Fractional Integral Inequalities Research Articles

Certain new proportional and Hadamard proportional fractional integral inequalities

The main goal of this paper is estimating certain new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions by employing proportional fractional integral (PFI) and Hadamard proportional fractional integral. We establish certain inequalities concerning one- and two-parameter proportional and Hadamard proportional fractional integrals. We also discuss certain particular cases.

New generalization of reverse Minkowski's inequality for fractional integral

In this research, we introduce some new fractional integral inequalities of Minkowski’s type by using Riemann-Liouville fractional integral operator. We replace the constants that appear on Minkowski’s inequality by two positive functions. Further, we establish some new fractional inequalities related to the reverse Minkowski type inequalities via Riemann-Liouville fractional integral. Using this fractional integral operator, some special cases of reverse Minkowski type are also discussed.

Caputo–Fabrizio fractional Hermite–Hadamard type and associated results for strongly convex functions

The study of fractional integral inequalities has attracted the interests of many researchers due to their potential applications in various fields. Estimates obtained via strongly convex functions produce better and sharper bounds when compared to convex functions. To this end, we establish some new Hermite–Hadamard and Fejer types inequalities by means of the Caputo–Fabrizio fractional integral operators for strongly convex functions. In particular, we prove among other things that if $$\omega :\mathfrak {I}\rightarrow {\mathbb {R}}$$ is a strongly convex function with modulus $$c>0$$ and $$\alpha ,\beta \in \mathfrak {I}$$ with $$\alpha <\beta $$ , then $$\begin{aligned} \begin{aligned}&\omega \left( \frac{\alpha +\beta }{2}\right) +\frac{c}{12}(\beta -\alpha )^2\\&\quad \le \frac{B(\mu )}{\mu (\beta -\alpha )}\left[ {^{cf}{\mathcal {I}}}_{\alpha }^{\mu }\omega (s)+^{cf}{\mathcal {I}}_{\beta }^{\mu }\omega (s)-\frac{2(1-\mu )}{B(\mu )}\omega (s)\right] \\&\quad \le \frac{\omega (\alpha )+\omega (\beta )}{2}-\frac{c}{6}(\beta -\alpha )^2, \end{aligned} \end{aligned}$$ where $$\mu \in (0,1]$$ , $$s\in \mathfrak {I}$$ and $$B(\mu )>0$$ is a normalization function. Some applications to special means have also been investigated.

Refinements and Generalizations of Some Fractional Integral Inequalities via Strongly Convex Functions

In this article, we have established the Hadamard inequalities for strongly convex functions using generalized Riemann–Liouville fractional integrals. The findings of this paper provide refinements of some fractional integral inequalities. Furthermore, the error bounds of these inequalities are given by using two generalized integral identities.

Generalized fractional Ostrowski type integral inequalities for logarithmically h-convex function

The authors have tried to prove a fractional integral identity generalizing already some existing identities. Based on, some Ostrowski-type fractional integral inequalities for r-times differentiable functions have been discussed, generalizing some existing results in the literature. As a consequence, some new Ostrowski-type, Hermite–Hadamard type and mid point type inequalities have been derived.

Hadamard and Fejér–Hadamard Inequalities for Further Generalized Fractional Integrals Involving Mittag-Leffler Functions

In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.

Inequalities for a Unified Integral Operator for Strongly α , m -Convex Function and Related Results in Fractional Calculus

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex and α , m -convex functions. A new definition of function, namely, strongly α , m -convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.

Conformable fractional Hermite-Hadamard type inequalities for product of two harmonic 𝑠-convex functions

In this paper, we establish some conformable fractional Hermite-Hadamard type integral inequalities via harmonic s s -convexity, and the estimates of the products of two harmonic s s -convex functions are also considered.

New Ostrowski type conformable fractional inequalities concerning differentiable generalized relative semi-$(r; m, p, q, h_1, h_2)$-preinvex mappings

In this article, we first presented a new integral identity concerning differentiable mappings defined on m-invex set. By using the notion of generalized relative semi-$(r; m, p, q, h_1, h_2)$-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Ostrowski type conformable fractional integral inequalities are established. It is pointed out that some new special cases can be deduced from main results of the article.

General Raina fractional integral inequalities on coordinates of convex functions

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an (l_{1},h_{1})-(l_{2},h_{2})-convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the (l_{1},h_{1})-(l_{2},h_{2})-convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

LOCAL FRACTIONAL OSTROWSKI-TYPE INEQUALITIES INVOLVING GENERALIZED h-CONVEX FUNCTIONS AND SOME APPLICATIONS FOR GENERALIZED MOMENTS

In this paper, we establish some local fractional Ostrowski-type integral inequalities for generalized [Formula: see text]-convex functions on real linear fractal set [Formula: see text]. We present two examples to illustrate our main results. As applications, we establish some local fractional integral inequalities involving generalized moments of continuous random variables to estimate the bounds of generalized moments.

Some General Integral Operator Inequalities Associated with φ -Quasiconvex Functions

This paper deals with generalized integral operator inequalities which are established by using φ -quasiconvex functions. Bounds of an integral operator are established which have connections with different kinds of known fractional integral operators. All the results are deducible for quasiconvex functions. Some fractional integral inequalities are deduced.

Some k-fractional extension of Gr\xfcss-type inequalities via generalized Hilfer\u2013Katugampola derivative

In this paper, we prove several inequalities of the Grüss type involving generalized k-fractional Hilfer–Katugampola derivative. In 1935, Grüss demonstrated a fascinating integral inequality, which gives approximation for the product of two functions. For these functions, we develop some new fractional integral inequalities. Our results with this new derivative operator are capable of evaluating several mathematical problems relevant to practical applications.

On the weighted fractional integral inequalities for Chebyshev functionals

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function mathcal{G} in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of omega (theta ) and mathcal{G}(theta ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of omega (theta ) and mathcal{G}(theta ).

Refinements of some integral inequalities for unified integral operators

In this paper we are presenting the refinements of integral inequalities established for convex functions. Consequently, we get refinements of several fractional integral inequalities for different kinds of fractional integral operators.

Some inequalities for multiplicative tempered fractional integrals involving the $ \lambda $-incomplete gamma functions

In this paper, we introduce a class of the multiplicative tempered fractional integral operators. Then, we investigate two Hermite–Hadamard type inequalities for this class. By using the established identity and the multiplicative convexity, we establish some integral inequalities for the multiplicative tempered fractional integrals involving the $ \lambda $-incomplete gamma functions. And our results obtained in the present paper generalize some results given by Budak and Tunc (2020) and Ali et al. (2019). Also, we provide three examples to demonstrate the simplicities of the calculations.

Hermite-Hadamard-Fejer type inequalities via fractional integral of a function concerning another function

In this paper, we at first develop a generalized integral identity by associating Riemann-Liouville (RL) fractional integral of a function concerning another function. By using this identity estimates for various convexities are accomplish which are fractional integral inequalities. From our results, we obtained bounds of known fractional results which are discussed in detail. As applications of the derived results, we obtain the mid-point-type inequalities. These outcomes might be helpful in the investigation of the uniqueness of partial differential equations and fractional boundary value problems.

Dynamical significance of generalized fractional integral inequalities via convexity

<abstract> The main goal of this paper is to develop the significance of generalized fractional integral inequalities via convex functions. We obtain the new version of fractional integral inequalities with the extended Wright generalized Bessel function acting as a kernel for the convex function, which deals with the Hermite-Hadamard type and trapezoid type inequalities. Moreover, we establish new mid-point type and trapezoid type integral inequalities for $ (\eta_{1}, \eta_{2}) $-convex function related to Hermite-Hadamard type inequality. We establish new version of integral inequality for $ (\eta_{1}, \eta_{2}) $-convex function related to Fejér type. The results discussed in this paper are a generalized version of many inequalities in literature. </abstract>

Hermite-Hadamard-Mercer type inequalities for fractional integrals

In the present note, we proved Hermite-Hadamard-Mercer inequalities for fractional integrals and we established some new fractional inequalities connected with the right and left-sides of Hermite-Hadamard-Mercer type inequalities for differentiable mappings whose derivatives in absolute value are convex.

Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions

&lt;abstract&gt;&lt;p&gt;In this paper Hadamard type inequalities for strongly $ (\alpha, m) $-convex functions via generalized Riemann-Liouville fractional integrals are studied. These inequalities provide generalizations as well as refinements of several well known inequalities. The established results are further connected with fractional integral inequalities for Riemann-Liouville fractional integrals of convex, strongly convex and strongly $ m $-convex functions. By using two fractional integral identities some more Hadamard type inequalities are proved.&lt;/p&gt;&lt;/abstract&gt;