Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Some Refinement of Holder’s and Its Reverse Inequality

Holder’s inequality, its refinement, and reverse have received considerable attention in the theory of mathematical analysis and differential equations. In this paper, we give some refinements of Holder’s inequality and its reverse using a simple analytical technique of algebra and calculus. Our results show many results related to holder’s inequality as special cases of the inequalities presented.

New Majorized Results on Hilbert's Integral Inequality

In this paper, some new majorized results on Hilbert’s integral inequality are showed.

On an Extension and a Refinement of Van der Corput's Inequality

By introducing a parameter α, we give an extension of Van der Corput＇s inequality. Also a new strengthened version of it is considered.

THE HADAMARD INEQUALITY FOR CONVEX FUNCTION VIA FRACTIONAL INTEGRALS

In this paper, we establish several inequalities for some differantiable mappings that are connected with the Riemann-Liouville fractional integrals. The analysis used in the proofs is fairly elementary.

A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane

In this paper, by introducing some parameters and estimating the weight coefficient, we give a new Hilbert’s inequality with the integral in whole plane and with a non-homogeneous and the equivalent form is given as well. The best constant factor is calculated by the way of Complex Analysis.

GENERALIZED ROSENTHAL'S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.

Bohr's Inequality on the Unit Ball B^n

Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.

Necessary and Sufficient Conditions of Doubly Weighted Hardy-Littlewood-Sobolev Inequality

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.

On Reverses of the Blaschke-Santalo Inequality for Convex Bodies

Reisner proved a reverse of the Blaschke-Santal5 inequality for zonoid bodies, Bourgain and Milman showed another reverse of the Blaschke-Santal5 inequality for centered convex bodies. In this paper, two reverses of the Blaschke-Santal5 inequality for convex bodies are given by the Petty projection inequality and above two reverses. Further, using above methods, we also obtain two analogues of the Petty＇s conjecture for projection bodies, respectively.

On an Extension of Hibert's Inequality and Applications

This paper gives a new generalization of Hilbert＇s inequality with a best constant factor involving the β function. An applications, we consider the equivalent form and some particular results.

Refinements to Hadamard’s Inequality for Log-Convex Functions

In this paper we show that a log-convex function satisfies Hadamard's inequality, as well as we give an extension for this result in several directions.

DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.

Inequalities relating to L_p-version of Petty's conjectured projection inequality

Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.

N次幂下S-凸函数和其性质研究

以凸函数、S-凸函数、平方凸函数、平方S-凸函数的定义和性质为研究基础,根据从平方凸函数到平方S-凸函数的演变推进过程,采用了凸函数概念的代数不等式形式,给出了N次幂下S-凸函数的概念,并对其判别定理和性质进行了讨论,定义并证明了N次幂下S-凸函数的Jensen型不等式。研究表明,N次幂下S-凸函数包含了凸函数、平方凸函数、平方S-凸函数的性质,是凸函数的一种广义凸函数的推广,开辟了全新的推广途径。

Rearrangement Inequality and Chebyshev＇s Sum Inequality on Positive Tensor Products of Orlicz Sequence Space with Banach Lattice

Let φ be an Orlicz function that has a complementary function φ＊ and let lφ be an Orlicz sequence space. We prove a similar version of Rearrangement Inequality and Chebyshev＇s Sum Inequality in lφ FX, the Fremlin projective tensor product of lφ with a Banach lattice X, and in lφ iX, the Wittstock injective tensor product of lφ with a Banach lattice X.

Some Improvement on Hilbert's Inequality and Its Application

A new improvement of Hilbert's inequality for double series can be establishedby means of a strengthened Cauchy's inequality. As application, a quite sharp result onFejer-Riesz's inequality is obtained.

LOOMIS-WHITNEY'S INEQUALITY ABOUT MIXED BODY

In this article, the authors establish two theorems for mixed body, which are the generalizations of the well-known Loomis-Whitney＇s inequality.

Extension of reverse Hilbert-type inequality with some parameters

By introducing some parameters and estimating the weight function,we obtain an extension of reverse Hilbert-type inequality with the best constant factor.As applications,we build its equivalent forms and some particular results.

The relation between Hardy's non-locality and violation of Bell inequality

We give an analytic quantitative relation between Hardy＇s non-locality and Bell operator. We find that Hardy＇s non-locality is a sufficient condition for the violation of Bell inequality, the upper bound of Hardy＇s non-locality allowed by information causality just corresponds to Tsirelson bound of Bell inequality and the upper bound of Hardy＇s non- locality allowed by the principle of no-signaling just corresponds to the algebraic maximum of Bell operator. Then we study the CabeUo＇s argument of Hardy＇s non-locality （a generalization of Hardy＇s argument） and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy＇s non-locality under the constraint of information causality with the aid of the above derived relation between Hardy＇s non-locality and Bell operator.