The best possibility of the grand Furuta inequality

Let be invertible bounded linear operators on a Hilbert space satisfying , and let be real numbers satisfying Furuta showed that if , then . This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality
and the Ando-Hiai inequality ( ).

In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if , then there exist invertible matrices with which do not satisfy .

     

Some Bonnesen-style inequalities for higher dimensions

We discuss the higher dimensional Bonnesen-style inequalities.Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in R n(n ≥ 3) are known.The results obtained in this paper are for general domains,convex or non-convex,in Rn.     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

Best possibility of the Furuta inequality

Let , and . Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . In this paper, we prove that the range and is best possible with respect to the Furuta inequality, that is, if or , then there exist which satisfy but .

     

Refinements of generalized triangle inequalities

In this paper, we discuss refinements of the well-known triangle inequality and it is reverse inequality for strongly integrable functions with values in a Banach space X. We also discuss refinement of a generalized triangle inequality of the second kind for Lp functions. For both cases, the attainability of the equality is also investigated.     

The Furuta inequality with negative powers

Let be bounded linear operators on a Hilbert space satisfying . Furuta showed the operator inequality
as long as positive real numbers satisfy and . In this paper, we show this inequality is valid if negative real numbers satisfy a certain condition. Also, we investigate the optimality of that condition.

     

On Mulholland's inequality

H.P. Mulholland has presented a sufficient condition for a generalization of the Minkowski inequality and another such condition was given by R.M. Tardiff. We show that Mulholland's condition implies Tardiff's, but that the converse is false.

     

用函数模型证明不等式

不等式的证明很具技巧性,本文利用构造函数模型的方法,将不等式的证明转化为函数模型,使得解题简洁明了.     

A note on periodic solutions of nonautonomous second-order systems

A multiplicity theorem is obtained for periodic solutions of nonautonomous second-order systems with partially periodic potentials by the minimax methods.

     

关于平面凸多边形的一个Bonnesen型不等式

本文探索了关于平面凸多边形的Bonnesen型不等式.利用分析方法,先构造一个解析函数的不等式,进而得到了一个关于平面凸多边形的Bonnesen型不等式.     

逆向型Hilbert-Pachpatte不等式

建立了逆向型Hilbert-Pachpatte不等式,推广和改进了离散型和连续型Pach- patte不等式的逆.     

关于曲率积分不等式的注记

本文主要研究平面卵形线的曲率积分不等式.利用积分几何中凸集的支持函数以及外平行集的性质,得到了Gage等周不等式与曲率的熵不等式的一个积分几何的简化证明;进一步地,我们得到了一个新的关于曲率积分的不等式.     

A note on Hardy-type inequalities

We use a theorem of Cartlidge and the technique of Redheffer's ``recurrent inequalities" to give some results on inequalities related to Hardy's inequality.

     

柯西-布涅柯夫斯基不等式在优化中的应用

本文给出了柯西—布涅柯夫斯基不等式在非线性规划中应用的两个例子．     

两平面凸域的对称混合等周不等式

设K_k(k=i,j)为欧氏平面R~2中面积为A_k,周长为P_k的域,它们的对称混合等周亏格(symmetric mixed isoperimetric deficit)为σ(K_i,K_j)=P_i~2P_j~2-16π~2A_iA_j.根据周家足,任德麟(2010)和Zhou,Yue(2009)中的思想,用积分几何方法,得到了两平面凸域的Bonnesen型对称混合不等式及对称混合等周不等式,给出了两域的对称混合等周亏格的一个上界估计.还得到了两平面凸域的离散Bonnesen型对称混合不等式及两凸域的对称混合等周亏格的一个上界估计,并应用这些对称混合(等周)不等式估计第二类完全椭圆积分.     

Extensions of Heinz-Kato-Furuta inequality

We give an extension of Lin's recent improvement of a generalized Schwarz inequality, which is based on the Heinz-Kato-Furuta inequality. As a consequence, we can sharpen the Heinz-Kato-Furuta inequality.

     

Generalized Bebiano-Lemos-Providência inequalities and their reverses

We improve Bebiano-Lemos-Providência inequality: For A,B?0

     

Generalized Transportation-Cost Inequalities and Applications

For μ: = e ^V(x)dx a probability measure on a complete connected Riemannian manifold, we establish a correspondence between the Entropy-Information inequality and the transportation-cost inequality for μ(f ²) = 1, where Φ and Ψ are increasing functions. Moreover, under the curvature–dimension condition, a Sobolev type HWI (entropy-cost-information) inequality is established. As applications, explicit estimates are obtained for the Sobolev constant and the diameter of a compact manifold, which either extend or improve some corresponding known results. Supported in part by NNSFC(10721091) and the 973-project in China.     

Weighted Modular Inequalities for Hardy Type Operators

Given weight functions , w, and v, the weighted modular inequality is characterized. Here Qis a strictly increasing function with Q(0) = 0, Q() = and2Q(x) Q(C x), P is a Young's function, and T is the Hardy operatoror a Hardy type operator. In particular, a characterizing conditionfor the Hardy type operator to map L^p(w) to L^q(v) when 0 <q < 1 p < is deduced. In addition, a new proof for theMaz'ja-Sinnamon theorem is given, and weighted Lorentz norminequalities for Hardy type operators are established. 1991Mathematics Subject Classification: primary 26D15, 42B25; secondary26A33, 46E30.     

再谈Minc-Sathre不等式的上下界

用调和平均值、均值不等式之间的关系及对数平均不等式,对Minc-Sathre不等式的上下界进行改进,使结论更精确.