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This study aims to investigate the properties of fractional calculus theory (FCT) in the complex domain. We focus on the relationship between the theories of special functions (SFT) and FCT, which have seen recent advancements and have led to various successful applications in fields such as engineering, mathematics, physics, biology, and other allied disciplines. Our main contribution is the development of a special function, specifically the confluent hypergeometric function (CHF) on the complex domain. By deriving various implementations of fractional order derivatives and integral operators using this function, we present a new class of special functions combining certain cases of Mittag-Leffler and confluent hypergeometric functions. Moreover, a new numerical technique for solving linear and nonlinear multi-order fractional differential equations has been developed using the proposed class of functions and the point collocation method. Graphical results are shown to demonstrate the efficacy of this proposed technique and its applicability.

We show that the fractional integral operator $I_{\alpha }$ , $0 , and the fractional maximal operator $M_{\alpha }$ , $0\le \alpha , are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator $M_\alpha $ are extensions of the work of Tang ['Choquet integrals, weighted Hausdorff content and maximal operators', Georgian Math. J. 18 (3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator $I_{\alpha }$ are essentially new. [ABSTRACT FROM AUTHOR]

This study aims to investigate the properties of fractional calculus theory (FCT) in the complex domain. We focus on the relationship between the theories of special functions (SFT) and FCT, which have seen recent advancements and have led to various successful applications in fields such as engineering, mathematics, physics, biology, and other allied disciplines. Our main contribution is the development of a special function, specifically the confluent hypergeometric function (CHF) on the complex domain. By deriving various implementations of fractional order derivatives and integral operators using this function, we present a new class of special functions combining certain cases of Mittag-Leffler and confluent hypergeometric functions. Moreover, a new numerical technique for solving linear and nonlinear multi-order fractional differential equations has been developed using the proposed class of functions and the point collocation method. Graphical results are shown to demonstrate the efficacy of this proposed technique and its applicability. [ABSTRACT FROM AUTHOR]

We prove the boundedness of the generalized fractional maximal operator Mα and the generalized fractional integral operator Iα on weak Choquet spaces with respect to Hausdorff content over quasi-metric measure spaces. [ABSTRACT FROM AUTHOR]

In this article, we obtained a new fractional integral identity, and using it, we established some new Simpson's type inequalities for s-convex functions in the second sense through the fractional integral operators of a function with respect to another function. Some applications of the obtained results to special means are also provided. Our results are more generalized in nature. [ABSTRACT FROM AUTHOR]

A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the subclass of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well known results are illustrated. [ABSTRACT FROM AUTHOR]

We study the fractional integral and Orlicz fractional maximal operator estimates in the second kind of Orlicz-Morrey spaces. This problem originates from the Adams and Olsen inequality in Morrey spaces. We establish the boundedness corresponding to the Adams-type and the Olsen-type inequality of the fractional operator in Orlicz-Morrey spaces under conditions of Young functions. Moreover, we construct and observe the examples of the Young functions' triplet that satisfies the condition. This paper reinforces the boundedness theory for the fractional integrals and Orlicz fractional maximal operators in Orlicz-Morrey spaces. [ABSTRACT FROM AUTHOR]

Let $ 0 < \alpha < n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.

We establish the mapping properties of the fractional integral operators on the Hardy local Morrey spaces with variable exponents by using the extrapolation theory. The local Morrey spaces with variable exponents are generalizations and extensions of the local Morrey spaces, the Lebesgue spaces with variable exponents and the Morrey spaces with variable exponents. The Hardy local Morrey spaces with variable exponents are the Hardy spaces built on the local Morrey spaces with variable exponents. Our main result extends and generalizes the mapping properties of the fractional integral operators on the Hardy spaces, the local Morrey spaces, the Hardy spaces with variable exponents and the local Morrey spaces with variable exponents. We obtain our main result by extending the J.L. Rubio de Francia extrapolation theory to the local Morrey spaces with variable exponents. This method was originally developed by J.L. Rubio de Francia on the weighted Lebesgue spaces and recently it has been extended to a number of function spaces such as the Morrey spaces with variable exponents, the local Morrey spaces with variable exponents and the Morrey-Banach spaces. We further extend it to the local Morrey spaces with variable exponents. By using the mapping properties of the fractional integral operators on the weighted Hardy spaces, we establish the mapping properties of the fractional integral operators on the Hardy local Morrey spaces with variable exponents. [ABSTRACT FROM AUTHOR]

In this paper, several applications of the Hermite-Hadamard inequality for fractional integrals using harmonic convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Beta function. [ABSTRACT FROM AUTHOR]

Embeddings in the framework of grand variable exponent function spaces are studied. In particular, embeddings from grand variable exponent Hajłasz-Sobolev-Morrey spaces to variable exponent H ¨older spaces are established. The regularity of a fractional integral operator defined with respect to a non-doubling measure is also investigated. In particular, mapping properties of this operator from a grand variable exponent Morrey space to a grand variable parameter H ¨older space are studied. The results are proved under the log-H ¨older continuity condition on the exponents. The spaces are defined, generally speaking, on quasi-metric measure spaces, however, the results are new even for Euclidean spaces. [ABSTRACT FROM AUTHOR]

We establish a factorization theorem for the Hardy space in terms of a multilinear fractional integral operator. As an application, a characterization of Lipschitz spaces via the boundedness of commutators of the multilinear fractional integral operator is obtained. [ABSTRACT FROM AUTHOR]

We consider the Fefferman–Stein inequality for weak Orlicz–Morrey spaces and the commutators [b,T]$[b,T]$ and [b,Iρ]$[b,I_{\rho }]$ on weak Orlicz–Morrey spaces, where T is a Calderón–Zygmund operator, Iρ$I_{\rho }$ is a generalized fractional integral operator and b is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness from of [b,T]$[b,T]$ and [b,Iρ]$[b,I_{\rho }]$ from a weak Orlicz Morrey space to another weak Orlicz–Morrey space. We use the Fefferman–Stein inequality to prove the boundedness of the commutators. Since weak Orlicz–Morrey spaces contain the weak Lebesgue, weak Orlicz and weak Morrey spaces as special cases, our results contain the bounedness on these function spaces which are also new results. [ABSTRACT FROM AUTHOR]

We extend the two weighted norm inequalities for the potential type operators to Herz spaces. As an application of this result, we have the two weighted norm inequalities of the fractional integral operators on Herz spaces. [ABSTRACT FROM AUTHOR]

We give boundedness results for the fractional maximal operator and the Riesz potential operator in the framework of Stummel spaces with variable exponents. [ABSTRACT FROM AUTHOR]

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In this paper, we study the boundedness for the fractional integral operator I α on generalized Morrey spaces associated with ball Banach function spaces. Moreover, the boundeness for the commutator of I α on these spaces is also established. Our main results extend various previous results on some concrete generalized Morrey spaces associated with ball Banach function spaces. [ABSTRACT FROM AUTHOR]

Abstract In this investigation, we unfold the Jensen–Mercer ( J − M $\mathtt{J-M}$ ) inequality for convex stochastic processes via a new fractional integral operator. The incorporation of convex stochastic processes, the J − M $\mathtt{J-M}$ inequality and a fractional integral operator having an exponential kernel brings a new direction to the theory of inequalities. With this in mind, estimations of Hermite–Hadamard–Mercer ( H − H − M $\mathtt{H-H-M}$ )-type fractional inequalities involving convex stochastic processes are presented. In the context of the new fractional integral operator, we also investigate a novel identity for differentiable mappings. Then, a new related H − H − M $\mathtt{H-H-M}$ -type inequality is presented using this identity as an auxiliary result. Applications to special means and matrices are also presented. These findings are particularly appealing from the perspective of optimization, as they provide a larger context to analyze optimization and mathematical programming problems.

Abstract In this study, a new definition of the fractional integral operator is first proposed, which generalizes some well-known fractional integral operators. Then, by using this newly generalized fractional integral operator, we proved several new Hermite-Hadamard-type inequalities for convex functions. Finally, we provided some corollaries to show that the current results extend and enrich the previous results in the literature.

Impulsive is the affinity to do something without thinking. In this effort, we model a mathematical formula types integro-differential equation (I-DE) to describe this behavior. We investigate periodic boundary value issues in Banach spaces for fractional a class of I-DEs with non-quick impulses. We provide numerous sufficient conditions of the existence of mild outcomes for I-DE utilizing the measure of non-compactness, the method of resolving domestic, and the fixed point result. Lastly, we illustrate a set of examples, which is given to demonstrate the investigations key findings. Our findings are generated some recent works in this direction.

We study the boundedness of commutators generated by BMO functions and the fractional integral operator on Orlicz-Morrey spaces of the second kind. To show this problem, we investigate the Fefferman-Stein type inequality concerning the Hardy-Littlewood maximal function and the sharp maximal operator and the boundedness of the Orlicz-fractional maximal operator in the Orlicz-Morrey spaces. [ABSTRACT FROM AUTHOR]

In this article, a generalization of the well-known Stein-Weiss inequality for the fractional integral operator on functions with different integrability properties in the radial and the angular direction in local Morrey spaces is established. We find that some conditions can be relaxed for the Stein-Weiss inequality for local mixed radial-angular Morrey spaces.

This article is aimed at establishing some results concerning integral inequalities of the Opial type in the fractional calculus scenario. Specifically, a generalized definition of a fractional integral operator is introduced from a new Raina-type special function, and with certain results proposed in previous publications and the choice of the parameters involved, the established results in the work are obtained. In addition, some criteria are established to obtain the aforementioned inequalities based on other integral operators. Finally, a more generalized definition is suggested, with which interesting results can be obtained in the field of fractional integral inequalities.

Impulsive is the affinity to do something without thinking. In this effort, we model a mathematical formula types integro-differential equation (I-DE) to describe this behavior. We investigate periodic boundary value issues in Banach spaces for fractional a class of I-DEs with nonquick impulses. We provide numerous sufficient conditions of the existence of mild outcomes for I-DE utilizing the measure of non-compactness, the method of resolving domestic, and the fixed point result. Lastly, we illustrate a set of examples, which is given to demonstrate the investigations key findings. Our findings are generated some recent works in this direction. [ABSTRACT FROM AUTHOR]

In this study, a new definition of the fractional integral operator is first proposed, which generalizes some well-known fractional integral operators. Then, by using this newly generalized fractional integral operator, we proved several new Hermite-Hadamard-type inequalities for convex functions. Finally, we provided some corollaries to show that the current results extend and enrich the previous results in the literature. [ABSTRACT FROM AUTHOR]

In this investigation, we unfold the Jensen–Mercer ( J − M ) inequality for convex stochastic processes via a new fractional integral operator. The incorporation of convex stochastic processes, the J − M inequality and a fractional integral operator having an exponential kernel brings a new direction to the theory of inequalities. With this in mind, estimations of Hermite–Hadamard–Mercer ( H − H − M )-type fractional inequalities involving convex stochastic processes are presented. In the context of the new fractional integral operator, we also investigate a novel identity for differentiable mappings. Then, a new related H − H − M -type inequality is presented using this identity as an auxiliary result. Applications to special means and matrices are also presented. These findings are particularly appealing from the perspective of optimization, as they provide a larger context to analyze optimization and mathematical programming problems. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to present a new numerical method based on Hahn hybrid functions (HHFs) for solving of Black–Scholes option pricing distributed order time‐fractional partial differential equation. To this end, HHFs are introduced and their fractional integral operator with some properties of HHFs is calculated. In the next, with the help of fractional integral operator of HHFs, Gauss–Legendre quadrature formula and collocation method, distributed order time‐fractional Black–Scholes model is reduced to a system of algebraic equations. Furthermore, convergence analysis of the mentioned scheme is discussed. Finally, some test problems have been included to confirm the validity and efficiency of the mentioned numerical scheme. Moreover, Black–Scholes equations are studied through a bibliometric viewpoint. [ABSTRACT FROM AUTHOR]

The aim of this research is to combine the concept of inequalities with fractional integral operators, which are the focus of attention due to their properties and frequency of usage. By using a novel fractional integral operator that has an exponential function in its kernel, we establish a new Hermite-Hadamard type integral inequality for an LR-convex interval-valued function. We also prove new fractional-order variants of the Fejér type inequalities and the Pachpatte type inequalities in the setting of pseudo-order relations. By showing several numerical examples, we further validate the accuracy of the results that we have derived in this study. We believe that the results, presented in this article are novel and that they will be beneficial in encouraging future research in this field.

In this paper, some extrapolations on ball Banach function spaces are established. As applications, we obtain the boundedness of fractional spherical maximal operators, fractional integral operators with rough kernels, spherical maximal operators, strongly singular integral operators and maximal Bochner–Riesz operators on ball Banach function spaces. Finally, we apply these boundedness results to two concrete examples of ball Banach function spaces, namely, variable Lebesgue spaces and mixed-norm Lebesgue spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the boundedness of some operator on grand generalized Morrey space $\mathcal{L}^{p),\varphi,\phi}_{\mu}(G)$ over non-homogeneous spaces, where $G\subset$ $\mathbb{R}^{n}$ is a bounded domain. Under assumption that functions $\varphi$ and $\phi$ satisfy certain conditions, the authors prove that the Hardy-Littlewood maximal operator, fractional integral operators and $\theta$-type Calder\'{o}n-Zygmund operators are bounded on the non-homogeneous grand generalized Morrey space $\mathcal{L}^{p),\varphi,\phi}_{\mu}(G)$. Moreover, the boundedness of commutator $[b,T^{G}_{\theta}]$ which is generated by $\theta$-type Calder\'{o}n-Zygmund operator $T_{\theta}$ and $b\in\mathrm{RBMO}(\mu)$ on spaces $\mathcal{L}^{p),\varphi,\phi}_{\mu}(G)$ is also established.

In this paper, a Gronwall inequality involving-fractional integral operator is obtained as a generalization of [23]. An example is listed to show the applications. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to deal with the boundedness of the fractional maximal and fractional integral operators in Herz spaces with variable exponent on the hyperplane. We also discuss the integrability of the fractional maximal and fractional integrals in such spaces. [ABSTRACT FROM AUTHOR]

This paper investigates the composition structures of certain fractional integral operators whose kernels are certain types of generalized hypergeometric functions. It is shown how composition formulas of these operators can be closely related to the various Erdélyi-type hypergeometric integrals. We also derive a derivative formula for the fractional integral operator and some applications of the operator are considered for a certain Volterra-type integral equation, which provide two generalizations to Khudozhnikov's integral equation (see below). Some specific relationships, examples, and some future research problems are also discussed. [ABSTRACT FROM AUTHOR]

We obtain some criteria for a bilinear inequality of a class of operators of fractional integration that imply the corresponding criteria and bilinear inequalities for the Riemann–Liouville, Hadamard, and Erdelyi–Kober operators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new subclasses of the class of uniformly convex functions, using the generalize fractional derivative and integral operators, introduced in 2020 by Issa and Darus. Then, we give coefficient inequalities and we study some analytic properties for this new subclasses. [ABSTRACT FROM AUTHOR]

The aim of this research is to combine the concept of inequalities with fractional integral operators, which are the focus of attention due to their properties and frequency of usage. By using a novel fractional integral operator that has an exponential function in its kernel, we establish a new Hermite-Hadamard type integral inequality for an LR-convex interval-valued function. We also prove new fractional-order variants of the Fejér type inequalities and the Pachpatte type inequalities in the setting of pseudo-order relations. By showing several numerical examples, we further validate the accuracy of the results that we have derived in this study. We believe that the results, presented in this article are novel and that they will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

In 2019, Salim et al proved the vector-valued inequality for maximal operator with rough kernel on Lebesgue spaces and Morrey spaces. This results extend Fefferman-Stein inequality (1971). In 1970's, Adams introduced another variant of Morrey spaces, which called as Morrey-Adams spaces. In this article, we prove vector-valued inequality for maximal operator and fractional integral operator with rough kernel on Morrey-Adams spaces. [ABSTRACT FROM AUTHOR]

The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR--convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions (IV-Fs) linked to Hermite–Hadamard (H-H) and Hermite–Hadamard–Fejér (H-H-Fejér) type inequalities via interval-valued Riemann–Liouville fractional integrals (IV-RL-fractional integrals). We also attain some related inequalities for the product of two LR--convex IV-Fs. These findings enable us to identify a new class of inclusions that may be seen as significant generalizations of results proved by Iscan and Chen. Some examples are included in our findings that may be used to determine the validity of the results. The findings in this work can be seen as a considerable advance over previously published findings. [ABSTRACT FROM AUTHOR]

In present paper, we establish new reverses of Minkowski, Hölder and Hardy type inequalities by using ψ-Riemann-Liouville fractional integral operator. [ABSTRACT FROM AUTHOR]

Abstract Lately, many studies were offered to introduce the population dynamics of COVID-19. In this investigation, we extend different physical conditions of the growth by employing fractional calculus. We study a system of coupled differential equations, which describes the dynamics of the infection spreading between infected and asymptomatic styles. The healthy population properties are measured due to the social meeting. The result is associated with a macroscopic law for the population. This dynamic system is appropriate to describe the performance of growth rate of the infection and to verify if its control is appropriately employed. A unique solution, under self-mapping possessions, is investigated. Approximate solutions are presented by utilizing fractional integral of Chebyshev polynomials. Our methodology is based on the Atangana–Baleanu calculus, which provides various activity results in the simulation. We tested the suggested system by using live data. We found positive action in the graphs.

Abstract In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with BMO functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with BMO functions and the associated maximal operators.

Let G = (RN,◦,δλ) be a homogeneous group, Q be the homogeneous dimension of G, X0,X1,...,Xm be left invariant real vector fields on G and satisfy H ¨ormander's rank condition on RN. Assume that X1,...,Xm (m ≤ N - 1) are homogeneous of degree one and X0 is homogeneous of degree two with respect to the family of dilations (δλ)λ>0. Consider the following hypoelliptic operator with drift on G ... where (aij) is a constant matrix satisfying the elliptic condition in Rm and a0 ≠ 0. In this paper, for this class of operators we obtain generalized weighted Sobolev-Morrey estimates by establishing boundedness of a large class of sublinear operators Tα, α ∈ [0,Q) generated by Calder 'on-Zygmund operators (α = 0) and generated by fractional integral operator (α > 0) on generalized weighted Morrey spaces and proving interpolation results in generalized weighted Sobolev-Morrey spaces on G. [ABSTRACT FROM AUTHOR]

We present unified versions of Minkowski-type fractional integral inequalities with the help of fractional integral operator based on a unified Mittag-Leffler function. These inequalities provide new as well as already known fractional versions of Minkowski-type inequalities. [ABSTRACT FROM AUTHOR]

Let α∈(0,n)$\alpha \in (0,n)$ and let φ1,φ2:Rn×[0,∞)→[0,∞)$\varphi _1,\varphi _2:\mathbb {R}^n\times [0,\infty)\rightarrow [0,\infty)$ be Musielak–Orlicz functions such that φ1(x,·)$\varphi _1(x,\cdot)$, φ2(x,·)$\varphi _2(x,\cdot)$ are Orlicz functions and φ1(·,t)$\varphi _1(\cdot ,t)$, φ2(·,t)$\varphi _2(\cdot ,t)$ are Muckenhoupt A∞(Rn)$A_\infty (\mathbb {R}^n)$ weights. In this paper, we give the necessary and sufficient condition for the boundedness of the fractional integral operator Iα$I_\alpha$ from the Musielak–Orlicz Hardy space Hφ1(Rn)$H^{\varphi _1}(\mathbb {R}^n)$ into the Musielak–Orlicz Hardy space Hφ2(Rn)$H^{\varphi _2}(\mathbb {R}^n)$. We also give the necessary and sufficient condition for the boundedness of Iα$I_\alpha$ from Hφ1(Rn)$H^{\varphi _1}(\mathbb {R}^n)$ into the Musielak–Orlicz space Lφ2(Rn)$L^{\varphi _2}(\mathbb {R}^n)$. Our results generalize the main results in [5, 20, 21]. [ABSTRACT FROM AUTHOR]