Hermite-Hadamard Type Inequality via Generalized Superquadratic Functions

Depth images are often accompanied by unavoidable and unpredictable noise. Depth image denoising algorithms mainly attempt to fill hole data and optimise edges. In this paper, we study in detail the problem of effectively filtering the data of depth images under noise interference. The classical filtering algorithm tends to blur edge and texture information, whereas the fractional integral operator can retain more edge and texture information. In this paper, the Grünwald–Letnikov-type fractional integral denoising operator is introduced into the depth image denoising process, and the convolution template of this operator is studied and improved upon to build a fractional integral denoising model and algorithm for depth image denoising. Depth images from the Redwood dataset were used to add noise, and the mask constructed by the fractional integral denoising operator was used to denoise the images by convolution. The experimental results show that the fractional integration order with the best denoising effect was −0.4 ≤ ν ≤ −0.3 and that the peak signal-to-noise ratio was improved by +3 to +6 dB. Under the same environment, median filter denoising had −15 to −30 dB distortion. The filtered depth image was converted to a point cloud image, from which the denoising effect was subjectively evaluated. Overall, the results prove that the fractional integral denoising operator can effectively handle noise in depth images while preserving their edge and texture information and thus has an excellent denoising effect.

Fractional calculus, particularly fractional integral operators, has gained significant attention in recent years due to its wide-ranging applications in various scientific disciplines.This paper provides a comprehensive review of the latest advancements in fractional integral operators, covering theoretical developments, practical applications, and computational methods.We explore the mathematical foundations of fractional integrals, discuss their applications in diverse fields such as signal processing, mathematical physics, and image processing, and examine state-of-the-art computational techniques for their efficient computation.Additionally, we highlight current research trends and future directions in the field of fractional integral operators. I. INTRODUCTIONFractional calculus, a branch of mathematics that generalizes the concepts of differentiation and integration to non-integer orders, has garnered increasing attention in recent years due to its versatile applications across diverse scientific disciplines.Central to the theory of fractional calculus are fractional integral operators, which play a pivotal role in describing non-local and memory-dependent phenomena in complex systems.In this paper, we present a comprehensive review of the latest advancements in fractional integral operators, encompassing theoretical developments, practical applications, and computational methods.Theoretical Foundations of Fractional Integrals, Fractional integral operators, such as the Riemann-Liouville and Caputo integrals, are fundamental tools in fractional calculus.We begin by examining the mathematical foundations of fractional integrals, including their definitions, properties, and relationships with fractional derivatives [1,2].Additionally, we explore important theoretical results related to fractional integrals, such as their behaviour in function spaces and their connection to fractional differential equations [3,4].Applications of Fractional Integrals, Fractional integral operators find applications across a wide range of scientific disciplines, including signal processing, mathematical physics, and image processing.Recent research has demonstrated their utility in modelling anomalous diffusion phenomena [5], describing fractal behaviour in complex systems [6], and analysing non-local effects in viscoelastic materials [7].Moreover, fractional integrals have been applied to noise reduction, feature extraction, and data compression in signal and image processing applications [8,9].Computational Methods for Fractional Integrals, Efficient computation of fractional integrals is essential for their practical use in scientific research and engineering applications.We survey state-of-the-art computational methods for approximating fractional integrals, including numerical integration techniques, spectral methods, and fractional calculus software packages [10,11].Additionally, we discuss challenges and considerations in implementing these methods and provide insights into their computational complexity and accuracy.Current Research Trends and Future Directions, despite significant advancements in the theory and applications of fractional integral operators, several research avenues remain unexplored.Current research trends in the field include the application of fractional calculus to machine learning and data science [12], as well as the development of novel computational methods for solving fractional differential equations [13].Chinta Mani Tiwari (2006).Given A note on Dirac delta function [14], again Chinta Mani Tiwari.(2007).Worked on the Neutrix product of three distributions [15], recently Chinta Mani Tiwari (2023).explained Generalized function and distribution [19].Future research directions may focus on exploring emerging applications and addressing theoretical challenges in the analysis of complex systems using fractional integral operators.In summary, fractional integral operators represent a powerful mathematical tool with widespread applications in science and engineering.By providing a comprehensive overview of theoretical foundations,

The article proposes a generalized polynomial projection method for solving linear integral equations with a fractional Riemann - Liouville integral in the main part. Fractional-integral equations have numerous applications in various applied problems, in particular, in problems of plasma diagnostics, optical and X-ray diffraction, in ultrasonic measurements in moving media, in metallurgy, biology, microscopy, seismology, in astrophysics and other fields of science.The equation we are considering is an equation of the first kind and therefore, generally speaking, refers to incorrectly set by Hadamard equations. The latter circumstance is connected with the fact that in known function spaces the fractional integral Riemann-Liouville operator is completely continuous. All this imposes its own characteristics on the construction and study of approximate methods for solving integral equations with a fractional integral operator in the main part. It should be noted that the operator of the projection method is not necessarily projective, which allows the construction of computational schemes using the methods of summation of Fourier series and interpolation polynomials. A theoretical and functional substantiation of the proposed projection method was carried out, based on the correct formulation of the equation in a pair of specially selected different Holder spaces. In particular, it proved the unique solvability of the system of linear algebraic equations of the method and the convergence of the constructed approximations to the exact solution in the norm of the space of Holder functions and, as a consequence, in the uniform metric. This also implies the substantiation of specific well-known projection methods such as the Galerkin methods, colocations, and subdomains.

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ,m)-convex functions, m-harmonic harmonically convex functions, (s,r)-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α,m)-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ-convex functions, MT-convex functions, (s,m)-convex functions, p-convex functions, h-convex functions, σ-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ,s-convex functions, modified (p,h)-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α,h−m)−p-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k,s)-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

A Fractional integral operator is one of the operators in mathematical analysis. Fractional integral operator itself maps any real-valued function into the integral form of the division of the at function. One of the expansion of fractional integral operator is generalized fractional integral operator. Morrey space is an extension of Lebesgue space. Morrey space is the set of all Lebesgue measurable functions, whose norm is finite over Morrey space. In this study, we will discuss the inequalities of the generalized fractional integral operator on a generalized non-homogeneous Morrey space. We proved of this inequality using the Chebyshev inequality and the Holder inequality.

In this paper, we derive some subordination and superordination results for the generalized "Srivastava-Attiya" fractional integral operator. Some interesting corollaries for this operator is also obtained.

Performing a thought study of fractional inequalities by means of convexity and fractional operators has conspicuous work in the field of analysis. The main object of this paper is to discuss the coordinated convexity, pre-invexity, and also establish fractional double integral operators (FDIO) having generalized Bessel-Maitland function as used its kernel. We develop a new generation of Hermite-Hadamard (H-H) and trapezoid-type inequalities through different types of coordinated convexities and pre-invexities with successfully implementation of newly designed fractional double integral operators. Moreover, we extract some corollaries, which are generalization of well-known inequalities for different coordinated convexities that show a strong consolidation of our main results.

<abstract><p>In this article, a new fractional mathematical model is presented to investigate the contagion of the human immunodeficiency virus (HIV). This model is constructed via recent improved fractional differential and integral operators. Other operators like Caputo, Riemann-Liouville, Katugampola, Jarad and Hadamard are being extended and generalized by these improved fractional differential and integral operators. Banach's and Leray-Schauder nonlinear alternative fixed point theorems are utilized to examine the existence and uniqueness results of the proposed fractional HIV model. Moreover, different kinds of Ulam stability for the fractional HIV model are established. It is simple to recognize that the extracted results can be reduced to some results acquired in multiple works of literature.</p></abstract>

<abstract> In this paper, the author studies the boundedness for a large class of sublinear operators $ T_\alpha, \alpha\in[0, n) $ generated by Calderón-Zygmund operators ($ \alpha=0 $) and generated by fractional integral operator ($ \alpha&gt;0 $) on generalized mixed Morrey spaces $ M^\varphi_{\vec{q}}({\mathbb R}^n) $. Moreover, the boundeness for commutators of $ T_\alpha, \alpha\in[0, n) $ on generalized mixed Morrey spaces $ M^\varphi_{\vec{q}}({\mathbb R}^n) $ is also studied. As applications, we obtain the boundedness for Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operators, fractional integral operator, fractional maximal operator and their commutators on generalzied mixed Morrey spaces. </abstract>

The main aim of this article is to design a novel framework to study a generalized fractional integral operator that unifies two existing fractional integral operators. To ensure the suitable selection of the operator and with the discussion of special cases, it is shown that our considered fractional integral generalizes the well-known Atangana–Baleanu fractional integral (AB-fractional integral) and the ABK-fractional integral. Conditions are stated for the generalized AB-fractional integral operator (GAB-fractional integral operator) to be bounded in the space Xcp (γ1,γ2). We also provide a fractional product-integration formula for this operator. Furthermore, we generalize the reverse Minkowski’s inequality and the reverse Hölder-type inequality by utilizing the GAB-fractional integral operator. Additionally, some other types of integral inequalities are established, and several special cases are noted. The concepts in this article may influence further research in fractional calculus.

The aim of the present paper is to optimize the values of different parameters related to the image denoising algorithm involving Caputo Fabrizio fractional integral operator of non-singular type with the Mittag-Leffler function in generalized form. The algorithm aims to find the coefficients of a kernel to remove the noise from images. The optimization of kernel coefficients are done on the basis of different numerical parameters like Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structure Similarity Index measure (SSIM) and Image Enhancement Factor (IEF). The performance of the proposed algorithm is investigated through above-mentioned numeric parameters and visual perception with the other prevailed algorithms. Experimental results demonstrate that the proposed optimized kernel based on generalized fractional operator performs favorably compared to state of the art methods. The uniqueness of the paper is to highlight the optimized values of performance parameters for different values of fractional order. The novelty of the presented work lies in the development of a kernel utilizing coefficients from a fractional integral operator, specifically involving the Mittag-Leffler function in a more generalized form.

In this paper, we prove that if a multilinear operator $\mathcal{T}$ and its multilinear commutator $\mathcal{T}_{\Sigma\vec{b}}$ and iterated commutator $\mathcal{T}_{\Pi\vec{b}}$ for $\vec{b}\in(\mathbb{R}^n)^m$ are bounded on product weighted Lebesgue space, then $\mathcal{T}$, $\mathcal{T}_{\Sigma\vec{b}}$ and $\mathcal{T}_{\Pi\vec{b}}$ are also bounded on product weighted Amalgam space. As its applications, we show that multilinear Littlewood-Paley functions and multilinear Marcinkiewicz integral functions with kernels of convolution type and non-convolution type, and their multilinear commutators and iterated commutators are bounded on product weighted Amalgam space. We also consider multilinear fractional type integral operators and their commutators' behaviors on weighted amalgam space. In order to deal with the endpoint case, we introduce the amalgam-Campanato space and show that fractional integral integral operator are bounded operators from product amalgam space to amalgam-Campanato space. What should point out is that even if in the linear case, our results for fractional integral operator are also new.

We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some new identities involving these new integral operators and obtained new fractional integral inequalities of the midpoint and trapezoidal type for functions whose derivatives are bounded or convex.

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green’s function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area.

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.