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.     

Ostrowski type inequalities on H-type groups

The main aim of this paper is to establish an Ostrowski type inequality on H-type groups using the L^∞ norm of the horizontal gradient. The work has been motivated by the work of Anastassiou and Goldstein in [G.A. Anastassiou, J.A. Goldstein, Higher order Ostrowski type inequalities over Euclidean domains, J. Math. Anal. Appl. 337 (2008) 962-968].     

Multidimensional Ostrowski inequalities, revisited

New very general multidimensional Ostrowski type inequalities are established, some of them prove to be sharp. They involve the · and ·_p norms of the engaged mixed partial of nth order n1. In establishing them, other important multivariate results of Montgomery type identity are developed and presented for the first time.     

Another generalization of weighted Ostrowski type inequality for mappings of bounded variation

A new generalization of weighted Ostrowski type inequality for mappings of bounded variation with a unified sharp bound is established.     

Lyapunov-type integral inequalities for certain higher order differential equations

In this paper, first we will give a short survey of the most basic results on Lyapunov inequality, and next we obtain this-type integral inequalities for certain higher order differential equations. Our results are sharper than some results of Yang (2003) [20].     

On Anastassiou's Generalizations of the Ostrowski Inequality and Related Results

We provide a generalization of a recent result of Anastassiou related to thewell-known Ostrowski inequality, as well as some related results. Ourresults subsume, extend, and consolidate a number of known results.     

Markov-type inequalities on certain irrational arcs and domains

Let denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set KR^d set Then the Markov factors on K are defined by (Here, as usual, S^d-1 stands for the Euclidean unit sphere in R^d.) Furthermore, given a smooth curve ΓR^d, we denote by D_TP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by Let . We prove that for every irrational number α>0 there are constants A,B>1 depending only on α such that for every sufficiently large n.Our second result presents some new bounds for M_n(Ω_α), where (d=2,α>1). We show that for every α>1 there exists a constant c>0 depending only on α such that M_n(Ω_α)n^clogn.     

Hardy's inequalities in function spaces containing derivatives of noninteger order

A theorem on Hardy's inequality in function spaces containing derivatives of noninteger order is proved. Translated fromMatematichcskie Zametki, Vol. 63, No. 5, pp. 673–678, May, 1998. The author wishes to thank Professor V. A. Kondrat'ev for his attention to this work.     

Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities

In this note we provide simple and short proofs for a class of inequalities of Caffarelli-Kohn-Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted L² spaces. These may be useful in studying solvability of problems involving new singular PDEs.     

Characterization of chaotic order and its applications to furuta's type operator inequalities

Very recently, we obtained a simple characterization of the chaotic order log A≥log B among positive invertible operators AB on a Hilbert space. In this note, we discuss Furuta's type operator inequalities as applications of our characterization of the chaotic order.     

Higher order Willmore hypersurfaces in Euclidean space

Let x ： Mn^n→ R^n＋1 be an n（≥2）-dimensional hypersurface immersed in Euclidean space Rn＋1. Let σi（0≤ i≤ n） be the ith mean curvature and Qn = ∑i=0^n（-1）^i＋1 （n^i）σ1^n-iσi. Recently, the author showed that Wn（x） = ∫M QndM is a conformal invariant under conformal group of R^n＋1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.     

Transfinitely valued Euclidean domains have arbitrary indecomposable order type

We prove that for every indecomposable ordinal there exists a (transfinitely valued) Euclidean domain whose minimal Euclidean norm is of that order type. Conversely, any such norm must have indecomposable type, and so we completely characterize the norm complexity of Euclidean domains. Modifying this construction, we also find a finitely valued Euclidean domain with no multiplicative integer valued norm.     

Higher order numerical discretizations for exterior and biharmonic type PDEs

A higher order numerical discretization technique based on Minimum Sobolev Norm (MSN) interpolation was introduced in our previous work. In this article, the discretization technique is presented as a tool to solve two hard classes of PDEs, namely, the exterior Laplace problem and the biharmonic problem. The exterior Laplace problem is compactified and the resultant near singular PDE is solved using this technique. This finite difference type method is then used to discretize and solve biharmonic type PDEs. A simple book keeping trick of using Ghost points is used to obtain a perfectly constrained discrete system. Numerical results such as discretization error, condition number estimate, and solution error are presented. For both classes of PDEs, variable coefficient examples on complicated geometries and irregular grids are considered. The method is seen to have high order of convergence in all these cases through numerical evidence. Perhaps for the first time, such a systematic higher order procedure for irregular grids and variable coefficient cases is now available. Though not discussed in the paper, the idea seems to be easily generalizable to finite element type techniques as well.     

Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half-space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.     

Multidimensional Hermite-Hadamard inequalities and the convex order

The problem of establishing inequalities of the Hermite-Hadamard type for convex functions on n-dimensional convex bodies translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order. We present two results of partial generality which unify and extend the most part of the multidimensional Hermite-Hadamard inequalities existing in the literature, at the same time that lead to new specific results. The first one fairly applies to the most familiar kinds of polytopes. The second one applies to symmetric random vectors taking values in a closed ball for a given (but arbitrary) norm on Rn. Related questions, such as estimates of approximation and extensions to signed measures, also are briefly discussed.     

Sub-elliptic global high order Poincaré inequalities in stratified Lie groups and applications

Sharp Poincaré inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Carnot-Carathéodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hörmander's condition). In this paper, we investigate the validity of sharp global Poincaré inequalities of both first order and higher order on the entire nilpotent stratified Lie groups or on unbounded extension domains in such groups. We will show that simultaneous sharp global Poincaré inequalities also hold and weighted versions of such results remain to be true. More precisely, let G be a nilpotent stratified Lie group and f be in the localized non-isotropic Sobolev space , where 1?p<Q/m and Q is the homogeneous dimension of the Lie group G. Suppose that the mth sub-elliptic derivatives of f is globally Lp integrable; i.e., is finite (but assume that lower order sub-elliptic derivatives are only locally Lp integrable). We denote the space of such functions as Bm,p(G). We prove a high order Poincaré inequality for f minus a polynomial of order m−1 over the entire space G or unbounded extension domains. As applications, we will prove a density theorem stating that smooth functions with compact support are dense in Bm,p(G) modulus a finite-dimensional subspace.     

New Lyapunov‐type inequalities for a class of even‐order linear differential equations

In this paper, we obtain some new Lyapunov‐type inequalities for a class of even‐order linear differential equations, the results are new and generalize and improve some early results in this field.