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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]

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]

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]

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]

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]

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]

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]

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 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]

Motivated by certain results known from the literature, in this paper we give several new dynamic inequalities for B-monotone functions with respect to time scales nabla calculus. If the time scale represents the set of real numbers, our results reduce to integral inequalities known from the literature. On the other hand, in the setting of positive integers, we obtain new discrete inequalities for B-monotone sequences. [ABSTRACT FROM AUTHOR]

In this study, we firstly derive a Wirtinger-type result, which gives the connection in between the integral of square of a function and the integral of square of its Caputo fractional derivatives with the help of left-sided and right-sided fractional Taylor's Formulas. Afterward, we provide a more general inequality involving Caputo fractional derivatives for L r norm with r > 1 via Hölder's inequality. Also, similar inequalities for Riemann–Liouville fractional derivatives are presented by means of a relation between Caputo fractional derivatives and Riemann–Liouville fractional derivatives. [ABSTRACT FROM AUTHOR]

We present Hermite–Hadamard–Fejér type inequalities for strongly MφMψ -convex functions. Some refinements of them and bounds for the integral mean of the product of two functions are also obtained. [ABSTRACT FROM AUTHOR]

In this article we study various forms of the Hardy inequality for affine connections on a complete noncompact Riemannian manifold, including the two-weight Hardy inequality, the improved Hardy inequality, the Rellich inequality, the Hardy–Poincaré inequality and the Heisenberg–Pauli–Weyl inequality. Our results improve and include many previously known results as special cases. [ABSTRACT FROM AUTHOR]

Our main focus in this paper is to explore the mapping properties for the n-dimensional fractional Hausdorff operator H Φ , β from L p ( R n , | x | α) to L q ( R n , | x | γ) , where p , q < 1 (p , q ≠ 0) , α , γ ∈ R , 0 ≤ β < n and Φ is a nonnegative measurable function on R n . For p , q < 1 (p , q ≠ 0) satisfying some additional assumptions, we give sufficient conditions for the validity of the reverse fractional Hausdorff inequality H Φ , β f L q ( R n , | x | γ) ≥ C ‖ f ‖ L p ( R n , | x | α) for some positive constant C and all nonnegative functions f ∈ L p ( R n , | x | α) . For the particular case 0 < p = q < 1 , we obtain the sharp reverse fractional Hausdorff inequality. As applications, we establish the sharp reverse inequalities for the n-dimensional fractional Hardy operator and its adjoint operator, and also the n-dimensional fractional Hardy–Littlewood–Pólya operator on power-weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Adams inequalities with exact growth conditions are derived for Riesz-like potentials on metric measure spaces. The results extend and improve those obtained recently on R n by the second author, for Riesz-like convolution operators. As a consequence, we will obtain new sharp Moser–Trudinger inequalities with exact growth conditions on R n , the Heisenberg group, and Hadamard manifolds. On R n such inequalities will be used to prove the existence of radial ground states solutions for a class of quasilinear elliptic equations. [ABSTRACT FROM AUTHOR]

In this article, we will demonstrate some Hardy’s inequalities by utilizing Hölder inequality, integration by parts, and chain rule of the conformable fractional calculus. When α=1\alpha =1, we can obtain some of classical Copson’s and Hardy’s inequalities. In the end, we will obtain an application of Hardy’s inequality utilizing the conformable fractional calculus.

By using the weight coefficients and the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given. The equivalent statements of the best value related to several parameters are provided. The equivalent forms, some inequalities in particular parameters, and the operator expressions are obtained.

We present Hermite–Hadamard–Fejér type inequalities for strongly MφMψ -convex functions. Some refinements of them and bounds for the integral mean of the product of two functions are also obtained.

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]

This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences. [ABSTRACT FROM AUTHOR]

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker L p -Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we define the p -adic mixed Morrey type spaces and study the boundedness of p -adic fractional integral operators and their commutators on these spaces. More precisely, we first obtain the boundedness of p -adic fractional integral operators and their commutators on p -adic mixed central Morrey spaces. Moreover, we further extend these results on p -adic generalized mixed Morrey spaces, when a symbol function b belongs to the p -adic generalized mixed Campanato spaces. [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 {\frac{1}{2}<\alpha<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]

Milne-type inequalities for \(h\)-convex functions involving conformable operators are established. Additionally, new results are presented that generalize various known inequalities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 (| ∇ u | 2 + V (x) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ , where 0 < β < 2 , V (x) ≥ 0 and may vanish on an open set in R 2 . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 : (0.2) − Δ u + γ u = f (u) | x | β , where the nonlinearity f has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, "Concentration-compactness principle of singular Trudinger-Moser inequalities in R n and n -Laplace equations," Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 : (0.3) − Δ u + V (x) u = f (u) | x | β . [ABSTRACT FROM AUTHOR]

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces. [ABSTRACT FROM AUTHOR]

We use a new function class called B-function to establish a novel version of Hermite–Hadamard inequality for weighted ψ-Hilfer operators. Additionally, we prove two new identities involving weighted ψ-Hilfer operators for differentiable functions. Moreover, by employing these equalities and the properties of the B-function, we derive several trapezoid- and midpoint-type inequalities for h-convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making specific choices of the function h. [ABSTRACT FROM AUTHOR]

Unfortunately, eleven of our provinces were severely affected due to two severe earthquakes that occurred in our country, the Republic of Turkey, on February 6, 2023. As a result, thousands of buildings were destroyed and tens of thousands of our citizens lost their lives. From past to present, such disasters have occurred in many parts of our world and will continue to happen. In order to raise awareness for researchers and academicians reading our article, we will give a new definition of convexity in this article, and we will call it "earthquake convexity". In this paper, we study some algebraic properties of the earthquake convexity. Then we compare the results obtained with both Hölder, Hölder–İşcan inequalities and power-mean, improved power-mean integral inequalities and show that the results obtained with Hölder–İşcan and improved power-mean inequalities are better than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Titchmarsh theorem for Laguerre hypergroup K = [ 0 , + ∞ [ × R , via moduli of continuity of higher orders. [ABSTRACT FROM AUTHOR]

We characterize the good weights for some weighted weak-type iterated Hardy-Copson inequalities to hold. [ABSTRACT FROM AUTHOR]

We start providing a quantitative stability theorem for the rigidity of an overdetermined problem involving harmonic functions in a punctured domain. Our approach is inspired by and based on the proof of rigidity established in Enciso and Peralta-Salas (Nonlinear Anal 70(2):1080–1086, 2009), and reveals essential differences with respect to the stability results obtained in the literature for the classical overdetermined Serrin problem. Secondly, we provide new weighted Poincaré-type inequalities for vector fields. These are crucial tools for the study of the quantitative stability issue initiated in Poggesi (Soap bubbles and convex cones: optimal quantitative rigidity, 2022. arXiv:2211.09429) concerning a class of rigidity results involving mixed boundary value problems. Finally, we provide a mean value-type property and an associated weighted Poincaré-type inequality for harmonic functions in cones. A duality relation between this new mean value property and a partially overdetermined boundary value problem is discussed, providing an extension of a classical result obtained in Payne and Schaefer (Math Methods Appl Sci 11(6):805–819, 1989). [ABSTRACT FROM AUTHOR]

This paper investigates a technique that uses Riemann-Liouville fractional integrals to study several Euler-Maclaurin-type inequalities for various function classes. Afterwards, we provide our results by using special cases of obtained theorems and This paper is to derive examples. Moreover, we give some Euler-Maclaurin-type inequalities for bounded functions by fractional integrals. Furthermore, we construct some fractional Euler-Maclaurin-type inequalities for Lipschitzian functions. Finally, we offer some Euler-Maclaurin-type inequalities by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

In this article, we apply the Fourier transform to prove the Hyers-Ulam and Hyers-Ulam-Rassias stability for the first- and second-order nonlinear differential equations with initial conditions. Additionally, we extend the results to investigate the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of these differential equations using the proposed method.

We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.