It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace eigenvalue are obtained. Necessary and sufficient conditions ensuring integral inequalities between a difference seminorm and the $L_p$-norm of the gradient are found.
We characterize the fractional Sobolev inequality with fractional isocapacitary and isoperimetric inequalities. We give a sufficient condition and examples so that the fractional capacity of the closure of an open set is bounded above by the fractional perimeter of its interior.
We survey some classical inequalities due to Maz'ya relating isocapacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz'ya's lower bound for the $q$-capacity ($q>1$) in terms of the 1-capacity (or isoperimetric) profile. We then proceed to describe results by Buser, Bakry, Ledoux and most recently by the author, which show that under suitable convexity assumptions on the measure-metric space, Maz'ya's inequality for capacities may be reversed, up to dimension independent numerical constants: a matching lower bound on 1-capa…
Motivared by Carleman's proof of the isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville type theorems.
We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.
We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.
In the paper we prove several inequalities involving two isotonic linear functionals. We consider inequalities for functions with variable bounds, for Lipschitz and H\" older type functions etc. These results give us an elegant method for obtaining a number of inequalities for various kinds of fractional integral operators such as for the Riemann-Liouville fractional integral operator, the Hadamard fractional integral operator, fractional hyperqeometric integral and corresponding q-integrals.
In this paper, we establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite- Hadamard integral inequality for mappings whose derivatives are (h -($α$?;m))-convex.The generalized integral inequalities contribute some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.
In this paper, we establish some new integral inequalities for $(α, m)-$convex functions and quasi-convex functions, respectively. Our results in special cases recapture known results.
This paper presents a number of Kantorovich type integral inequalities involving tensor products of continuous fields of bounded linear operators on a Hilbert space. Kantorovich type inequality in which the product is replaced by an operator mean is also considered. Such inequalities include discrete inequalities as special cases. Moreover, some generalizations of an additive Gruss integral inequality for operators are obtained.
In this paper, some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, which are super-multiplicative or super-additive, are given. These results not only generalize those of Set (2012) and Tunc (2012), but also provide new estimates on these types of Ostrowski inequalities for fractional integrals.
The purpose of this paper is to establish the weighted norm inequalities of one-sided oscillatory integral operators by the aid of interpolation of operators with change of measures.
In this paper, we established some new Hadamard-type integral inequalities for functions whose derivatives of absolute values are m-convex and (α,m)-convex functions via Riemann-Liouville fractional integrals.
In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor's theorem with the integral remainder.
We set up some weighted norm inequalities for fractional oscillatory integral operators. As applications, the corresponding results for commutators formed by $BMO(\mathbb{R}^{n})$ functions and the operators are established.
In this paper, we establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose derivatives are $s$-$(α,m)$-convex.The generalised integral inequalities contribute some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.
In this paper, we have derived certain classical inequalities, namely, Young's, Hölder's, Minkowski's and Hermite-Hadamard inequalities for pseudo-integral (also known as $g$-integral). For Young's, Hölder's, Minkowski's inequalities, both the cases $p>1$ as well as $p<1,\,p\ne 0$ have been covered. Moreover, in the case of Hermite-Hadamard inequality, a refinement has also been proved and as a special case, $g$-analogue of geometric-logarithmic-arithmetic inequality has been deduced.
New identity for fractional integrals have been defined. By using of this identity, some new Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral have been developed. Our results have some relationships with the result of Avci et al., proved in AKO.
Fractional spline wavelet systems are considered in the work. Molecular structure of elements of such systems admits estimates connecting norms of fractional integrals' images and pre--images in Besov spaces.
In this paper, we give a detailed survey on norm inequalities for inner product type integral transformers. We first consider unitarily invariant norms and operator valued functions. We then give results on norm inequalities for inner product type integral transformers in terms of Landau inequality, Gr\" uss inequality. Lastly, we explore some of the applications in quantum theory.
The aim of this paper is to prove the boundedness of the oscillation and variation operators for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces.
In this note we extend some new estimates of the integral $\int_a^b (x-a)^p(b-x)^qf(x)dx$ for functions when a power of the absolute value is $P-$convex.
Weighted $L^p-L^r$ inequalities with arbitrary measurable non-negative weights for positive quasilinear integral operators with Oinarov's kernel on the semiaxis are characterized. Application to the boundedness of maximal operator in the Lorentz $Γ-$spaces is given.
We introduce the concept of Gaussian integral isoperimetric transference and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of $L^{q}$ spaces on the unit $n-$dimensional cube our results extend the recent inequalities that were obtained in \cite{FKS} using extrapolation.
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
In this paper, we introduce the concept of operator arithmetic-geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results. Moreover, some unitarily invariant norm inequalities are established.
In this paper, weighted norm inequalities with $A_p$ weights are established for the multilinear singular integral operators whose kernels satisfy $L^{r'}$-Hörmander regularity condition. As applications, we recover a weighted estimate for the multilinear Fourier multiplier obtained by Fujita and Tomita, and obtain several new weighted estimates for the multilinear Fourier multiplier as well.
The aim of this paper is to establish Hermite-Hadamard, Hermite-Hadamard-Fejér, Dragomir-Agarwal and Pachpatte type inequalities for new fractional integral operators with exponential kernel. These results allow us to obtain a new class of functional inequalities which generalizes known inequalities involving convex functions. Furthermore, the obtained results may act as a useful source of inspiration for future research in convex analysis and related optimization fields.
In this paper, we extend some estimates of the left hand side of a Hermite- Hadamard type inequality for nonconvex functions whose derivatives absolute values are preinvex and log-preinvex.
We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content $\mathcal{H}_\infty^δ$. In particular, if $Ω$ is a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <δ\le n$, we prove that the corresponding $(δp/(δ-p),p)$-Poincaré-Sobolev inequalities hold for all continuously differentiable functions defined on $Ω$ whenever $δ/n < p < δ$. We prove also that the $(p,p)$-Poincaré inequality is valid for all $p>δ/n$.
Applying methods of Real Analysis and Functional Analysis, we build two weight functions with parameters and provide two kinds of parameterized Yang-Hilbert-type integral inequalities with the best constant factors. Equivalent forms, the reverses, and the operator expressions are also given. In particular, the Hardy-type inequalities and Hardy-type operators are studied. Additionally, a number of examples with two kinds of particular kernels are considered.
Let $f \in M_+(\mathbb{R}_+)$, the class of nonnegative, Lebesgure-measurable functions on $\mathbb{R}_+=(0, \infty)$. We deal with integral operators of the form \[ (T_Kf)(x)=\int_{\mathbb{R}_+}K(x,y)f(y)\, dy, \quad x \in \mathbb{R}_+, \] with $K \in M_+(\mathbb{R}_+^2)$.
In this paper, we establish some new Hadamard type inequalities using elementary well known inequalities for functions whose inequalities absolute values are α-, m-, (α,m)-logarithmically convex.
We prove weighted $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given.
In this paper, some Hermite-Hadamard type inequalities are established for harmonically $(α,m)$-convex functions via fractional integrals and some Hermite-Hadamard type inequalities are obtained for these classes of functions.
We prove sharp maximal inequalities for $L^q$-valued stochastic integrals with respect to any Hilbert space-valued local martingale. Our proof relies on new Burkholder-Rosenthal type inequalities for martingales taking values in an $L^q$-space.
In this paper, an integral identity for twice differentiable functions is generalized. Then, by using convexity of |f''| or q-th power of |f''| and with the aid of power mean and Holder's inequalities we achieved some new results. We also gave some applications to quadrature formulas and some special means. Therewithal, by choosing (alpha=1/2) in our main results, we obtained some findings in [13].
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a Banach vector space generated by $m$ $L_μ^p$-spaces for $1\leq p<+\infty.$ Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by $m$ $L_μ^\infty$-spaces instead.
In this paper we obtain some weighted generalizations of Ostrowski type inequalities on time scales involving combination of weighted Δ-integral means, i.e., a weighted Ostrowski type inequality on time scales involving combination of weighted Δ-integral means, two weighted Ostrowski type inequalities for two functions on time scales, and some weighted perturbed Ostrowski type inequalities on time scales. We also give some other interesting inequalities and recapture some known results as special cases.
Khabibullin's conjecture deals with two linear integral inequalities for some non-negative continuous function $q(t)$. The integral in the first of these two inequalities converts $q(t)$ into another function of one variable $g(t)$. This integral yields the direct conversion formula. An inverse conversion formula means a formula expressing $q(t)$ back through $g(t)$. Such an inverse conversion formula is derived.
In this paper, we obtain some new inequalities of Hermite-Hadamard type and Simpson type for functions whose third derivatives belong to Godunova-Levin class.
In this paper, we establish some new Hadamard type inequalities for s-logarithmically convex functions in the second sense via fractional integrals by using Lemma 1 which has been proved by Sarikaya et al. in the paper [3].
In this paper, we describe s-logarithmically convex functions in the first and second sense which are connected with the ordinary logatihmic convex and s-convex in the first and second sense. Afterwards, some new inequalities related to above new definitions are given.
In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes. We extend various classic inequalities, such as the Brunn-Minkowski and the Alexandrov-Fenchel inequality, to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselv…
