The Moser-Trudinger-Onofri Inequality

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks. Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.     

A Multidimensional Integral Inequality Related to Hilbert-Type Inequality

We present a multidimensional integral inequality related to the Hilbert-type inequality and the Carlson-type inequality. As an application, we obtain a sharper form of the Hilbert-type inequality. Carlson-type inequality is also considered.     

REFINEMENTS OF THE FAN－TODD‘S INEQUALITIES

51. IntroductionIn recent years, refinements or interpolations have played an important role on severaltypes of inequalities with new results deduced as a consequence. Please refer to the papers[2, 8, 9, 12], etc. The aim of this paper is to furnish refinements of the Cauchy's and Bessel'sinequalties as shown in Section 2, and also refinements of the Fan-Todd's inequality and theFan-Todd's determinantal inequality in Sections 3 and 4, with an improved condition forequality derived.First of…     

对一道积分不等式题的再思考

利用提升维度的方法并结合几何图形直观分析,给出一道一元函数积分均值不等式的新证明,并将原不等式推广至形式较为对称的不等式,使得原不等式成为新不等式的特例.最后证明新不等式与函数单调递减的定义等价.     

REFINEMENTS OF THE FAN-TODD'S INEQUALITIES

Refinements to inequalities on inner product spaces are presented. In this respect, inequalities dealt with in this paper are: Cauchy's inequality, Bessel's inequality, Fan-Todd's inequality and Fan-Todd's determinantal inequality. In each case, a strictly increasing function is put forward, which lies between the smaller and the larger quantities of each inequality. As a result, an improved condition for equality of the Fan-Todd's determinantal inequality is deduced.     

Strict feasibility and stable solvability of bifunction variational inequalities

In this paper, we study strict feasibility of a bifunction variational inequality. It is proved that a monotone bifunction variational inequality has a nonempty and bounded solution set if and only if it is strictly feasible. Stable solvability of the bifunction variational inequality is discussed under strict feasibility assumption when the domain set is perturbed. Our results generalize earlier results on the classical variational inequality to the case of the bifunction variational inequality.     

Two new forms of half-discrete Hilbert inequality

In this paper, we introduce two new forms of the half-discrete Hilbert inequality. The first form is a sharper form of the half-discrete Hilbert inequality and is related to Hardy inequality. In the second one, we give a differential form of this inequality.     

A multilinear generalisation of the Cauchy-Schwarz inequality

We prove a multilinear inequality which in the bilinear case reduces to the Cauchy-Schwarz inequality. The inequality is combinatorial in nature and is closely related to one established by Katz and Tao in their work on dimensions of Kakeya sets. Although the inequality is ``elementary" in essence, the proof given is genuinely analytical insofar as limiting procedures are employed. Extensive remarks are made to place the inequality in context.

     

Heinz's inequality and Bernstein's inequality

The purpose of the present account is to sharpen Heinz's inequality, and to investigate the equality and the bound of the inequality. As a consequence of this we present a Bernstein type inequality for nonselfadjoint operators. The Heinz inequality can be naturally extended to a more general case, and from which we obtain in particular Bessel's equality and inequality. Finally, Bernstein's inequality is extended to eigenvectors, and shows that the bound of the inequality is preserved.

     

HARDY-SOBOLEV INEQUALITIES WITH GENERAL WEIGHTS AND REMAINDER TERMS

The Hardy-Sobolev inequality with general weights is established, and it is shown that the constant is optimal. The two weights in this inequality are determined by a Bernoulli equation. In addition, the authors obtain the Hardy-Sobolev inequality with general weights and remainder terms. By choosing special weights, it turns to be many versions of the Hardy-Sobolev inequality and the Caffarelli-Kohn-Nirenberg inequality with remainder terms in the literature.     

Rado不等式的多参数推广及应用

通过引入加权幂平均以及参数α、β、μ建立一个新的R ado型不等式,由于所得不等式中含多个参数,因此是一个非常广泛的结果,可以通过对参数的适当取值得到一些已知结果的改进(如著名的Popov iciu不等式的改进),同时也可获得许多新的不等式.最后,我们应用所得结果给出加权幂平均不等式以及加权平均不等式的加细形式.     

A Feng Qi type inequality for Sugeno integral

In this paper, a Feng Qi type inequality for Sugeno integral is shown. The studied inequality is based on the classical Feng Qi type inequality for Lebesgue integral. Moreover, a generalized Feng Qi type inequality for Sugeno integral is proved with several examples given to illustrate the validity of the proposed inequalities.     

Dual mixed Orlicz–Brunn–Minkowski inequality and dual Orlicz mixed quermassintegrals

The dual Orlicz–Brunn–Minkowski inequality is extended from volume to dual quermassintegrals. As application, a dual mixed log-Brunn–Minkowski inequality is obtained. Moreover, dual Orlicz mixed quermassintegrals are defined and a dual Orlicz–Minkowski inequality and a dual mixed log-Minkowski inequality are established.     

Local Poincaré inequalities in non-negative curvature and finite dimension

In this paper, we derive a new set of Poincaré inequalities on the sphere, with respect to some Markov kernels parameterized by a point in the ball. When this point goes to the boundary, those Poincaré inequalities are shown to give the curvature-dimension inequality of the sphere, and when it is at the center they reduce to the usual Poincaré inequality. We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature-dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure. This inequality is optimal in the case of the spheres.     

Radon不等式的新推广

利用Ho。lder不等式、Young不等式、Chebyshev不等式、幂平均不等式建立Radon不等式的指数推广形式,得到一个具有广泛应用价值的不等式.指出文[7]中给出的关于Radon不等式的推广结果是错误的,并在本文中作了修正.     

A Sobolev Interpolation Inequality and a Gross-Sobolev Logarithmic Inequality

Mathematical Notes - A sharp integral inequality is proved and used to obtain a Sobolev interpolation inequality. Further, a new proof of a Gross-Sobolev logarithmic inequality is constructed on...     

A Generalization of the Logarithmic Gross–Sobolev Inequality

Doklady Mathematics - A sharp integral inequality is proved that is used to derive a Sobolev interpolation inequality. A generalization of the logarithmic Sobolev inequality is proposed based on...     

Matrix versions of the Cauchy and Kantorovich inequalities

Summary A version of Cauchy's inequality is obtained which relates two matrices by an inequality in the sense of the Loewner ordering. In that ordering a symmetric idempotent matrix is dominated by the identity matrix and this fact yields a simple proof.A consequence of this matrix Cauchy inequality leads to a matrix version of the Kantorovich inequality, again in the sense of Loewner.     

一个关于多个算子的保序不等式(英文)

本文研究了Furuta型算子不等式问题.利用Lwner-Heinz不等式和Uchiyama不等式,把关于两个算子的保序不等式推广为多个算子的情形,从而推广了Furuta的结果.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.