Conformable Integral Operators Research Articles

Derivation of bounds of integral operators via convex functions

Convex functions play a vital role in the derivation of inequalities. In this paper these functions are used to obtain certain bounds of a unified integral operator. A Hadamard inequality for these operators is established. Further bounds of several kinds of fractional and conformable integral operators are deduced in particular.

Some unified bounds for exponentially $tgs$-convex functions governed by conformable fractional operators

In the article, we introduce the concept of the exponentially $tgs$-convex function and discover two new conformable fractional integral identities concerning the first-order differentiable convex mappings. By using these identities, we establish several new right-sided Hermite-Hadamard type inequalities for the exponentially $tgs$-convex functions via conformable fractional integrals. Our outcomes for conformable fractional integral operators are also applied to some special means.

New Hermite-Hadamard type inequalities for exponentially convex functions and applications

The investigation of the proposed techniques is effective and convenient for solving the integrodifferential and difference equations. The present investigation depends on two highlights; the novel Hermite-Hadamard type inequalities for $\mathcal{K}$-conformable fractional integral operator in terms of a new parameter $\mathcal{K}>0$ and weighted version of Hermite-Hadamard type inequalities for exponentially convex functions in the classical sense. By using an integral identity together with the Holder-Iscan and improved power-mean inequality we establish several new inequalities for differentiable exponentially convex functions. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. Our contribution expands some innovative studies in this line. Moreover, two suitable examples are presented to demonstrate the novelty of the results established, the first one about the contributions of the modified Bessel functions and the other is about $\sigma$-digamma function. Finally, various applications for some special means as arithmetic, geometric and logarithmic are given.

New extensions of Chebyshev-Pólya-Szegö type inequalities via conformable integrals

Recently, several papers related to integral inequalities involving various fractional integral operators have been presented. In this work, motivated essentially by the previous works, we prove some new Polya-Szego inequalities via conformable fractional integral operator and use them to prove some new fractional Chebyshev type inequalities concerning the integral of the product of two functions and the product of two integrals which are improvement of the results in the paper [Ntouyas, S.K., Agarwal, P. and Tariboon, J., On Polya-Szego and Chebyshev type inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal (see [ 9 ])].

A study on Hermite–Hadamard-type inequalities via new fractional conformable integrals

Due to their wide range of applications in many areas of modern analysis, the splendid Hermite–Hadamard inequalities for convex function are very attractive for researcher community. Many authors have been working on their variants, refinements and extensions. The aim of this research is to establish a novel identity and Hermite–Hadamard-type inequalities for convex functions linked with the renowned conformable integral operators. We have pointed some particular cases of the results obtained here in order to show that they reduce some well-known results to relatively simple ones.

Some inequalities of the Grüss type for conformable $${\varvec{k}}$$-fractional integral operators

In the paper, the authors establish several new inequalities of the Gruss type for conformable k-fractional integral operators. These inequalities generalize some known results.

Some new inequalities for generalized fractional conformable integral operators

The present paper aims to establish certain new classes of integral inequalities for a class of n (nin mathbb{N}) positive continuous and decreasing functions by utilizing the generalized fractional conformable integral operators (FCIO) recently defined by Khan and Khan. From these results, we also derive several particular cases.

Chebyshev type inequalities via generalized fractional conformable integrals

Our aim in this present paper is to establish several Chebyshev type inequalities involving generalized fractional conformable integral operator recently introduced by T.U. Khan and M.A. Khan (J. Comput. Appl. Math. 346:378–389, 2019). Also, we present Chebyshev type inequalities involving Riemann–Liouville type fractional conformable integral operators as a particular result of our main result.

Some inequalities via fractional conformable integral operators

In this paper, we adopt conformable fractional integral to develop integral inequalities such as Minkowski and Hermite–Hadamard inequalities. Our results are the generalization of the inequalities obtained by Dahmani and Bougoffa cited in the literature.

Certain Chebyshev-Type Inequalities Involving Fractional Conformable Integral Operators

Since an interesting functional by P.L. Chebyshev was presented in the year 1882, many results, which are called Chebyshev-type inequalities, have been established. Some of these inequalities were obtained by using fractional integral operators. Very recently, a new variant of the fractional conformable integral operator was introduced by Jarad et al. Motivated by this operator, we aim at establishing novel inequalities for a class of differentiable functions, which are associated with Chebyshev’s functional, by employing a fractional conformable integral operator. We also aim at showing important connections of the results here with those including Riemann–Liouville fractional and classical integrals.

The Minkowski inequality involving generalized k-fractional conformable integral

In the research paper, the authors exploit the definition of a new class of fractional integral operators, recently proposed by Jarad et al. (Adv. Differ. Equ. 2017:247, 2017), to define a new class of generalized k-fractional integral operators and develop a generalization of the reverse Minkowski inequality involving the newly introduced fractional integral operators. The two new theorems correlating with this inequality, including statements and verifications of other inequalities via the suggested k-fractional conformable integral operators, are presented.

Inequalities for a Unified Integral Operator and Associated Results in Fractional Calculus

Integral operators are useful in real analysis, mathematical analysis, functional analysis and other subjects of mathematical approach. The goal of this paper is to study a unified integral operator via convexity. By using convexity and conditions of unified integral operators, bounds of these operators are obtained. Furthermore consequences of these results are discussed for fractional and conformable integral operators.

Some new generalizations for m-convexity via new conformable fractional integral operators

Some new generalizations for m-convexity via new conformable fractional integral operators

Ostrowski type inequalities via new fractional conformable integrals

In this present study, firstly, some necessary definitions and some results related to Riemann-Liouville fractional and new fractional conformable integral operators defined by Jarad et al. [ 13 ] are given. As a second, a new identity has been proved. By using this identity, new Ostrowski type inequalities has obtained involving fractional conformable integral operators. Also, some new inequalities has established for AG-convex functions via fractional conformable integrals in this study. Relevant connections of the results presented here with those earlier ones are also pointed out.

Conformable fractional integral inequalities of Chebyshev type

A number of Chebyshev type inequalities involving various fractional integral operators have, recently, been presented. Here, motivated essentially by the earlier works and their applications in diverse research subjects, we aim to establish several Chebyshev type inequalities involving generalized new conformable fractional integral operator.

Some Inequalities of Čebyšev Type for Conformable k-Fractional Integral Operators

In the article, the authors present several inequalities of the Čebyšev type for conformable k-fractional integral operators.

Generalized conformable fractional operators

In 2015, Abdeljawad [1] has put an open problem, which is stated as: “Is it hard to fractionalize the conformable fractional calculus, either by iterating the conformable fractional derivative (Grunwald–Letnikov approach) or by iterating the conformable fractional integral of order 0<α≤1 (Riemann approach)?. Notice that when α=0 we obtain Hadamard type fractional integrals”. In this article we claim that yes it is possible to iterate the conformable fractional integral of order 0<α≤1 (Riemann approach), such that when α=0 we obtain Hadamard fractional integrals. First of all we prove Cauchy integral formula for repeated conformable fractional integral and proceed to define new generalized conformable fractional integral and derivative operators (left and right sided). We also prove some basic properties which are satisfied by these operators. These operators (integral and derivative) are the generalizations of Katugampola operators, Riemann–Liouville fractional operators, Hadamard fractional operators. We apply our results to a simple function. Also we consider a nonlinear fractional differential equation using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear problem. At the end, we give conclusion and point out an open problem.

Certain Hermite-Hadamard type inequalities associated with conformable fractional integral operators

The aim of this article is to obtain some new Hermite-Hadamard type inequalities for convex functions via conformable fractional integral. The results presented here would provide extensions of those given in earlier works.