Search

In this paper, we establish a novel identity by leveraging fractional integral operators of Riemann-type. Subsequently, we employ this identity to investigate Bullen's fractional integral inequalities within the framework of classical convex functions. The validity of these innovative inequalities is verified through the visualization of 2D and 3D graphs. Furthermore, we obtain certain well-known results as special cases to our main findings and express them as remarks. Lastly, we derive differences in mean estimates, which serve as valuable tools for representing complex information, facilitating decision-making and aiding analysis in various fields.

In this paper, we are going to deal with fractional operators (FOs) with non-singular kernels which is not an easy task because of its restriction at the origin. In this work, we first show the boundedness of the extended form of the modified Atangana-Baleanu (A-B) Caputo fractional derivative operator. The generalized Laplace transform is evaluated for the introduced operator. By using the generalized Laplace transform, we solve some fractional differential equations. The corresponding form of the Atangana-Baleanu Caputo fractional integral operator is also established. This integral operator is proved bounded and obtained its Laplace transform. The existence and Hyers-Ulam stability is explored. In the last results, we studied the relation between our defined operators. The operators in the literature are obtained as special cases for these newly explored FOs.

The extension of interval-valued and real-valued functions known as fuzzy interval-valued function (FIVF) has made substantial contributions to the theory of interval analysis. In this article, we explore the importance of h-Godunova-Levin fuzzy convex and preinvex functions and also develop the new generation of the Hermite-Hadamard and trapezoid-type fuzzy fractional integral by the implementation of generalized fuzzy fractional operators having modified version of the Bessel-Maitland E1vBMF function as its kernel. Moreover, we extract some well-known inequalities from our main results.

Abstract In this paper, we explore a class of Hermite–Hadamard integral inequalities for convex and m-convex functions. The Hölder inequality is used to create this class, which has a wide range of applications in optimization theory. Some trapezoid-type inequalities and midpoint error estimates are investigated. Inequalities for several q-special functions are highlighted. As particular cases, we have included several previous results.

In this paper, we describe a new generalization of the multivariate Mittag-Leffler (M-L) function in terms of generalized Pochhammer symbol and study its properties. We provide a few differential and fractional integral formulas for the generalized multivariate M-L function. Furthermore, by using the generalized multivariate M-L function in the kernel, we present a new generalization of the fractional integral operator. Finally, we describe some fundamental characteristics of generalized fractional integrals.

Abstract In this article, we introduce a class of functions U ( p ) $\mathfrak{U}(\mathfrak{p})$ with integral representation defined over a measure space with σ-finite measure. The main purpose of this paper is to extend the Minkowski and related inequalities by considering general kernels. As a consequence of our general results, we connect our results with various variants for the fractional integrals operators. Such applications have wide use and importance in the field of applied sciences.

In this paper, we prove several new integral inequalities for the k-Hilfer fractional derivative operator, which is a fractional calculus operator. As a result, we have a whole new set of fractional integral inequalities. For the generalized fractional derivative, we also use Young's inequality to find new forms of inequalities. Such conclusions for this novel and generalized fractional derivative are extremely useful and valuable in the domains of differential equations and fractional differential calculus, both of which have a strong connections to real-world situations. These findings may stimulate additional research in a variety of fields of pure and applied sciences.

The fractional calculus (FC) has been extensively studied by researchers due to its vast applications in sciences in the last few years. In fractional calculus, multivariate Mittag–Leffler functions are considered the powerful extension of the classical Mittag–Leffler functions. This paper defines the generalized fractional integral operator with multivariate Mittag–Leffler (M-L) function. We prove certain basic properties of the proposed operators, such as an expansion of an infinite series of Riemann–Liouville integrals, Laplace transform (LT), semigroup property, composition with Riemann–Liouville integrals. Also, we present the fractional differential operators and their properties. The application of the proposed operators like the fractional kinetic differential and the time-fractional heat equation are also discussed.

In this research study, some striking features of single-valued fixed point theorems on multiplicative metric spaces have been established. Our displayed work consists of some unique fixed point theorems under generalized contraction with maximum and minimum conditions. In support of our work, we demonstrate some illustrative examples to justify all the conditions of our main theorems. In addition, a nonlinear integral equation is presented as an application to express the validity of our work. The offered outcomes in this study extend and improve many of the results proved in recent decades.

Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for θ=0.5,0.7,0.9,1.0. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative.

Many researchers in recent years have studied fractional integrals and derivatives. Some authors recently introduced generalized fractional integrals, the so-called unified fractional integrals. In this article, we establish certain new integral inequalities by employing the unified fractional integral operators. In fact, for a class of n (n∈N), positive continuous and decreasing functions on [v1,v2], certain new classes of integral inequalities are discussed. The inequalities established in this manuscript are more general forms of the classical inequalities given in the literature. The existing classical inequalities can be rectified by imposing the conditions stated in remarks. By imposing certain conditions on ℏ and Λ available in the literature, many new forms of fractional integral inequalities can be produced.

In this paper, a new class of Hermite-Hadamard type integral inequalities using a strong type of convexity, known as n-polynomial exponential type s-convex function, is studied. This class is established by utilizing the Hölder's inequality, which has several applications in optimization theory. Some existing results of the literature are obtained from newly explored consequences. Finally, some novel limits for specific means of positive real numbers are shown as applications.

In recent years, integral inequalities are investigated due to their extensive applications in several domains. The aim of the paper is to investigate certain new fractional integral inequalities which include Hermite-Hadamard inequality and different forms of trapezoid type inequalities related to Hermite-Hadamard inequality for h-Godunova-Levin preinvex function. Moreover, we compare our obtained results with the existing work in the literature and are represented by corollaries.

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green’s function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that the second derivative of the functions is monotone, absolutely convex, and concave. A section relating the results of exploration to generalized means and trapezoid formulas is included in the applications. We anticipate that the method presented in this study will inspire further research in this field.

The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The Ostrowski’s type inequality is frequently used to investigate errors in numerical quadrature rules and computations. In this work, Ostrowski-type inequality is demonstrated using the generalized fractional integral operators. From an application perspective, we present the bounds of the fractional Hadamard inequalities. The results that are being presented involve a number of fractional inequalities that are already known and have been published.

In this paper, we describe generalized fractional integral operator and its inverse with generalized Bessel-Maitland function (BMF-Ⅴ) as its kernel. We discuss its convergence, boundedness, its relation with other well known fractional operators (Saigo fractional integral operator, Riemann-Liouville fractional operator), and establish its integral transform. Moreover, we have given the relationship of BMF-Ⅴ with Mittag-Leffler functions.

The main goal of this paper is to describe the new version of extended Bessel–Maitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann–Liouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized Fox–Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.

In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.

Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. In this article, we elaborate the convexities and pre-invexities in aspects of FIVFs and investigate the existence of fuzzy fractional integral operators (FFIOs) having a generalized Bessel–Maitland function as their kernel. Using the class of convexities and pre-invexities FIVFs, we prove some Hermite–Hadamard (H-H) and trapezoid-type inequalities by the implementation of FFIOs. Additionally, we obtain other well known inequalities having significant behavior in the field of fuzzy interval analysis.

The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using specific functions to represent the inequalities. This approach allows us to demonstrate the validity of our findings and further solidify our conclusions. Moreover, we find that some previously published results emerge as special consequences of our main findings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity.

Several inequalities for convex functions are derived in this paper using the monotonicity properties of functions and a generalized weighted-type fractional integral operator, which allows the integration of a function κ with respect to another function in fractional order. Additionally, it is clear that the results were generalizations of the previously presented findings. In addition, different types of inequalities are obtained using the basic features of mathematical analysis. Finally, we believe that the methodology used in this work will inspire additional research in this field.

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 (η1,η2)-convex function related to Hermite-Hadamard type inequality. We establish new version of integral inequality for (η1,η2)-convex function related to Fejér type. The results discussed in this paper are a generalized version of many inequalities in literature.

In this article, we develop a new class of trapezium-type inequalities up to twice differentiable h-convex mappings for fractional integrals of Riemann-type. We conclude numerous existing results in literature from our general inequalities. Based on our consequences, we will obtain some quadrature formulas as applications.

Abstract 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.

In this article, we establish the inequalities of the Redheffer-type involving generalized Fox-Wright function. Furthermore, as a consequence, new Redheffer-type inequalities for generalized hypergeometric functions and the four-parametric generalized Mittag-Leffler functions are also discussed by using the suitable values of exponents in generalized inequalities.

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.

Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.

Abstract 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 G $\mathcal{G}$ 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 ω ( θ ) $\omega (\theta )$ and G ( θ ) $\mathcal{G}(\theta )$ 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 ω ( θ ) $\omega (\theta )$ and G ( θ ) $\mathcal{G}(\theta )$ .

Integral inequalities play a key role in applied and theoretical mathematics. The purpose of inequalities is to develop mathematical techniques in analysis. The goal of this paper is to develop a fractional integral operator having a non-singular function (generalized multi-index Bessel function) as a kernel and then to obtain some significant inequalities like Hermit Hadamard Mercer inequality, exponentially (s−m)-preinvex inequalities, Pólya-Szegö and Chebyshev type integral inequalities with the newly developed fractional operator. These results describe in general behave and provide the extension of fractional operator theory (FOT) in inequalities.

Abstract The primary objective of this present paper is to establish certain new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing the generalized weighted fractional integral involving another function Ψ in the kernel. The inequalities presented in this paper cover some new inequalities involving all other type weighted fractional integrals by applying certain conditions on ω ( θ ) $\omega (\theta )$ and Ψ ( θ ) $\Psi (\theta )$ . Also, the Pólya–Szegö and Chebyshev type integral inequalities for all other type fractional integrals, such as the Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, conformable fractional integral, and Hadamard fractional integral, are the special cases of our main results with certain choices of ω ( θ ) $\omega (\theta )$ and Ψ ( θ ) $\Psi (\theta )$ . Additionally, examples of constructing bounded functions are also presented in the paper.

Abstract In this paper, we establish some new integral inequalities involving general kernels. We obtain the related broad range of fractional integral inequalities. Also, we apply the Young inequality to find new forms of inequalities for generalized kernels. These new and motivated results generalize the results for fractional integrals such that fractional integral of a function with respect to an increasing function, Riemann–Lioville fractional integrals, Erdélyi–Kober fractional integrals, Hadamard fractional integrals, generalized factional integral integrals in addition to the corresponding k-fractional integrals.

In this paper, we discuss the extended generalized multi-index Bessel function by using the extended beta type function. Then we investigate its several properties including integral representation, derivatives, beta transform, Laplace transform, Mellin transforms, and some relations of extension of extended generalized multi-index Bessel function (E1GMBF) with the Laguerre polynomial and Whittaker functions. Further, we also discuss the composition of the generalized fractional integral operator having Appell function as a kernel with the extension of extended generalized multi-index Bessel function and establish these results in terms of Wright functions.

Abstract In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for ( s + 1 ) $(s+1)$ -convex functions involving the ( k , s ) $(k,s)$ -Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical Hermite–Hadamard and trapezoid inequalities.

Abstract In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.

Abstract The aim of this present investigation is establishing Minkowski fractional integral inequalities and certain other fractional integral inequalities by employing the Marichev–Saigo–Maeda (MSM) fractional integral operator. The inequalities presented in this paper are more general than the existing classical inequalities cited.

Abstract We study a generalized left sided tempered fractional (GTF)-integral concerning another function Ψ in the kernel. Then we investigate several kinds of inequalities such as Grüss-type and certain other related inequalities by utilizing the GTF-integral. Additionally, we present various special cases of the main result. By utilizing the connection between GTF-integral and Riemann–Liouville integral concerning another function Ψ in the kernel, certain distinct particular cases of the main result are also presented. Furthermore, certain other inequalities can be formed by applying various kinds of conditions on the function Ψ.

Abstract 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.

The principal aim of this paper is to introduce $\mathtt{k}$-fractional derivative operator by using the definition of $\mathtt{k}$-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to $\mathtt{k}$-hypergeometric and $\mathtt{k}$-Appell's functions. Also, we investigate the $\mathtt{k}$-fractional derivative of $\mathtt{k}$-Mittag-Leffler and the Wright hypergeometric functions.

The principal aim of this article is to establish certain generalized fractional integral inequalities by utilizing the Marichev-Saigo-Maeda (MSM) fractional integral operator. Some new classes of generalized fractional integral inequalities for a class of n (n ∈ $\mathbb{N}$) positive continuous and decreasing functions on [a, b] by using the MSM fractional integral operator also derived.

Abstract The main aim of this present paper is to establish fractional conformable inequalities for the weighted and extended Chebyshev functionals. We present some special cases of our main result in terms of the Riemann–Liouville fractional integral operator and classical inequalities.

Abstract Fractional kinetic equations (FKEs) including a wide variety of special functions have been widely and successfully applied in describing and solving many important problems of physics and astrophysics. In this paper, we derive the solutions for FKEs including the class of functions with the help of Sumudu transforms. Many important special cases are then revealed and analyzed. The use of the class of functions to obtain the solution of FKEs is fairly general and can be efficiently used to construct several well-known and novel FKEs.

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities for h-Godunova–Levin convexity and h-Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex, s-Godunova–Levin function, P-function, and s-convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities.

In this article, we demonstrated various Hermite–Hadamard and Fejér type inequalities for modified h-convex functions. We showed several inequalities for the products of two modified h-convex functions. New identities related to inequalities in various forms are also established for different values of the h(φt) function. We believe that the approach presented in this paper will inspire more research in this area.

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.

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched.

In the recent era of research, the field of integral inequalities has earned more recognition due to its wide applications in diverse domains. The researchers have widely studied the integral inequalities by utilizing different approaches. In this present article, we aim to develop a variety of certain new inequalities using the generalized fractional integral in the sense of multivariate Mittag-Leffler (M-L) functions, including Grüss-type and some other related inequalities. Also, we use the relationship between the Riemann-Liouville integral, the Prabhakar integral, and the generalized fractional integral to deduce specific findings. Moreover, we support our findings by presenting examples and corollaries.

Abstract The present paper aims to establish certain new classes of integral inequalities for a class of n ( n∈N $n\in \mathbb{N}$) 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.

Abstract In this paper, we introduce a new approach to the fractional derivation which generalizes the classical Hadamard fractional derivative. We prove some properties of this new approach and also establish some results by addressing some standard functions.

Abstract In the article, we introduce the generalized proportional Hadamard fractional integrals and establish several inequalities for convex functions in the framework of the defined class of fractional integrals. The given results are generalizations of some known results.

Abstract In the last few years, various researchers studied the so-called conformable integrals and derivatives. Based on that notion some authors used modified conformable derivatives (proportional derivatives) to generate nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, which contain exponential functions in their kernels. Our aim in this paper is to establish some new integral inequalities by utilizing the fractional proportional-integral operators. In fact, certain new classes of integral inequalities for a class of n ( n∈N $n\in \mathbb{N}$) positive continuous and decreasing functions on [a,b] $[a,b]$ are presented. The inequalities presented in this paper are more general than the existing classical inequalities.