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This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R + n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques. [ABSTRACT FROM AUTHOR]

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order β approaches 1, in addition to the fractional integrals we examined. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear fractional-order boundary value problem   H D a , g α 1 , β , μ x (t) + f (t , x (t) , H D a , g α 2 , β , μ x (t)) = 0 , for t ∈ [ a , b ] , α 1 ∈ (1 , 2 ] , α 2 ∈ (0 , 1 ] , β ∈ [ 0 , 1 ] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and   H D a , g α , β , μ represents the tempered-Hilfer fractional derivative of order α > 0 with parameter μ ∈ R + and type β ∈ [ 0 , 1 ] . The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. [ABSTRACT FROM AUTHOR]

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order βξ instead of β1; for details, see Lemma 2.2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity. [ABSTRACT FROM AUTHOR]

In this research paper, we utilize the q-derivative concept to formulate specific differential and integral operators denoted as R q n , m , λ , F q n , m , λ and G q n , m , λ . These operators are introduced with the aim of generalizing the class of Ruscheweyh operators within the set of univalent functions. We extract certain properties and characteristics of the set of differential subordinations employing specific techniques. By utilizing the newly defined operators, this paper goes on to establish subclasses of analytic functions defined on an open unit disc. Additionally, we delve into the convexity properties of the two recently introduced q-integral operators, F q n , m , λ and G q n , m , λ . Special cases of the primary findings are also discussed. [ABSTRACT FROM AUTHOR]

This paper was concerned with the existence and uniqueness results for a coupled system of nonlinear generalized fractional differential equations supplemented with a new class of nonlocal coupled multipoint boundary conditions containing Riemann-Stieltjes and generalized fractional integrals. The nonlinearities in the given system depend on the unknown functions as well as their lower order generalized fractional derivatives. We made use of the Leray-Schauder alternative and Banach contraction mapping principle to obtain the desired results. An illustrative example was also discussed. The paper concluded with some interesting observations. [ABSTRACT FROM AUTHOR]

We analyze the well-posedness of certain field-only boundary integral equations (BIEs) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field E\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}(x) and (2) scalar quantity E\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}(x)\cdot x are radiative solutions of the Helmholtz equation, we see that novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIEs. In this paper we use the combined field methodology of Burton and Miller within the fieldonly BIE approach, and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are nonzero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one--a property similar to second-kind integral equations. [ABSTRACT FROM AUTHOR]

Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities. [ABSTRACT FROM AUTHOR]

We investigate fractional operators with homogeneous kernel in Morrey spaces. In particular, we prove that fractional integral operators and fractional maximal operators with homogeneous kernel are bounded from the Calderón product of Morrey spaces to certain Morrey spaces. Our results can be seen as a generalization of a recent result on the relation between the boundedness of (classical) fractional operators and interpolation of Morrey spaces. What is new about this paper is not only the passage from the classical fractional integral operators to the rough integral operators. Even the case of fractional integral operators, handled in earlier papers, is significantly simplified. [ABSTRACT FROM AUTHOR]

Singular integral operators play an important role in approximation theory and harmonic analysis. In this paper, we consider a weighted Lebesgue space L p , w , define a modified Gauss–Weierstrass singular integral on it, and study direct and inverse approximation properties of the operator followed by a Korovkin-type approximation theorem for a function f ∈ L p , w . We use the modulus of continuity of the functions to measure the rate of convergence. [ABSTRACT FROM AUTHOR]

Let S n − 1 denote unit sphere in R n equipped with the normalized Lebesgue measure. Let Φ ∈ L s (S n − 1) be a homogeneous function of degree zero such that ∫ S n − 1 Φ (y ′) d σ (y ′) = 0 , where y ′ = y / | y | for any y ≠ 0 . The commutators of variable Marcinkiewicz fractional integral operator is defined as [ b , μ Φ ] β m (f) (x) = (∫ 0 ∞ | ∫ | x − y | ≤ s Φ (x − y) [ b (x) − b (y) ] m | x − y | n − 1 − β (x) f (y) d y | 2 d s s 3) 1 2. In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces K ˙ p (⋅) α (⋅) , q) , θ (R n) . [ABSTRACT FROM AUTHOR]

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman's integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down ( U · D ) relations and over newly defined class U · D -ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of U · D -relations, recent findings have been developed using fuzzy Aumman's, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields. [ABSTRACT FROM AUTHOR]

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right ℏ -pre-invex interval-valued mappings ( C · L · R - ℏ -pre-invex Ι · V - M ), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite–Hadamard and Pachpatte's type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes. [ABSTRACT FROM AUTHOR]

In this paper, the time-independent Klein-Gordon equation in R 3 is treated with a decomposition of the operator Δ − γ 2 I by the Clifford algebra C l (V 3 , 3) . Some properties of integral operators associated the kind of equations and some Riemann-Hilbert boundary value problems for perturbed Dirac operators are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we extend the study of the Cesáro function spaces to the Cesáro-Morrey function spaces. We establish a general principle on the boundedness of integral operators on the Cesáro-Morrey function spaces. By applying this principle, we have the boundedness of the Erdélyi-Kober fractional integrals and the Hadamard fractional integrals on the Cesáro-Morrey function spaces. In addition, we also extend the study of Tandori spaces to Tandori-Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subordination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities. [ABSTRACT FROM AUTHOR]

This paper presents a new faster iteration approach that is based upon the well-known three-step fixed point iteration scheme called SP-iterative scheme for a broad and larger class of non-linear BVPs. The scheme is constructed using Green's function based integral operator and the new modified scheme in the paper will be called SP–Green's iterative scheme. We present a fixed point convergence theorem using some assumption on the sequences of scalars in our scheme. We also prove the weak w 2 -stability for our scheme and present a numerical example to support our scheme. We provide eventually applications of our scheme to a second-order BVP and a singular BVP of isothermal gas sphere. [ABSTRACT FROM AUTHOR]

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s = 1 , λ = α ,   σ (0) = 1 , and w = 0 . In conclusion, the methodology described in this article is expected to stimulate further research in this area. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

In this paper, we study the small‐λ$\lambda$ spectral asymptotics of an integral operator K$\mathcal {K}$ defined on two multi‐intervals J$J$ and E$E$, when the multi‐intervals touch each other (but their interiors are disjoint). The operator K$\mathcal {K}$ is closely related to the multi‐interval finite Hilbert transform (FHT). This case can be viewed as a singular limit of self‐adjoint Hilbert–Schmidt integral operators with so‐called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when dist(J,E)>0$\text{dist}(J,E)>0$, and K$\mathcal {K}$ is of the Hilbert–Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that U=J∪E$U=J\cup E$ is a single interval (although part of our analysis is valid in a more general situation). We show that the eigenvalues of K$\mathcal {K}$, if they exist, do not accumulate at λ=0$\lambda =0$. Combined with the results in an earlier paper by the authors, this implies that Hp$H_p$, the subspace of discontinuity (the span of all eigenfunctions) of K$\mathcal {K}$, is finite dimensional and consists of functions that are smooth in the interiors of J$J$ and E$E$. We also obtain an approximation to the kernel of the unitary transformation that diagonalizes K$\mathcal {K}$, and obtain a precise estimate of the exponential instability of inverting K$\mathcal {K}$. Our work is based on the method of Riemann–Hilbert problem and the nonlinear steepest descent method of Deift and Zhou. [ABSTRACT FROM AUTHOR]

In this paper, we consider the commutators associated with the fractional maximal function and sharp maximal function when symbol function b in Lipschitz spaces. We give some characterizations of the Lipschitz spaces via the boundedness of these commutators on total Morrey spaces. [ABSTRACT FROM AUTHOR]

We prove a weighted Lp boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space Lp(Rn ×Rm, ω1(x)dx, ω2(y)dy), provided that the weights ω1 and ω2 are certain radial weights and that the kernels are rough in the optimal space L(log L)(Sn-1 × Sm-1). In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted Lp boundedness of the related square and maximal functions. Our weighted Lp inequalities extend as well as generalize previously known Lp boundedness results. [ABSTRACT FROM AUTHOR]

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F ˜ λ , we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F ˜ λ. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlocal fractional coupled system of (k , ψ) -Hilfer fractional differential equations, which involve, in boundary conditions, (k , ψ) -Hilfer fractional derivatives and (k , ψ) -Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel'skiĭ's fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. [ABSTRACT FROM AUTHOR]

The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them. [ABSTRACT FROM AUTHOR]

It is difficult to give a fractional global threshold (FGT) that works well on all images as the image contents are totally different. This paper describes an interesting use of fractional calculus in the field of digital image processing. In the proposed method, the fractional global threshold-based edge detector (FGTED) is established using the Riemann–Liouville fractional integral operator. FGTED is used to find the microedges in minimum time for any input digital images. The results demonstrate that the FGTED outperforms conventional techniques for detecting microtype edges. The image with a higher entropy was produced by the FGT value-based approach. Tables and images are used to summarize the output performance analysis of various images using structural similarity index measure, F-score (F-measure), precision and recall, signal-to-noise ratio, peak signal-to-noise ratio, and computational time. The FGTED can be used to detect very thin or microtype edges more accurately in minimum time without training or prior knowledge. [ABSTRACT FROM AUTHOR]

In this paper we obtain sufficient condition for univalence of analytic functions defined by Hurwitz-Lerach Zeta function. [ABSTRACT FROM AUTHOR]

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Lp boundedness of generalized Marcinkiewicz operators along surfaces of revolution on product spaces under very weak conditions on the the singular kernels. Our results generalize and improve many previously known results. [ABSTRACT FROM AUTHOR]

Let 0 < α < n and b be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator (TΩ,α)bm generated by the function b and the fractional integral operator with the homogeneous kernel TΩ,α on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator (TΩ,α)bm on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator (Tα)bm generated by the function b and the fractional integral operator Tα on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator [b, Tα] generated by the function b and the fractional integral operator Tα on variable Lebesgue spaces and mixed Morrey spaces. [ABSTRACT FROM AUTHOR]

Denoising algorithms for obtaining the effective data of depth images affected by random noise mainly focus on the processing of gray images. These algorithms are not distinct from traditional image-processing methods, and there is no way to evaluate the effectiveness of denoising after the point cloud transformation of denoised depth images. In this paper, the principle of fractional-order integral denoising is studied in detail and inverse distance weighted interpolation is introduced into a denoising model, which is based on the G–L (Grünwald–Letnikov) fractional-order integral to construct a fractional-order integral with an inverse distance weighted denoising model. The model is used to solve the blurring problem caused by sharp changes at the edge and achieves an excellent denoising effect. By using the optimized fractional-order integral denoising operator to construct a denoising model for depth images, the results of the experiments demonstrate that the fractional-order integral of the best denoising effect achieved by the model is −0.6 ≤ ν ≤ −0.4, and the peak signal-to-noise ratio is improved from +6 to +13 dB. In the same condition, median denoising has a distortion of −30 to −15 dB. The depth image that has been denoised is converted into an image of point clouds, and subjective evaluation indicates that the noise is effectively removed. On the whole, the results demonstrate that the fractional-order integral denoising operator with inverse distance weight shows the high efficiency and the outstanding effect in removing noise from depth images while maintaining the image related to the edge and texture information. [ABSTRACT FROM AUTHOR]

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using G A -convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions. [ABSTRACT FROM AUTHOR]

Some fundamental properties of the Muckenhoupt class A p of weights and the Gehring class G q of weights on time scales and some relations between them will be proved in this paper. To prove the main results, we will apply an approach based on proving some properties of integral operators on time scales with powers and certain mathematical relations connecting the norms of Muckenhoupt and Gehring classes. The results as special cases cover the results for functions following David Cruz-Uribe, C. J. Neugebauer, and A. Popoli, and when the time scale equals the positive integers, the results for sequences are essentially new. [ABSTRACT FROM AUTHOR]

In this paper, we employ a q -Noor integral operator to perform a q -analogue of certain fractional integral operator defined on an open unit disc. Then, we make use of the Hadamard convolution product to discuss several related results. Also, we derive a class of convex functions by utilizing the q -fractional integral operator and apply the inspired presented theory of the differential subordination, to geometrically explore the most popular differential subordination properties of the aforementioned operator. In addition, we discuss an exciting inclusion for the given convex class of functions. Over and above, we investigate the q -fractional integral operator and obtain some applications for the differential subordination. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate L p bounds of these Maximal operators under the assumption Ω ∈ L q (S m − 1 × S n − 1) for some q > 1 , and then we employ these bounds along with Yano's extrapolation argument to obtain the L p boundedness of the aforementioned integral operators under a weaker condition in which Ω belongs to either the space B q (0 , 2 τ ′ − 1) (S m − 1 × S n − 1) or to the space L (l o g L) 2 / τ ′ (S m − × S n − 1) . Our results extend and improve many previously known results. [ABSTRACT FROM AUTHOR]

Let A p n be the class of functions f z given by f z = z p + a p + n z p + n + a p + n + 1 z p + n + 1 + ⋯ which are analytic in the open unit disk U. For f z ∈ A p n , new integral operators O − j f z and O j f z j = 0 , 1 , 2 , ⋯. are considered. The operators O − j f z and O j f z satisfy O j O − j f z = O − j O j f z = f z and O − j ∗ O j f z = O j ∗ O − j f z = f ∗ f z for the convolution ∗ of O − j f z and O j f z. In the present paper, the dominants for both operators O − j f z and O j f z and subordinations for O − j f z and O j f z are discussed. Also, new subclass T p γ m , δ , ρ ; m , j concerning with m different boundary points is defined and discussed. Moreover, some interesting problems of T p γ m , δ , ρ ; m , j associated with O j f z are obtained. Furthermore, some interesting examples for our results are considered. [ABSTRACT FROM AUTHOR]

The paper is essentially motivated by the demonstrated potential for applications of the presented results in numerous widespread research areas, such as the mathematical, physical, engineering, and statistical sciences. The main object is to introduce and investigate a class of fractional integral operators involving a certain general family of multiindex Mittag-Leffler functions in their kernel. Among other results obtained in the paper, we establish several interesting expressions for the composition of well-known fractional integral and fractional derivative operators, such as (e.g.) the Riemann–Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above-mentioned fractional integral operator involving the general family of multiindex Mittag-Leffler functions in its kernel. Our main result is a generalization of the results obtained in earlier investigations in this field. We also present some potentially useful integral representations for the product of two members of the general family of multiindex Mittag-Leffler functions in terms of the well-known Fox–Wright hypergeometric function pΨ q with p numerator and q denominator parameters. [ABSTRACT FROM AUTHOR]

We discuss de Branges–Rovnyak spaces ℋ ⁢ (b) generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ (b) , the restriction of the shift operator S on H 2 to ℋ ⁢ (b) , is a strict 2 ⁢ m -isometry and proved that such spaces ℋ ⁢ (b) are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ (b) coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213]. [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]

In this paper, we present a new fixed-point result to draw conclusions about the existence and uniqueness of the solution for a nonlinear Fredholm integral equation of the second kind with non-differentiable Nemytskii operator. To do this, we will transform the problem of locating a fixed point for an integral operator into the problem of locating a solution of an integral equation. Thus, assuming conditions on the Nemytskii operator, we will obtain a global convergence domain for the solution of the considered integral equation, taking for this a uniparametric family of derivativefree iterative processes with quadratic convergence. This result provides us a new fixed-point result for the integral operator considered. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse Laplace transform of the Volterra μ-function (the Bromwich integral) and evaluate it with respect to the Hankel, Schläfli and Bourguet complex contours. In this sense, we establish the generalized Ramanujan's integral representations of the Volterra μ-function for general variations of the parameters. We also discuss the asymptotic analysis of this function with large parameters using the steepest descent method. Further, we show that the solution of Volterra integral equation with differentiated-order fractional integral operator is the Volterra μ-function. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by applying q-calculus. Coefficient inequalities, extreme points, distortion bounds, covering results, convolution condition and convex combination are determined for this class. Finally, we discuss a class preserving integral operator for this class. [ABSTRACT FROM AUTHOR]

We continue studying the σ -Ricci vector field u on a Riemannian manifold (N m , g) , which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold (N m , g) , m > 1 , of positive scalar curvature τ , admits a closed σ -Ricci vector field u such that the vector u − ∇ σ is an eigenvector of T with eigenvalue τ m − 1 , if and only if it is isometric to the m-sphere S α m . In the second result, we show that if a compact and connected T-manifold (N m , g) , m > 2 , admits a σ -Ricci vector field u with σ ≠ 0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature R i c u , u that has a suitable lower bound, then (N m , g) is isometric to the m-sphere S α m , and the converse also holds. Finally, we show that a compact and connected Riemannian manifold (N m , g) admits a σ -Ricci vector field u with σ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature R i c u , u has a lower bound depending on a positive constant α , if and only if (N m , g) is isometric to the m-sphere S α m . [ABSTRACT FROM AUTHOR]

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator D q , λ m ρ , σ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc U. Furthermore, we analyze novel findings with respect to the inverse function (μ − 1) within the class of q-starlike functions in U. The findings in this paper are easy to understand and show a connection between present and past studies. [ABSTRACT FROM AUTHOR]

In this paper, we prove strong type and weak type inequalities of the Hardy-Littlewood maximal operator(M) and fractional Hardy- Littlewood maximal operator(Mα) on variable sequence spaces Ιp(·)(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity. [ABSTRACT FROM AUTHOR]

In this paper, we determine the Hardy spaces of certain integral operators on normalised analytic functions defined in the open unit disk in the complex plane with the prior knowledge of the Hardy spaces of the functions or their derivatives in the integral. [ABSTRACT FROM AUTHOR]