Fractional Sobolev-type inequalities

Here we present univariate Sobolev-type fractional inequalities involving fractional derivatives of Canavati, Riemann–Liouville and Caputo types. The results are general L _p inequalities forward and converse on a closed interval. We give an application to a fractional ODE. We present also the mean Sobolev-type fractional inequalities.     

On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.     

Reduction of Opial-type inequalities to norm inequalities

Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the nonincreasing rearrangement are presented.

     

On general Bernstein and Nikol’ski type inequalities

The goal of this paper is to establish the relations between general Bernstein and Nikol’ski type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a family of functions equipped with Bernstein type inequality which satisfies Nikol’ski type inequality is found.     

Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy-entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré-type inequalities with weights are derived. Optimal constants are compared to the usual Poincaré constant.     

Laeng-Morpurgo-Type Uncertainty Inequalities for the Laguerre Transform

In this work,we prove Clarkson-type and Nash-type inequalities for the Laguerre transform■on M=[0,∞)×R.By combining these inequalities,we show Laeng-Morpurgo-type uncertainty inequalities.We establish also a local-type uncertainty inequalities for the Laguerre transform■,and we deduce a Heisenberg-Pauli-Weyl-type inequality for this transform.     

Functional inequalities for the Wright functions

In this paper, our aim is to show some mean value inequalities for the Wright function, such as Turán-type inequalities, Lazarevi?-type inequalities, Wilker-type inequalities and Redheffer-type inequalities. Moreover, we prove monotonicity of ratios for sections of series of Wright functions, the result is also closely connected with Turán-type inequalities. In the end of the paper, we present some other inequalities for the Wright function.     

关于Polya-Szeg不等式

本文给出两个改进的不等式,使改进后的每一个新的不等式均含有Polay-Szego两个不等式改进在内．     

Some inequalities involving π and an application to Hilbert's inequality

Some inequalities involving π are proved. As an application, we give some strengthened Hilbert's inequalities.     

Variational Formulas of Poincaré-type Inequalities for Birth-Death Processes

Abstract In author’s one previous paper, the same topic was studied for one dimensional diffusions. As a continuation, this paper studies the discrete case, that is the birth-death processes. The explicit criteria for the inequalities, the variational formulas and explicit bounds of the corresponding constants in the inequalities are presented. As typical applications, the Nash inequalities and logarithmic Sobolev inequalities are examined. Research supported in part by NSFC (No. 10121101), 973 Project and RFDP     

关于广义变分不等式与广义拟变分不等式

本文在非常一般的框架下,建立了极大极小不等式,广义变分不等式和广义拟变分不等式,证明了解的存在定理,且它们是在非紧集上得到的,从而推广和改进了[3～13]中的相应结果.     

Operator inequalities and reverse inequalities related to the Kittaneh–Manasrah inequalities

This work is concerned with exploring more refinement forms of the Young inequalities and the Kittaneh–Manasrah inequalities. We deduce the Operator version inequalities and reverse version inequalities related to the Kittaneh–Manasrah inequalities.     

关于两个新型Hilbert不等式的推广

众所周知,Hilbert不等式是一个有重要应用的不等式,1998年,B．G．Pachpatte给出了类似Hilbert级数不等式的两个新不等式,本文的主要工作是推广了这两个新个新不等式。     

On first order interpolation inequalities with weights on the Heisenberg group

In this paper, sufficient and necessary conditions for the first order interpolation inequalities with weights on the Heisenberg group are given. The necessity is discussed by polar coordinates changes of the Heisenberg group. Establishing a class of Hardy type inequalities via a new representation formula for functions and Hardy-Sobolev type inequalities by interpolation, we derive the sufficiency. Finally, sharp constants for Hardy type inequalities are determined.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002     

Grüss and Ostrowski type inequalities

In this paper we use a method originated in [S. S. Dragomir, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure Appl. Math. 4 (2) (2003) Article 42] to establish some Grüss and Ostrowski type inequalities.     

一类新的Radon型不等式及其应用

给出一类新的R adon型不等式,它们在代数不等式研究中有着广泛的应用,利用它们可直接得到一大批新的分式型不等式,也可运用它们证明或推广许多不等式.     

Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations

In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnack's inequality) for a class of degenerate elliptic equations with measurable coefficients.     

On Variance and Covariance for Bounded Linear Operators

In this paper we initiate a study of covariance and variance for two operators on a Hilbert space, proving that the c-v (covariance-variance) inequality holds, which is equivalent to the Cauchy-Schwarz inequality. As for applications of the c-v inequality we prove uniformly the Bernstein-type inequalities and equalities, and show the generalized Heinz-Kato-Furuta-type inequalities and equalities, from which a generalization and sharpening of Reid's inequality is obtained. We show that every operator can be expressed as a p-hyponormal-type, and a hyponormal-type operator. Finally, some new characterizations of the Furuta inequality are given. Received April 9, 2000, Revised July 20, 2000, Accepted August 8, 2000     

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.