformable integral operator for convex functions. First, we present a new integral identity correlated with the k-fractional conformable operator for the first-order derivative of a given function. Employing this new identity, the authors have proved some generalized inequalities of Hermite–Hadamard type via Hölder’s inequality and the power mean inequality. Inequalities have a strong correlation with convex and symmetric convex functions. There exist expansive properties and strong correlations between the symmetric function and various areas of convexity, including convex functions, probability theory, and convex geometry on convex sets because of their fascinating properties in the mathematical sciences. The results of this paper show that the methodology can be directly applied and is computationally easy to use and exact.

New integral inequalities for geometrically convex functions via conformable fractional integral operators

We establish several basic inequalities versions of the Hermite-Hadamard type inequalities for GA− and GG−convexity for conformable fractional integrals. Several special cases are also discussed, which can be deduced from our main result.

Inequalities for $tgs$-convex functions via some conformable fractional integrals

In this research article, we establish several Hermite-Hadamard type inequalities for tgs-convex functions via conformable fractional integrals and new fractional conformable integral operators.

On Opial-type inequalities via a new generalized integral operator

Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.

Unified integral operator inequalities via convex composition of two functions

In this paper we have established inequalities for a unified integral operator by using convexity of composition of two functions. The obtained results are directly connected to bounds of various fractional and conformable integral operators which are already known in literature. A generalized Hadamard integral inequality is obtained which further leads to its various versions for associated fractional integrals. Further, some implicated results are discussed.

On quantum hybrid fractional conformable differential and integral operators in a complex domain

On quantum hybrid fractional conformable differential and integral operators in a complex domain

On new integral inequalities using mixed conformable fractional integrals

During the past two decades or so, fractional integral operators have been one of the most important tools in the development of inequalities theory. By this means, a lot generalized intergral inequalities involving various the fractional integral operators have been presented in the literature. Very recently, mixed conformable fractional integral operators has been introduced by T. Abdeljawad and with the help of these operators some new integral inequalities are obtained. The main aim of the paper is to establish some new Chebyshev type fractional integral inequalities by using mixed conformable fractional integral operators.

Derivation of Bounds of an Integral Operator via Exponentially Convex Functions

In this paper, bounds of fractional and conformable integral operators are established in a compact form. By using exponentially convex functions, certain bounds of these operators are derived and further used to prove their boundedness and continuity. A modulus inequality is established for a differentiable function whose derivative in absolute value is exponentially convex. Upper and lower bounds of these operators are obtained in the form of a Hadamard inequality. Some particular cases of main results are also studied.

New generalized reverse Minkowski and related integral inequalities involving generalized fractional conformable integrals

This paper gives some novel generalizations by considering the generalized conformable fractional integrals operator for reverse Minkowski type and reverse Hölder type inequalities. Furthermore, novel consequences connected with this inequality, together with statements and confirmation of various variants for the advocated generalized conformable fractional integral operator, are elaborated. Moreover, our derived results are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions. The numerical approximations of our consequence have several utilities in applied sciences and fractional integro-differential equations.

Inequalities for a Unified Integral Operator via α,m-Convex Functions

Recently, a unified integral operator has been introduced by Farid, 2020, which produces several kinds of known fractional and conformable integral operators defined in recent decades (Kwun, 2019, Remarks 6 and 7). The aim of this paper is to establish bounds of this unified integral operator by means of α,m-convex functions. The resulting inequalities provide the bounds of all associated fractional and conformable integral operators in a compact form. Also, the results of this paper hold for different kinds of convex functions connected with α,m-convex functions.

Bounds of a Unified Integral Operator via Exponentially s,m-Convexity and Their Consequences

Various known fractional and conformable integral operators can be obtained from a unified integral operator. The aim of this paper is to find bounds of this unified integral operator via exponentially s,m-convex functions. The resulting bounds provide compact formulas for the bounds of associated fractional and conformable integral operators. Several Hadamard-type inequalities have been produced from a compact version for unified integral operators for exponentially s,m-convex functions.

Integral inequalities for s-convex functions via generalized conformable fractional integral operators

We introduce new operators, the so-called left and right generalized conformable fractional integral operators. By using these operators we establish new Hermite–Hadamard inequalities for s-convex functions and products of two s-convex functions in the second sense. Also, we obtain two interesting identities for a differentiable function involving a generalized conformable fractional integral operator. By applying these identities we give Hermite–Hadamard and midpoint-type integral inequalities for s-convex functions. Different special cases have been identified and some known results are recovered from our general results. These results may motivate further research in different areas of pure and applied sciences.

A modification to the conformable fractional calculus with some applications

In the conformable fractional calculus, TαTβ≠TβTα and IαIβ≠IβIα, where Tα and Iα are conformable fractional differential and integral operators, respectively. Also, Tβ≠Tnα and Iβ≠Inα, where β=nα for some n∈N. In this work, we introduce a modification to the definition of conformable fractional derivative and conformable fractional integral. Our definition characterizes by the realization of the commutative property between the differential operators and the integral operators. The higher derivatives in the sense of the modified conformable fractional derivative are coinciding with the sequential derivatives. Finally, we present the solutions of the Cauchy-Euler and linear (with constant coefficients) fractional differential equations to show our new approach.

On boundedness of unified integral operators for quasiconvex functions

This work deals with the bounds of a unified integral operator with which several fractional and conformable integral operators are directly associated. By using quasiconvex and monotone functions we establish bounds of these integral operators. We prove their boundedness and continuity. The results of this paper generalize already published results and have direct consequences for fractional and conformable integrals

Derivation of bounds of several kinds of operators via (s,m)-convexity

The objective of this paper is to derive the bounds of fractional and conformable integral operators for (s,m)-convex functions in a unified form. Further, the upper and lower bounds of these operators are obtained in the form of a Hadamard inequality, and their various fractional versions are presented. Some connections with already known results are obtained.

Bounds of a unified integral operator for (<i>s</i>,<i>m</i>)-convex functions and their consequences

The unified integral operator presented in Definition 4 produces several kinds of known fractional and conformable integral operators. The goal of this paper is to obtain bounds of this unified integral operator by using the definition of (s, m)-convexity. The resulting inequalities in specific cases represent the bounds of many known fractional and conformable fractional integral operators in a compact form.