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In this paper, we first introduce the multiplicative conformable left and right fractional integrals, followed by the derivation of key properties, such as integrability, boundedness, continuity, and the semi-group property, for the newly defined multiplicative conformable fractional integrals. Then, we establish the Hermite–Hadamard inequalities in three distinct senses for multiplicative conformable fractional integrals. Moreover, we present several corresponding midpoint and trapezoidal inequalities for the obtained Hermite-Hadamard inequalities including multiplicative conformable fractional integrals. By special cases, we present the relations between newly obtained inequalities for multiplicative conformable fractional integrals and existing results for multiplicative Riemann–Liouville fractional integrals and multiplicative integrals. Furthermore, we give some new Hermite–Hadamard type, trapezoid type and midpoint type inequalities or multiplicative Riemann–Liouville fractional integrals. Finally, we give several examples and 3D graphs to illustrate the main results. [ABSTRACT FROM AUTHOR]

The concept of a convex function plays a crucial role in fields of mathematical analysis and inequality theory. The importance of convex functions is exemplified by Mercer's inequality. This inequality, which is derived through the application of convex functions, has been extensively studied and is a subject of considerable interest within the mathematical community. This paper presents new generalizations related to Mercer's inequality and compares the inequalities obtained for different values of n with those presented in the existing literature. Also, this study presents pioneering contributions to the field, and we believe that the results in this study will enhance further research on the topic. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a generalized fractional version of Hermite-Hadamard-Mercer type inequalities using h-convex functions by means of ψ-Hilfer fractional integral operators. Then, we give new identities of this type with special functions depending on ψ. Moreover, we establish some new fractional integral inequalities connected with the right- and left-hand sides of Hermite-Hadamard-Mercer inequalities involving differentiable mappings whose absolute values of the derivatives are h-convex. For the development of these novel integral inequalities, we utilize h-Mercer inequality and Hölder's integral inequality. These results offer the significant advantage of being convertible into classical integral inequalities and Riemann–Liouville fractional integral inequalities for convex functions, s-convex functions, and P-convex functions. [ABSTRACT FROM AUTHOR]

Inequalities involving fractional operators have also been an active area of research. These inequalities play a crucial role in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, firstly we established two new identities for the case of differentiable convex functions using generalized tempered fractional integral operators. By utilizing these identities, some novel inequalities like Simpson-type, Bullen-type, and trapezoidal-type are proved for differentiable s-convex functions. Additionally, from the obtained results, several special cases of the known results for different choices of parameters are recaptured. Finally, some applications to q-digamma and modified Bessel functions are given. [ABSTRACT FROM AUTHOR]

The paper discusses corrected Simpson-type inequalities on fractal sets. Based on an introduced identity, we establish some error bounds for the considered formula using the generalized s-convexity and s-concavity of the local fractional derivative. Finally, we present some graphical representations justifying the established theoretical framework as well as some applications. [ABSTRACT FROM AUTHOR]

In this paper, we mainly extend the classical Adams' inequality to its anisotropic type. By using the rearrangement argument, we establish best constants for anisotropic Adams' type inequality with exact growth in R 4 . We give the optimal growth rate of the exponential-type function in the whole space R 4 . Furthermore, using the same method, we also prove anisotropic Adams' type inequality on bounded domain Ω ⊂ R 4 . [ABSTRACT FROM AUTHOR]

The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions: sup ∫ 0 1 w (r) u ′ (r) β + 2 d λ α 1 / (β + 2) ≤ 1 ∫ 0 1 e μ α , θ , γ u β + 2 β + 1 1 − γ d λ θ < + ∞ , where 0 ≤ γ < 1, α = β + 1, μ α , θ , γ ≔ θ + 1 ω α 1 / α 1 − γ 1 1 − γ , w (r) = w 1 (r) = log 1 r γ β + 1 or w (r) = w 2 (r) = log e r γ β + 1 and λθ(E) = ωθ∫Erθdr for all E ⊂ R . The case γ > 1 and γ = 1 are also be considered in this part to improve our paper. Indeed, we have a continuous embedding X(w2) ↪ L∞(0, 1) for γ > 1 and a critical growth of double exponential type for γ = 1. Second, we apply the Lions type Concentration-Compactness principle for Trudinger–Moser inequalities and the precise estimate of normalized concentration limit for normalized concentrating sequence at origin to establish the existence of extremals for Trudinger–Moser inequalities when w (r) = w 1 (r) = log 1 r γ β + 1 and γ > 0 is sufficiently small. [ABSTRACT FROM AUTHOR]

Given a general complete Riemannian manifold M, we introduce the concept of "local Moser–Trudinger inequality on W 1 , n (M) ". We show how the validity of the Moser–Trudinger inequality can be extended from a local to a global scale under additional assumptions: either by assuming the validity of the Poincaré inequality, or by imposing a stronger norm condition. We apply these results to Hadamard manifolds. The technique is general enough to be applicable also in sub-Riemannian settings, such as the Heisenberg group. [ABSTRACT FROM AUTHOR]

The Peetre K-functional is a key object in the development of the real method of interpolation. In this paper we point out a less known relation to wavelet theory and its applications to approximation theory and engineering applications. As a new basis for further development of these studies we present some known properties in the form appropriate for further applications and then derive new information and prove some new results concerning the K-functional and its close relation to (almost) quasi-monotone functions, various indices and interpolation theory. In particular, we extend and unify some known function parameter generalizations of the standard real interpolation spaces (A 0 , A 1) θ , q . [ABSTRACT FROM AUTHOR]

This in-depth study looks at symmetric four-point Newton-Cotes-type inequalities with a focus on error estimates for numerical integration. The precision of these estimates is explored across various classes of functions, including those with bounded variation, bounded derivatives, Lipschitzian derivatives, convex derivatives, and others. The research synthesizes and extends existing knowledge, providing a nuanced understanding of how error bounds depend on the characteristics of integrated functions. Through a systematic review of seminal works, the study contributes to the practical application of numerical integration techniques, offering insight for researchers and practitioners to make informed choices based on the specific features of the functions involved. [ABSTRACT FROM AUTHOR]

In this research article, equality is proved to obtain corrected Euler–Maclaurin-type inequalities. Using this identity, we establish several corrected Euler–Maclaurin-type inequalities for the case of differentiable convex functions by means of conformable fractional integrals. Moreover, some corrected Euler–Maclaurin-type inequalities are given for bounded functions by fractional integrals. Additionally, fractional corrected Euler–Maclaurin-type inequalities are constructed for Lipschitzian functions. Finally, corrected Euler–Maclaurin-type inequalities are considered by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

This paper develops a novel Milne inequality for third-differentiable and h-convex functions using Riemann integrals. Furthermore, new Milne inequalities are proposed utilizing a summation parameter p ≥ 1 for s-convexity, convexity, and P-functions class. We examine cases when the third derivative functions are also bounded and Lipschitzian. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze two new results on the Mittag–Leffler–Ulam–Hyers stability for a class of nonlinear fractional reaction–diffusion equations with delay. We demonstrate that these equations exhibit Ulam–Hyers–Mittag–Leffler stability on a compact interval with respect to the Chebyshev and Bielecki norms, using fixed–point method. These results extend and encompass many previous findings while offering notable improvements. To illustrate the practical applications of our results, we provide three examples. Despite extensive literature on the Lyapunov, Ulam, and Mittag–Leffler stability of fractional equations with and without delays, there is limited research on the Mittag–Leffler–Ulam–Hyers stability of fractional equations with delay. Hence, a key aim of this work is to address this gap by exploring a class of nonlinear fractional reaction–diffusion equations with delay and establishing new results on their Mittag–Leffler–Ulam–Hyers stability. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the deformable fractional derivative. We achieve our results using classical fixed point theorems such as the Banach contraction principle, the Shaefer and the alternative Leray–Schauder theorems. We provide an example to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities to establish nonexistence of extremizers for and improve known sharp Hardy inequalities. Secondly, we establish geometrically interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean distance function, as well as generalize and improve several Hardy type inequalities in the spirit of Brezis and Marcus and spectral estimates of Davies. Lastly, we apply our results to obtain Sobolev inequalities for non-regular Riemannian metrics on geometric exterior domains. [ABSTRACT FROM AUTHOR]

In this study, we establish new generalizations and results for Simpson, midpoint, and trapezoid-type integral inequalities within the framework of multiplicative calculus. We begin by proving a new identity for multiplicatively differentiable functions. Using this identity, we then obtain a new Simpson-type inequality for multiplicatively ss-convex functions. Additionally, we derive novel integral inequalities related to the right and left sides of the Hermite-Hadamard inequality for multiplicatively ss-convex functions. We also demonstrate that some of the results obtained here improve upon known results, while others are generalizations. Finally, we present some applications to special means.

In this article, we first prove some auxiliary results in the form of lemmas using an improved Schwarz lemma at the boundary recently proved by Mercer. Furthermore, we establish some new inequalities for rational functions on the unit disk in the complex plane with prescribed poles and restricted zeros. It is of interest to know that in the bounds of theorems we prove, four extremal coefficients of the numerator polynomial are incorporated rather than usually two coefficients in the existing literature. Moreover, our results strengthen some recent results concerning the inequalities for rational functions and, in turn, produce refinements of some polynomial inequalities as particular cases.

In this paper, we mainly extend the classical Adams’ inequality to its anisotropic type. By using the rearrangement argument, we establish best constants for anisotropic Adams’ type inequality with exact growth in R4 ${\mathbb{R}}^{4}$ . We give the optimal growth rate of the exponential-type function in the whole space R4 ${\mathbb{R}}^{4}$ . Furthermore, using the same method, we also prove anisotropic Adams’ type inequality on bounded domain Ω⊂R4 ${\Omega}\subset {\mathbb{R}}^{4}$ .

In this paper we obtain Ulyanov-type inequalities between mixed moduli of smoothness of positive orders in mixed metrics in the case of limit values of the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present new Hardy-type inequalities with negative parameters on a time scale T . The adopted approach draws upon the use of a reversed Hölder dynamic inequality, a chain rule, and the integration by parts rule on time scales. In the continuous case, our results contain integral inequalities due to Benaissa and Budak, while in the discrete case, the obtained inequalities are essentially new. Additionally, we demonstrate the applicability of our results in the quantum case. [ABSTRACT FROM AUTHOR]

This manuscript explores the idea of a multiplicatively P-superquadratic function and its properties. Utilizing these properties, we derive the inequalities of Hermite–Hadamard type for such functions in the framework of multiplicative calculus. In addition to this, we derive integral inequalities of Hermite–Hadamard type for the product and quotient of multiplicatively P-superquadratic and multiplicatively P-subquadratic functions. Moreover, we develop the fractional version of Hermite–Hadamard type inequalities involving midpoints and end points for multiplicatively P-superquadratic functions with respect to multiplicatively Riemann–Liouville (R.L) fractional integrals. Graphical illustrations based on specific relevant examples validate the credibility of the findings. The study is further stimulating by being pushed with potential applications in terms of moment of random variables, special means, and modified Bessel functions of the first kind. Regarding superquadraticity, the results presented in this work are new, therefore they clearly provide extensions and improvements of the work available in the literature. [ABSTRACT FROM AUTHOR]

In this note, we define p‐adic mixed Lebesgue space and mixed λ‐central Morrey‐type spaces and characterize p‐adic mixed λ‐central bounded mean oscillation space via the boundedness of commutators of p‐adic Hardy‐type operators on p‐adic mixed Lebesgue space. Moreover, we also furnish boundedness of commutators of p‐adic Hardy‐type operators generated with p‐adic mixed λ‐central bounded mean oscillation on mixed λ‐central Morrey space. [ABSTRACT FROM AUTHOR]

In this paper, an identity involving Riemann–Liouville fractional integrals is newly produced. Some new extensions of integral inequalities are proved with the help of Hölder, power-mean integral inequalities and newly established identity. Whereas, the absolute values of first derivatives of real valued functions, which are appeared in these inequalities, are exponentially ( $ \alpha,m) $ α , m) -convex. Some special cases are discussed and graphical evidences are presented for validity of proved results. [ABSTRACT FROM AUTHOR]

In this study, midpoint-type integral inequalities for $ (s,m) $ (s , m) -convex function in the third sense, involving Caputo fractional derivatives and Caputo–Fabrizio integral operators, are demonstrated. Inequalities, containing Caputo–Fabrizio integral operators for functions, whose derivatives are $ (s,m) $ (s , m) -convex function in the third sense, are also established. This article utilized $ (s,m) $ (s , m) -convex function in the third sense to generalized midpoint-type integral inequalities which give more sharp inequalities. [ABSTRACT FROM AUTHOR]

In this work, Korkine and Sonin's identities are defined on arbitrary time scales. These identities are utilized to establish the Chebyshev and Grüss-type inequalities on time scales. By applying these inequalities, we determine the bounds of the remainders in Montgomery identities that incorporate Taylor's formula on time scales. Moreover, we derive discrete and quantum inequalities based on these results. [ABSTRACT FROM AUTHOR]

A novel subgradient evaluation method is proposed for nonsmooth convex relaxations of parametric solutions of ordinary differential equations (ODEs) arising in global dynamic optimization, assuming that the relaxations always lie strictly within interval bounds during integration. We argue that this assumption is reasonable in practice. These subgradients are computed as the unique solution of an auxiliary parametric affine ODE, analogous to classical forward/tangent sensitivity evaluation methods for smooth dynamic systems. Unlike established subgradient evaluation approaches for nonsmooth dynamic systems, this new method does not require smoothness or transversality assumptions, and is compatible with existing subgradient evaluation methods for closed-form convex functions, as implemented in subgradient evaluation software such as EAGO.jl and MC ++. Moreover, we show that a subgradient for a lower-bounding problem in global dynamic optimization can be directly evaluated using reverse/adjoint sensitivity analysis, which may reduce the overall computational effort for an overarching global optimization method. Numerical examples are presented, based on a proof-of-concept implementation in Julia. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper. [ABSTRACT FROM AUTHOR]

The goal of the present article is to offer several new retarded nonlinear inequalities of Gronwall, Bellman, and Pachpatte kind for a class of integral and integro-differential equations. These inequalities generalize and provide new formulations of some well-known results in the mathematical framework of integral and differential inequalities that have been derived currently as well as in earlier times. These results can be utilized to investigate diverse aspects, both qualitative and quantitative, of a class of aforementioned equations. We propose a few applications to ensure the effectiveness of these inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov's and Schaefer's fixed point theorems. An example illustrating our result is provided. [ABSTRACT FROM AUTHOR]

In this research, we investigate the error bounds associated with Milne’s formula, a well-known open Newton–Cotes approach, initially focused on differentiable convex functions within the frameworks of fractional calculus. Building on this work, we investigate fractional Milne-type inequalities, focusing on their application to the more refined class of twice-differentiable convex functions. This study effectively presents an identity involving twice differentiable functions and Riemann–Liouville fractional integrals. Using this newly established identity, we established error bounds for Milne’s formula in fractional and classical calculus. This study emphasizes the significance of convexity principles and incorporates the use of the Hölder inequality in formulating novel inequalities. In addition, we present precise mathematical illustrations to showcase the accuracy of the recently established bounds for Milne’s formula. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in investigating a weighted variant of Hermite–Hadamard type inequalities involving convex functionals. The approach undertaken makes it possible to refine and reverse certain inequalities already known in the literature. It also allows us to provide new weighted convex functional means and establish some related properties with respect to some standard means. [ABSTRACT FROM AUTHOR]

This paper gives a second way to solve the one-dimensional minimization problem of the form : min f ≢ 0 ∫ 0 ∞ f ′ ′ 2 x μ + 1 d x ∫ 0 ∞ x 2 f ′ 2 - ε f 2 x μ - 1 d x ∫ 0 ∞ f ′ 2 x μ d x 2 for scalar-valued functions f on the half line, where f ′ and f ′ ′ are its derivatives and ε and μ are positive parameters with ε < μ 2 4. This problem plays an essential part of the calculation of the best constant in Heisenberg's uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021). [ABSTRACT FROM AUTHOR]

In this manuscript, we demonstrate a graphical comparison analysis of the classical, quantum, and symmetric quantum derivatives for any continuous function, evaluated at z = 1 and q = 0.5 . We introduce the new notions of symmetric Hahn calculus and give the basic interpretations of derivative and integration of a function in Hahn symmetric quantum calculus. This enable us to to derive general Hölder's inequality in symmetric Hahn calculus and as an application, we extract corresponding symmetric Hahn Minkowski's inequality in general settings. Later, we construct an exciting Lemma in which we derive power rule of integration in symmetric Hahn calculus. An example is provided in which we give explicit analysis on four cases. Finally, these findings allow us to construct Hermite–Hadamard inequality and its related kinds in symmetric Hahn calculus. It is pertinent to mention that the particular case of our results recapture the results of symmetric quantum calculus. [ABSTRACT FROM AUTHOR]

The goal of this study is to derive the generalized Levinson-type inequalities in the form of Taylor representation for higher order convex functions. This study uses Taylor’s formula and several novel types of 3-convex Green functions, to establish the novel identities associated with Bullen-type inequalities for higher order convex functions. Moreover, various Levinson-type inequalities are proved for positive real weights using Green functions and Taylor’s formula. [ABSTRACT FROM AUTHOR]

This study explores the L p {L^{p}} -boundedness and compactness characteristics of SG pseudo-differential operators within the Sobolev space framework. We analyze different attributes of minimal-maximal pseudo-differential operators using the SG symbol σ ⁢ ( x , λ ) {\sigma(x,\lambda)} within the class S m 1 , m 2 {S^{m_{1},m_{2}}} , where m 1 {m_{1}} and m 2 {m_{2}} are real numbers. The investigation delves into various aspects of these operators, and we derive a weak solution for the SG pseudo-differential equation by employing the Dunkl transform as part of the aforementioned theoretical framework. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p = 1 and 1 ≤ q < ∞ . This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case 1 < p ≤ q < ∞ . The case p = 1 requires a different argument and does not follow as the limit of known inequalities for p > 1 . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case p = 1 and 1 ≤ q < ∞ . [ABSTRACT FROM AUTHOR]

In the current investigation, we offer the generalized version of q 1 ⁢ q 2 -Simpson's type inequalities via (α , m) -coordinated convex functions. To validate their generalized behavior, we demonstrate the link between our outcomes and the already derived ones. Moreover, we provide some application to special means of positive real numbers to support our findings. The principal outcomes raised in this investigation are extensions and generalizations of the comparable results in the history on Simpson's inequalities for coordinated convex functions. [ABSTRACT FROM AUTHOR]

In the present paper, some new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are established. The integral and discrete Hardy-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales. [ABSTRACT FROM AUTHOR]

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed. [ABSTRACT FROM AUTHOR]

This study establishes Newton-type inequalities for third differentiable and s-convex functions that use the Riemann integral. New Newton-type inequalities are also introduced using a summation parameter p ≥ 1 for various convexity cases. [ABSTRACT FROM AUTHOR]

The theory of inequalities is greatly influenced by interval‐valued concepts, and this contribution is explored from several perspectives and domains. The aim of this note is to develop several mathematical inequalities such as Hermite–Hadamard, Fejér, and the product version based on center radius CR‐order relations. Furthermore, we develop several nontrivial examples and remarks to support the main findings. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to studying some sharp bounds for Hardy-type operators on mixed radial-angular type function spaces. In addition, we will establish the sharp weak-type estimates for the fractional Hardy operator and its conjugate operator, respectively. [ABSTRACT FROM AUTHOR]

The aim of this note is to study the problem $ -{\rm div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u=0 $ − div (| ∇ u | p − 2 ∇u) + V | u | p − 2 u = 0 in Ω, where Ω is a bounded domain in $ \mathds {R}^n $ R n and the potential V is assumed to be not equivalent to zero and lies in $ L_p(\Omega) $ L p (Ω). Also, we establish the strong unique continuation property of the eigenfunctions for the p-Laplacian operator in the case where $ V\in L_p(\Omega) $ V ∈ L p (Ω). [ABSTRACT FROM AUTHOR]

Let Ω be a bounded non-smooth domain in R n that satisfies the measure density condition. In this paper, the authors study the interrelations of three basic types of Besov spaces B p , q s (Ω) , B ˚ p , q s (Ω) and B ~ p , q s (Ω) on Ω , which are defined, respectively, via the restriction, completion and supporting conditions with p , q ∈ [ 1 , ∞) and s ∈ (0 , 1) . The authors prove that B p , q s (Ω) = B ˚ p , q s (Ω) = B ~ p , q s (Ω) , if Ω supports a fractional Besov–Hardy inequality, where the latter is proved under certain conditions on fractional Besov capacity or Aikawa's dimension of the boundary of Ω . [ABSTRACT FROM AUTHOR]

In this research, we define an essential identity for differentiable functions in the framework of tempered fractional integral. By utilizing this identity, we deduce several modifications of fractional Milne-type inequalities. We provide novel expansions of Milne-type inequalities in the domain of tempered fractional integrals. The investigation emphasises important functional categories, including convex functions, bounded functions, Lipschitzian functions, and functions with bounded variation. [ABSTRACT FROM AUTHOR]

In this paper, we derive a quantum analogue of Hermite–Hadamard-type inequalities for twice differentiable convex functions whose second derivatives in absolute value are strongly (α , m) -convex. We obtain new bounds using the H o ¨ lder and power mean inequalities. Moreover, we provide suitable examples in support of our theoretical results. We correlate our findings with comparable results in the literature and show that the obtained results are refinements and improvements. [ABSTRACT FROM AUTHOR]

Let { ρ ν } ν ∈ (0 , ν 0) with ν 0 ∈ (0 , ∞) be a ν 0 -radial decreasing approximation of the identity on R n , X (R n) a ball Banach function space satisfying some extra mild assumptions, and Ω ⊂ R n a W 1 , X -extension domain. In this article, the authors prove that, for any f belonging to the inhomogeneous ball Banach Sobolev space W 1 , X (Ω) , lim ν → 0 + ∫ Ω | f (·) - f (y) | p | · - y | p ρ ν (| · - y |) d y 1 p X (Ω) p = 2 π n - 1 2 Γ (p + 1 2) Γ (p + n 2) ∇ f X (Ω) p , where Γ is the Gamma function and p ∈ [ 1 , ∞) is related to X (R n) . Using this asymptotics, the authors further establish a characterization of W 1 , X (Ω) in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation and some recently found profound properties of W 1 , X (R n) to overcome those difficulties caused by that the norm of X (R n) has no explicit expression and X (R n) might not be translation invariant. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which are all new. Particularly, when X (Ω) : = L p (Ω) with p ∈ (1 , ∞) , this characterization coincides with the celebrated results of J. Bourgain, H. Brezis, and P. Mironescu in 2001 and H. Brezis in 2002; moreover, this characterization is also new even when X (Ω) : = L q (Ω) with both q ∈ (1 , ∞) and p ∈ [ 1 , q) ∪ (q , n n - 1 ] . In addition, the authors give several specific examples of W 1 , X -extension domains as well as W ˙ 1 , X -extension domains. [ABSTRACT FROM AUTHOR]

Let n be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature Q = α 1 f (x 1) + ⋯ + α m f (x m) (with positive weights) of order at least n + 1 and for every n - convex function f, the value of Q on f lies between the values of Gauss-Legendre and Gauss-Lobatto quadratures of order n + 1 calculated for the same function f. We generalize this result in two directions, replacing Q by an integral with respect to a given measure and allowing the number n to any positive integer (for even n Gauss-Radau quadratures replace Gauss-Legendre and Gauss-Lobatto ones). [ABSTRACT FROM AUTHOR]