Fractional Integral Identity Research Articles

An Extension of Left Radau Type Inequalities to Fractal Spaces and Applications

In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained results not only represent an extension of certain previously established findings to fractal sets but also a refinement of these when the fractal dimension μ is equal to one. Finally, to support our findings, we present a practical application to demonstrate the effectiveness of our results.

The multi-parameterized integral inequalities for multiplicative Riemann–Liouville fractional integrals

Using the multiplicative Riemann–Liouville fractional integrals, for ⁎⁎differentiable functions, we present a fractional integral identity together with multi-parameters. Relying on it, and the fact that the function f⁎⁎ is multiplicatively convex or (lnf⁎⁎)q is convex for q>1, in particular considering the case 0<q≤1, a series of three-point Newton–Cotes type inequalities are obtained in this article. To enable the readers to comprehend the results better, we provide three examples to support the findings obtained here. Some applications in special means are offered as well.

On the multiparameterized fractional multiplicative integral inequalities

AbstractWe introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus.

On parameterized inequalities for fractional multiplicative integrals

Abstract In this article, we present a one-parameter fractional multiplicative integral identity and use it to derive a set of inequalities for multiplicatively s s -convex mappings. These inequalities include new discoveries and improvements upon some well-known results. Finally, we provide an illustrative example with graphical representations, along with some applications to special means of real numbers within the domain of multiplicative calculus.

On fractional Bullen-type inequalities with applications

&lt;p&gt;Integral inequalities in mathematical interpretations are a substantial and ongoing body of research. Because fractional calculus techniques are widely used in science, a lot of research has recently been done on them. A key concept in fractional calculus is the Caputo-Fabrizio fractional integral. In this work, we focus on using the Caputo-Fabrizio fractional integral operator to build a multi-parameter fractional integral identity. Using the obtained integral identity, certain generalized estimates of Bullen-type fractional inequalities have been generated. By establishing certain inequalities, this study advances the fields of fractional calculus and convex function research. Both graphical and numerical statistics are provided to show the correctness of our results. We finally provide applications to modified Bessel functions, $ \mathfrak{h} $-divergence measures, and probability density functions.&lt;/p&gt;

ON GENERAL LOCAL FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED H-PREINVEX FUNCTIONS ON YANG’S FRACTAL SETS

In this paper, based on Yang’s fractal theory, the Hermite–Hadamard’s inequalities for generalized [Formula: see text]-preinvex function are proved. Then, using the local fractional integral identity proposed by Sun [Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals 27(5) (2019) 1950071] as auxiliary function, some parameterized local fractional integral inequalities for generalized [Formula: see text]-preinvex functions are established. For the special cases of the parameters, some generalized Simpson-type, midpoint-type and trapezoidal inequalities are established. Finally, some applications of these inequalities in numerical integration are proposed.

A STUDY OF FRACTIONAL HERMITE–HADAMARD–MERCER INEQUALITIES FOR DIFFERENTIABLE FUNCTIONS

In this work, we prove a parameterized fractional integral identity involving differentiable functions. Then, we use the newly established identity to establish some new parameterized fractional Hermite–Hadamard–Mercer-type inequalities for differentiable function. The main benefit of the newly established inequalities is that these inequalities can be converted into some new Mercer inequalities of midpoint type, trapezoidal type, and Simpson’s type for differentiable functions. Finally, we show the validation of the results with the help of some mathematical examples and their graphs.

Analysis and Applications of Some New Fractional Integral Inequalities

This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis.

Hermite–Hadamard type inequalities for multiplicative Riemann–Liouville fractional integrals

In this paper, we present a multiplicative fractional integral identity. Based upon it, we establish the Hermite–Hadamard type inequalities for multiplicatively convex functions via multiplicative Riemann–Liouville fractional integrals. We also give some applications to special means of real numbers in the framework of multiplicative calculus. To make the results obtained here more intuitive for the readers, we provide some examples for suitable choices of multiplicative convex functions and their graphical descriptions.

Some New Fractional Hadamard and Pachpatte-Type Inequalities with Applications via Generalized Preinvexity

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus.

Some novel refinements of Hermite-Hadamard and Pachpatte type integral inequalities involving a generalized preinvex function pertaining to Caputo-Fabrizio fractional integral operator

&lt;abstract&gt;&lt;p&gt;In this article, we aim to introduce and explore a new class of preinvex functions called $ \mathfrak{n} $-polynomial $ m $-preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for differentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus.&lt;/p&gt;&lt;/abstract&gt;

On Simpson’s and Newton’s type inequalities in multiplicative fractional calculus

In this paper, we prove two multiplicative fractional integral identities involving multiplicative differentiable functions. Then, with the help of newly established identities, we establish multiplicative fractional versions of Simpson?s and Newton?s formulas type inequalities for differentiable multiplicative convex functions. It is also shown that the newly proved inequalities are extensions of some existing inequalities within the literature.

Fractional Maclaurin-type inequalities for multiplicatively convex functions and multiplicatively P-functions

In this paper, we present a fractional integral identity, and then based upon it we establish the Maclaurin?s inequalities for multiplicatively convex functions and multiplicatively P-functions via multiplicative Riemann-Liouville fractional integrals.

On the multi-parameterized inequalities involving the tempered fractional integral operators

In virtue of the conception of the tempered fractional integrals, put forward by Sabzikar et al. in the published article [J. Comput. Phys., 293: 14-28, 2015], we present a fractional integral identity together with multi-parameter. Based on it, we develop certain parameterized integral inequalities in association with differentiable mappings. Furthermore, we give two examples to verify the correctness of the derived findings.

Some extended fractional integral inequalities with applications

Here, an extended fractional integral identity has been established to construct some extended Simpson-type inequalities for differentiable convex functions and differentiable concave functions connected to Hermite-Hadamard inequality. Some applications to means, f?divergence measure, probability density function and approximate error to some quadrature rules are given.

Some New Parameterized Quantum Fractional Integral Inequalities Involving s-Convex Functions and Applications

Convexity performs the appropriate role in the theoretical study of inequalities according to the nature and behavior. Its significance is raised by the strong connection between symmetry and convexity. In this article, we consider a new parameterized quantum fractional integral identity. By applying this identity, we obtain as main results some integral inequalities of trapezium, midpoint and Simpson’s type pertaining to s-convex functions. Moreover, we deduce several special cases, which are discussed in detail. To validate our theoretical findings, an example and application to special means of positive real numbers are presented. Numerical analysis investigation shows that the mixed fractional calculus with quantum calculus give better estimates compared with fractional calculus or quantum calculus separately.

Parameterized Quantum Fractional Integral Inequalities Defined by Using n-Polynomial Convex Functions

Convexity performs the appropriate role in the theoretical study of inequalities according to the nature and behaviour. There is a strong relation between symmetry and convexity. In this article, we consider a new parameterized quantum fractional integral identity. Following that, our main results are established, which consist of some integral inequalities of Ostrowski and midpoint type pertaining to n-polynomial convex functions. From our main results, we discuss in detail several special cases. Finally, an example and an application to special means of positive real numbers are presented to support our theoretical results.

Corrected Dual-Simpson-Type Inequalities for Differentiable Generalized Convex Functions on Fractal Set

The present paper provides several corrected dual-Simpson-type inequalities for functions whose local fractional derivatives are generalized convex. To that end, we derive a new local fractional integral identity as an auxiliary result. Using this new identity along with generalized Hölder’s inequality and generalized power mean inequality, we establish some new variants of fractal corrected dual-Simpson-type integral inequalities. Furthermore, some applications for error estimates of quadrature formulas as well as some special means involving arithmetic and p-logarithmic mean are offered to demonstrate the efficacy of our findings.

On some new generalized fractional Bullen-type inequalities with applications

In this paper, we establish a Bullen-type generalized fractional integral identity for the differentiable functions and derive some new estimates for the Bullen-type fractional integral inequalities via the Raina fractional integrals. Further examples are given to indicate the validity of obtained results. Lastly, some error estimates for the quadrature rules are provided, and to find applications for our results to information theory, we proved a new generalization based on f-divergence measure.

Jensen-Mercer Type Inequalities in the Setting of Fractional Calculus with Applications

The main objective of this paper is to establish some new variants of the Jensen–Mercer inequality via harmonically strongly convex function. We also propose some new fractional analogues of Hermite–Hadamard–Jensen–Mercer-like inequalities using AB fractional integrals. In order to obtain some of our main results, we also derive new fractional integral identities. To demonstrate the significance of our main results, we present some interesting applications to special means and to error bounds as well.