Fractional Integral Identity Research Articles

Some New Refinements of Hermite–Hadamard-Type Inequalities Involvingψk-Riemann–Liouville Fractional Integrals and Applications

The main objective of this article is to establish some new fractional refinements of Hermite–Hadamard-type inequalities essentially using newψk-Riemann–Liouville fractional integrals, wherek>0. Using this new fractional integral, we also derive two new fractional integral identities. Applications of the obtained results are also discussed.

Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications

In this paper, we establish a local fractional integral identity with a parameter on Yang's fractal sets. Using this identity, by generalized power mean inequality and generalized Hölder inequality, two Hermite‐Hadamard type local fractional integral inequalities for generalized harmonically convex functions are established. By giving some special values to the parameter, some inequalities with specific form can be obtained. Some applications to generalized special means are given.

Fractional integral identity, estimation of its bounds and some applications to trapezoidal quadrature rule

The aim of this paper is to introduce a new extension of preinvexity called exponentially (m,?1,?2, h1,h2)-preinvexity. Some new integral inequalities of Hermite-Hadamard type for exponentially (m,?1,?2,h1,h2)-preinvex functions via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for exponentially (m,?1,?2,h1,h2)-preinvex functions via general fractional integrals are obtained. We show that the class of exponentially (m,?1,?2, h1,h2)-preinvex functions includes several other classes of preinvex functions. We shown by two basic examples the efficiency of the obtained inequalities on the base of comparing those with the other corresponding existing ones. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.

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.

Ostrowski type inequalities via the Katugampola fractional integrals

The main aim of this study is to reveal new generalized-Ostrowski-type inequalities using Katugampola fractional integral operator which generalizes Riemann-Liouville and Hadamard fractional integral operators into a single form. For this purpose, at first, a new fractional integral identity is generated by the researchers. Then, by using this identity, some inequalities for the class of functions whose certain powers of absolute values of derivatives are $p-$convex are derived. Some applications to special means for positive real numbers are also given. It is observed that the obtained inequalities are generalizations of some well known results.

Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex

In this paper, we establish a new fractional integral identity, and then wederive some new fractional Hermite-Hadamard type inequalities for functionswhose derivatives are s-preinvex.

Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel

In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals with exponential kernels for convex functions. Finally, some applications to special means of real number are presented.

Some k-fractional extensions of the trapezium inequalities through generalized relative semi-(m,h)-preinvexity

In this paper, we first introduce the concept of generalized relative semi--preinvex functions, and then a new k-Riemann–Liouville fractional integral identity for differentiable mapping is derived. With the help of this identity, we present some new bounds on trapezium inequalities via generalized relative semi--preinvexity. It is pointed out that some new and known special cases can be deduced from main results of the article.

Fractional Hermite-Hadamard type inequalities for functions whose derivatives are extended s-(?,m)-preinvex

In this paper, we introduce the class of extended s-(alpha,m)-preinvex functions. We establish a new fractional integral identity and derive some new fractional Hermite-Hadamard type inequalities for functions whose derivatives are in this novel class of function.

On the Hermite\u2013Hadamard type inequality for \u03c8-Riemann\u2013Liouville fractional integrals via convex functions

In this paper, we establish a new Hermite–Hadamard inequality involving left-sided and right-sided ψ-Riemann–Liouville fractional integrals via convex functions. We also show two basic ψ-Riemann–Liouville fractional integral identities including the first order derivative of a given convex function, and these will be used to derive estimates for some fractional Hermite–Hadamard inequalities. Finally, we give some applications to special means of real numbers.

Certain Bounds Related to Multi-Parameterized $k$ -Fractional Integral Inequalities and Their Applications

A $k$ -fractional integral identity with multiple parameters is investigated. Based on this identity, some estimation-type results related to $k$ -fractional integral inequalities for the first-order differentiable functions are obtained. These results are then applied to the estimation of cumulative distribution function and some other special means.

Generalization of different type integral inequalities for generalized (𝑠, 𝑚)-preinvex Godunova–Levin functions

Abstract In the present paper, the notion of generalized ( s , m ) {(s,m)} -preinvex Godunova–Levin function of second kind is introduced, and some new integral inequalities involving generalized ( s , m ) {(s,m)} -preinvex Godunova–Levin functions of second kind along with beta function are given. By using a new identity for fractional integrals, some new estimates on generalizations of Hermite–Hadamard, Ostrowski and Simpson type inequalities for generalized ( s , m ) {(s,m)} -preinvex Godunova–Levin functions of second kind via Riemann–Liouville fractional integral are established.

Generalization of Hermite-Hadamard Type Inequalities via Conformable Fractional Integrals

We establish a Hermite-Hadamard type identity and several new Hermite-Hadamard type inequalities for conformable fractional integrals and present their applications to special bivariate means.

Riemann-Liouville fractional trapezium-like inequalities via generalized (m, h 1 , h 2 )-preinvexity

In this paper, we derive a fractional integral identity concerning three times differentiable generalized preinvex mappings defined on minvex set. By using of this identity, we obtain new estimates on generalization of trapezium-like inequalities for functions whose third order derivatives are generalized (m, h₁, h₂)-preinvex via Riemann-Liouville fractional integrals. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

A fractional integral identity and its application to fractional Hermite-Hadamard type inequalities

In this paper, we establish a new important integral identity for a given function involving Riemann-Liouville fractional integrals. By using this fundamental integral identity, convexity of the functions and elementary inequalities such as Hölder inequality, several new fractional Hermite-Hadamard type inequalities are obtained. Finally, we apply these inequalities to Some special means.

New inequalities on Hermite-Hadamard utilizing fractional integrals

In the present article, firstly authors have established an integral identity for Riemann-Liouville fractional integrals. Secondly, some Hermite-Hadamard type integral inequalities utilizing this integral identity are obtained and presented results have some closely connection with [M. Z. Sarikaya, N. Aktan, On the generalization of some integral inequalities and their applications, Mathematical and Computer Modeling 54(9) (2011), 2175-2182].

Hermite-Hadamard type inequalities for n-times differentiable convex functions via Riemann-Liouville fractional integrals

In this paper, we establish an identity for n-times differentiable functions via Riemann-Liouville fractional integrals. By using this new identity, we have some new results about trapezoid type inequalities for n-times differentiable convex functions via Riemann-Liouville fractional integrals. The results, given here extended the results given the previous works.

Some New Fractional Fejér Type Inequalities for Convex Functions

Abstract In this paper, firstly, a new identity for conformable fractional integrals is established. Then by making use of the established identity, some new fractional Fejér type inequalities are established. The results presented here have some relationships with the results of Set et al. (2015), proved in [6].

Hermite–Hadamard type inequalities for conformable fractional integrals

In this paper, first, we prove an identity for conformable fractional integrals. Second, by using this identity we will present some integral inequalities connected with the left hand side of the Hermite–Hadamard type inequalities for conformable fractional integrals. At the end, applications to some special means and error estimates for the midpoint formula are provided.

Inequalities for \u03b1-fractional differentiable functions

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.