Some nonlinear dynamic inequalities on time scales

The aim of this paper is to investigate some nonlinear dynamic inequalities on time scales, which provide explicit bounds on unknown functions. The inequalities given here unify and extend some inequalities in (B G Pachpatte, On some new inequalities related to a certain inequality arising in the theory of differential equation, J. Math. Anal. Appl. 251 (2000) 736–751).     

Some new dynamic inequalities on time scales

In this paper, based on some known dynamic inequalities, we investigate certain new dynamic inequalities on time scales, which provide explicit bounds on unknown functions. Our results unify and extend some continuous inequalities and their corresponding discrete analogues.     

Sharp Inequalities for Ratios of Partition Functions of Schrödinger Operators

This paper derives inequalities for multiple integrals from which inequalities for ratios of integrals of heat kernels of certain Schrödinger operators follows. Such ratio inequalities imply inequalities for the partition functions of these operators which extend the spectral gap results proved by R. Bañuelos and P. Méndez-Hernández and B. Davis.     

Rearrangement Inequalities with Application to Ratios of Heat Kernels

We prove rearrangement inequalities for multiple integrals, using the polarization technique. Polarization refers to rearranging a function with respect to a hyperplane. Then we derive sharp inequalities for ratios of integrals of heat kernels of Schrödinger operators, using our polarization inequalities. These ratio inequalities imply inequalities for the partition functions and extend the results of R. Bañuelos, P.J. Méndez-Hernández and D. You. Mathematics Subject Classifications (2000) 26D15, 28A25, 35J10.     

Half-linear Volterra-Fredholm type integral inequalities on time scales and their applications

The main aim of this paper is to establish some new half-linear Volterra-Fredholm type integral inequalities on time scales. Our results not only extend and complement some known integral inequalities but also provide an effective tool for the study of qualitative properties of solutions of some dynamic equations.     

On the Bernstein Inequality for Rational Functions with a Prescribed Zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc.193(1974), 67–98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero atz=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist–Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.     

时标上的推广的Pachpatte型不等式及其在边值问题中的应用

该文利用单调化技巧研究了时标上的推广的Pachpatte型不等式, 该不等式右端有一个非常数项和三个包含未知函数与没有假设单调性的非线性函数的复合函数的积分项, 不等式左端是未知函数与非线性函数的复合函数. 所得不等式不仅把Pachpatte型不等式的离散形式和连续形式统一起来, 而且推广了已有的时标上的相应不等式. 最后, 用得到的结果研究时标上边值问题解的估计.     

Continuous rearrangement and symmetry of solutions of elliptic problems

This work presents new results and applications for the continuous Steiner symmetrization. There are proved some functional inequalities, e.g. for Dirichlet-type integrals and convolutions and also continuity properties in Sobolev spacesW ^1,p. Further it is shown that the local minimizers of some variational problems and the nonnegative solutions of some semilinear elliptic problems in symmetric domains satisfy a weak, ‘local’ kind of symmetry.     

The application of multi-dimensional Jensen’s inequality for G-martingale

Jensen’s inequalities play a key role in probability theory. In this paper, we aim to construct G-martingales by Jensen’s inequalities in the form of G-expectation, and extend the above results to the n-dimensional case.     

Stieltjes integral inequalities of Gronwall type and applications

Summary We obtain estimates for solutions of integral inequalities of Gronwall type involving Stieltjes integrals and their inverse inequalities. From these we obtain some new results for integral inequalities for Riemann integrals and functional integral inequalities. Extensions are also given to Bihari type integral inequalities.Research supported by NSERC Canada.     

Some inequalities of Qi type for double integrals

In the paper, the authors establish some new inequalities of Qi type for double integrals on a rectangle, from which some known integral inequalities of Qi type may be derived.     

Hadamard-type fractional differential equations for the system of integral inequalities on time scales

ABSTRACT

This paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining the properties of certain classes of partial dynamic equations and difference equations on time scales. As an application, a system of fractional differential equations is considered to explain the value of our results.     

短阵乘积和重排不等式的推广

In this paper, some inequalities for sequence rearrangement and matrix product areestablished. The authors extend and improve some known results, and show that there aresome errors in reference on inequalities for sequence rearrangement by examples,     

Weak sharpness for set-valued variational inequalities and applications to finite termination of iterative algorithms

In this paper, we introduce the notion of weak sharpness for set-valued variational inequalities in the n-dimensional Euclidean space and then present some characterizations of weak sharpness. We also give some examples to illustrate this notion. Under the assumption of weak sharpness, by using the inner limit of a set sequence we establish a sufficient and necessary condition to guarantee the finite termination of an arbitrary algorithm for solving a set-valued variational inequality involving maximal monotone mappings. As an application, we show that the sequence generated by the hybrid projection-proximal point algorithm proposed by Solodov and Svaiter terminates at solutions in a finite number of iterations. These obtained results extend some known results of classical variational inequalities.     

各向异性Heisenberg群上带余项的Hardy型不等式

利用一些非常精细的估计技巧,证明了各向异性Heisenberg群上的一类带余项的Hardy型不等式,推广了最近文献中关于Heisenberg群上的带余项的Hardy型不等式的结果.     

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.     

The Extreme Points of Subsets of <Emphasis Type="Italic">s</Emphasis>-Concave Probabilities and a Geometric Localization Theorem

We prove that the extreme points of the set of s-concave probability measures satisfying a linear constraint are some Dirac measures and some s-affine probabilities supported by a segment. From this we deduce that the constrained maximization of a convex functional on the s-concave probability measures is reduced to this small set of extreme points. This gives a new approach to a localization theorem due to Kannan, Lovász and Simonovits which happens to be very useful in geometry to obtain inequalities for integrals like concentration and isoperimetric inequalities. Roughly speaking, the study of such inequalities is reduced to these extreme points.     

Sharp maximal inequalities for stochastic integrals in which the integrator is a submartingale

We obtain sharp maximal inequalities for strong subordinates of real-valued submartingales. Analogous inequalities also hold for stochastic integrals in which the integrator is a submartingale. The impossibility of general moment inequalities is also demonstrated.

     

Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study.     

Clarkson and Random Clarkson Inequalities for Lr(X)

We first show how (p,p′) Clarkson inequality for a Banach space X is inherited by Lebesgue-Bochner spaces L_r(X), which extends Clarkson's procedure deriving his inequalities for L_p from their scalar versions. Fairly many previous and new results on Clarkson's inequalities, and also those on Rademacher type and cotype at the same time (by a recent result of the authors), are obtained as immediate consequences. Secondly we show that if the (p, p') Clarkson inequality holds in X, then random Clarkson inequalities hold in L_r(X) for any 1 ≤ r ≤ ∞; the converse is true if r = p'. As corollaries the original Clarkson and random Clarkson inequalities for L_p are both directly derived from the parallelogram law for scalars.